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<title>Chapter 20 Practical. z- and t-intervals | Fundamental statistical concepts and techniques in the biological and environmental sciences: With jamovi</title>
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<meta name="twitter:title" content="Chapter 20 Practical. z- and t-intervals | Fundamental statistical concepts and techniques in the biological and environmental sciences: With jamovi" />
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<ul class="summary">
<li><a href="./">Statistics with jamovi</a></li>
<li class="divider"></li>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html"><i class="fa fa-check"></i>Preface</a>
<ul>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html#structure"><i class="fa fa-check"></i>How this book is structured</a></li>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html#datasets"><i class="fa fa-check"></i>Datasets used in this book</a></li>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html#acknowledgements"><i class="fa fa-check"></i>Acknowledgements</a></li>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html#author"><i class="fa fa-check"></i>About the author</a></li>
</ul></li>
<li class="chapter" data-level="1" data-path="Chapter_1.html"><a href="Chapter_1.html"><i class="fa fa-check"></i><b>1</b> Background mathematics</a>
<ul>
<li class="chapter" data-level="1.1" data-path="Chapter_1.html"><a href="Chapter_1.html#numbers-and-operations"><i class="fa fa-check"></i><b>1.1</b> Numbers and operations</a></li>
<li class="chapter" data-level="1.2" data-path="Chapter_1.html"><a href="Chapter_1.html#logarithms"><i class="fa fa-check"></i><b>1.2</b> Logarithms</a></li>
<li class="chapter" data-level="1.3" data-path="Chapter_1.html"><a href="Chapter_1.html#order-of-operations"><i class="fa fa-check"></i><b>1.3</b> Order of operations</a></li>
</ul></li>
<li class="chapter" data-level="2" data-path="Chapter_2.html"><a href="Chapter_2.html"><i class="fa fa-check"></i><b>2</b> Data organisation</a>
<ul>
<li class="chapter" data-level="2.1" data-path="Chapter_2.html"><a href="Chapter_2.html#tidy-data"><i class="fa fa-check"></i><b>2.1</b> Tidy data</a></li>
<li class="chapter" data-level="2.2" data-path="Chapter_2.html"><a href="Chapter_2.html#data-files"><i class="fa fa-check"></i><b>2.2</b> Data files</a></li>
<li class="chapter" data-level="2.3" data-path="Chapter_2.html"><a href="Chapter_2.html#managing-data-files"><i class="fa fa-check"></i><b>2.3</b> Managing data files</a></li>
</ul></li>
<li class="chapter" data-level="3" data-path="Chapter_3.html"><a href="Chapter_3.html"><i class="fa fa-check"></i><b>3</b> <em>Practical</em>. Preparing data</a>
<ul>
<li class="chapter" data-level="3.1" data-path="Chapter_3.html"><a href="Chapter_3.html#transferring-data-to-a-spreadsheet"><i class="fa fa-check"></i><b>3.1</b> Transferring data to a spreadsheet</a></li>
<li class="chapter" data-level="3.2" data-path="Chapter_3.html"><a href="Chapter_3.html#making-spreadsheet-data-tidy"><i class="fa fa-check"></i><b>3.2</b> Making spreadsheet data tidy</a></li>
<li class="chapter" data-level="3.3" data-path="Chapter_3.html"><a href="Chapter_3.html#making-data-tidy-again"><i class="fa fa-check"></i><b>3.3</b> Making data tidy again</a></li>
<li class="chapter" data-level="3.4" data-path="Chapter_3.html"><a href="Chapter_3.html#tidy-data-and-spreadsheet-calculations"><i class="fa fa-check"></i><b>3.4</b> Tidy data and spreadsheet calculations</a></li>
<li class="chapter" data-level="3.5" data-path="Chapter_3.html"><a href="Chapter_3.html#summary"><i class="fa fa-check"></i><b>3.5</b> Summary</a></li>
</ul></li>
<li class="chapter" data-level="4" data-path="Chapter_4.html"><a href="Chapter_4.html"><i class="fa fa-check"></i><b>4</b> Populations and samples</a></li>
<li class="chapter" data-level="5" data-path="Chapter_5.html"><a href="Chapter_5.html"><i class="fa fa-check"></i><b>5</b> Types of variables</a></li>
<li class="chapter" data-level="6" data-path="Chapter_6.html"><a href="Chapter_6.html"><i class="fa fa-check"></i><b>6</b> Accuracy, precision, and units</a>
<ul>
<li class="chapter" data-level="6.1" data-path="Chapter_6.html"><a href="Chapter_6.html#accuracy"><i class="fa fa-check"></i><b>6.1</b> Accuracy</a></li>
<li class="chapter" data-level="6.2" data-path="Chapter_6.html"><a href="Chapter_6.html#precision"><i class="fa fa-check"></i><b>6.2</b> Precision</a></li>
<li class="chapter" data-level="6.3" data-path="Chapter_6.html"><a href="Chapter_6.html#systems-of-units"><i class="fa fa-check"></i><b>6.3</b> Systems of units</a></li>
</ul></li>
<li class="chapter" data-level="7" data-path="Chapter_7.html"><a href="Chapter_7.html"><i class="fa fa-check"></i><b>7</b> Uncertainty propagation</a>
<ul>
<li class="chapter" data-level="7.1" data-path="Chapter_7.html"><a href="Chapter_7.html#adding-or-subtracting-errors"><i class="fa fa-check"></i><b>7.1</b> Adding or subtracting errors</a></li>
<li class="chapter" data-level="7.2" data-path="Chapter_7.html"><a href="Chapter_7.html#multiplying-or-dividing-errors"><i class="fa fa-check"></i><b>7.2</b> Multiplying or dividing errors</a></li>
</ul></li>
<li class="chapter" data-level="8" data-path="Chapter_8.html"><a href="Chapter_8.html"><i class="fa fa-check"></i><b>8</b> <em>Practical</em>. Introduction to jamovi</a>
<ul>
<li class="chapter" data-level="8.1" data-path="Chapter_8.html"><a href="Chapter_8.html#summary_statistics_02"><i class="fa fa-check"></i><b>8.1</b> Summary statistics</a></li>
<li class="chapter" data-level="8.2" data-path="Chapter_8.html"><a href="Chapter_8.html#transforming_variables_02"><i class="fa fa-check"></i><b>8.2</b> Transforming variables</a></li>
<li class="chapter" data-level="8.3" data-path="Chapter_8.html"><a href="Chapter_8.html#computing_variables_02"><i class="fa fa-check"></i><b>8.3</b> Computing variables</a></li>
<li class="chapter" data-level="8.4" data-path="Chapter_8.html"><a href="Chapter_8.html#summary-1"><i class="fa fa-check"></i><b>8.4</b> Summary</a></li>
</ul></li>
<li class="chapter" data-level="9" data-path="Chapter_9.html"><a href="Chapter_9.html"><i class="fa fa-check"></i><b>9</b> Decimal places, significant figures, and rounding</a>
<ul>
<li class="chapter" data-level="9.1" data-path="Chapter_9.html"><a href="Chapter_9.html#decimal-places-and-significant-figures"><i class="fa fa-check"></i><b>9.1</b> Decimal places and significant figures</a></li>
<li class="chapter" data-level="9.2" data-path="Chapter_9.html"><a href="Chapter_9.html#rounding"><i class="fa fa-check"></i><b>9.2</b> Rounding</a></li>
</ul></li>
<li class="chapter" data-level="10" data-path="Chapter_10.html"><a href="Chapter_10.html"><i class="fa fa-check"></i><b>10</b> Graphs</a>
<ul>
<li class="chapter" data-level="10.1" data-path="Chapter_10.html"><a href="Chapter_10.html#histograms"><i class="fa fa-check"></i><b>10.1</b> Histograms</a></li>
<li class="chapter" data-level="10.2" data-path="Chapter_10.html"><a href="Chapter_10.html#barplots-and-pie-charts"><i class="fa fa-check"></i><b>10.2</b> Barplots and pie charts</a></li>
<li class="chapter" data-level="10.3" data-path="Chapter_10.html"><a href="Chapter_10.html#box-whisker-plots"><i class="fa fa-check"></i><b>10.3</b> Box-whisker plots</a></li>
</ul></li>
<li class="chapter" data-level="11" data-path="Chapter_11.html"><a href="Chapter_11.html"><i class="fa fa-check"></i><b>11</b> Measures of central tendency</a>
<ul>
<li class="chapter" data-level="11.1" data-path="Chapter_11.html"><a href="Chapter_11.html#the-mean"><i class="fa fa-check"></i><b>11.1</b> The mean</a></li>
<li class="chapter" data-level="11.2" data-path="Chapter_11.html"><a href="Chapter_11.html#the-mode"><i class="fa fa-check"></i><b>11.2</b> The mode</a></li>
<li class="chapter" data-level="11.3" data-path="Chapter_11.html"><a href="Chapter_11.html#the-median-and-quantiles"><i class="fa fa-check"></i><b>11.3</b> The median and quantiles</a></li>
</ul></li>
<li class="chapter" data-level="12" data-path="Chapter_12.html"><a href="Chapter_12.html"><i class="fa fa-check"></i><b>12</b> Measures of spread</a>
<ul>
<li class="chapter" data-level="12.1" data-path="Chapter_12.html"><a href="Chapter_12.html#the-range"><i class="fa fa-check"></i><b>12.1</b> The range</a></li>
<li class="chapter" data-level="12.2" data-path="Chapter_12.html"><a href="Chapter_12.html#the-inter-quartile-range"><i class="fa fa-check"></i><b>12.2</b> The inter-quartile range</a></li>
<li class="chapter" data-level="12.3" data-path="Chapter_12.html"><a href="Chapter_12.html#the-variance"><i class="fa fa-check"></i><b>12.3</b> The variance</a></li>
<li class="chapter" data-level="12.4" data-path="Chapter_12.html"><a href="Chapter_12.html#the-standard-deviation"><i class="fa fa-check"></i><b>12.4</b> The standard deviation</a></li>
<li class="chapter" data-level="12.5" data-path="Chapter_12.html"><a href="Chapter_12.html#the-coefficient-of-variation"><i class="fa fa-check"></i><b>12.5</b> The coefficient of variation</a></li>
<li class="chapter" data-level="12.6" data-path="Chapter_12.html"><a href="Chapter_12.html#the-standard-error"><i class="fa fa-check"></i><b>12.6</b> The standard error</a></li>
</ul></li>
<li class="chapter" data-level="13" data-path="Chapter_13.html"><a href="Chapter_13.html"><i class="fa fa-check"></i><b>13</b> Skew and kurtosis</a>
<ul>
<li class="chapter" data-level="13.1" data-path="Chapter_13.html"><a href="Chapter_13.html#skew"><i class="fa fa-check"></i><b>13.1</b> Skew</a></li>
<li class="chapter" data-level="13.2" data-path="Chapter_13.html"><a href="Chapter_13.html#kurtosis"><i class="fa fa-check"></i><b>13.2</b> Kurtosis</a></li>
<li class="chapter" data-level="13.3" data-path="Chapter_13.html"><a href="Chapter_13.html#moments"><i class="fa fa-check"></i><b>13.3</b> Moments</a></li>
</ul></li>
<li class="chapter" data-level="14" data-path="Chapter_14.html"><a href="Chapter_14.html"><i class="fa fa-check"></i><b>14</b> <em>Practical</em>. Plotting and statistical summaries in jamovi</a>
<ul>
<li class="chapter" data-level="14.1" data-path="Chapter_14.html"><a href="Chapter_14.html#reorganise-the-dataset-into-a-tidy-format"><i class="fa fa-check"></i><b>14.1</b> Reorganise the dataset into a tidy format</a></li>
<li class="chapter" data-level="14.2" data-path="Chapter_14.html"><a href="Chapter_14.html#histograms-and-box-whisker-plots"><i class="fa fa-check"></i><b>14.2</b> Histograms and box-whisker plots</a></li>
<li class="chapter" data-level="14.3" data-path="Chapter_14.html"><a href="Chapter_14.html#calculate-summary-statistics"><i class="fa fa-check"></i><b>14.3</b> Calculate summary statistics</a></li>
<li class="chapter" data-level="14.4" data-path="Chapter_14.html"><a href="Chapter_14.html#reporting-decimals-and-significant-figures"><i class="fa fa-check"></i><b>14.4</b> Reporting decimals and significant figures</a></li>
<li class="chapter" data-level="14.5" data-path="Chapter_14.html"><a href="Chapter_14.html#comparing-across-sites"><i class="fa fa-check"></i><b>14.5</b> Comparing across sites</a></li>
</ul></li>
<li class="chapter" data-level="15" data-path="Chapter_15.html"><a href="Chapter_15.html"><i class="fa fa-check"></i><b>15</b> Introduction to probability models</a>
<ul>
<li class="chapter" data-level="15.1" data-path="Chapter_15.html"><a href="Chapter_15.html#instructive-example"><i class="fa fa-check"></i><b>15.1</b> Instructive example</a></li>
<li class="chapter" data-level="15.2" data-path="Chapter_15.html"><a href="Chapter_15.html#biological-applications"><i class="fa fa-check"></i><b>15.2</b> Biological applications</a></li>
<li class="chapter" data-level="15.3" data-path="Chapter_15.html"><a href="Chapter_15.html#sampling-with-and-without-replacement"><i class="fa fa-check"></i><b>15.3</b> Sampling with and without replacement</a></li>
<li class="chapter" data-level="15.4" data-path="Chapter_15.html"><a href="Chapter_15.html#probability-distributions"><i class="fa fa-check"></i><b>15.4</b> Probability distributions</a>
<ul>
<li class="chapter" data-level="15.4.1" data-path="Chapter_15.html"><a href="Chapter_15.html#binomial-distribution"><i class="fa fa-check"></i><b>15.4.1</b> Binomial distribution</a></li>
<li class="chapter" data-level="15.4.2" data-path="Chapter_15.html"><a href="Chapter_15.html#poisson-distribution"><i class="fa fa-check"></i><b>15.4.2</b> Poisson distribution</a></li>
<li class="chapter" data-level="15.4.3" data-path="Chapter_15.html"><a href="Chapter_15.html#uniform-distribution"><i class="fa fa-check"></i><b>15.4.3</b> Uniform distribution</a></li>
<li class="chapter" data-level="15.4.4" data-path="Chapter_15.html"><a href="Chapter_15.html#normal-distribution"><i class="fa fa-check"></i><b>15.4.4</b> Normal distribution</a></li>
</ul></li>
<li class="chapter" data-level="15.5" data-path="Chapter_15.html"><a href="Chapter_15.html#summary-2"><i class="fa fa-check"></i><b>15.5</b> Summary</a></li>
</ul></li>
<li class="chapter" data-level="16" data-path="Chapter_16.html"><a href="Chapter_16.html"><i class="fa fa-check"></i><b>16</b> Central Limit Theorem</a>
<ul>
<li class="chapter" data-level="16.1" data-path="Chapter_16.html"><a href="Chapter_16.html#the-distribution-of-means-is-normal"><i class="fa fa-check"></i><b>16.1</b> The distribution of means is normal</a></li>
<li class="chapter" data-level="16.2" data-path="Chapter_16.html"><a href="Chapter_16.html#probability-and-z-scores"><i class="fa fa-check"></i><b>16.2</b> Probability and z-scores</a></li>
</ul></li>
<li class="chapter" data-level="17" data-path="Chapter_17.html"><a href="Chapter_17.html"><i class="fa fa-check"></i><b>17</b> <em>Practical</em>. Probability and simulation</a>
<ul>
<li class="chapter" data-level="17.1" data-path="Chapter_17.html"><a href="Chapter_17.html#probabilities-from-a-dataset"><i class="fa fa-check"></i><b>17.1</b> Probabilities from a dataset</a></li>
<li class="chapter" data-level="17.2" data-path="Chapter_17.html"><a href="Chapter_17.html#probabilities-from-a-normal-distribution"><i class="fa fa-check"></i><b>17.2</b> Probabilities from a normal distribution</a></li>
<li class="chapter" data-level="17.3" data-path="Chapter_17.html"><a href="Chapter_17.html#central-limit-theorem"><i class="fa fa-check"></i><b>17.3</b> Central limit theorem</a></li>
</ul></li>
<li class="chapter" data-level="18" data-path="Chapter_18.html"><a href="Chapter_18.html"><i class="fa fa-check"></i><b>18</b> Confidence intervals</a>
<ul>
<li class="chapter" data-level="18.1" data-path="Chapter_18.html"><a href="Chapter_18.html#normal-distribution-cis"><i class="fa fa-check"></i><b>18.1</b> Normal distribution CIs</a></li>
<li class="chapter" data-level="18.2" data-path="Chapter_18.html"><a href="Chapter_18.html#binomial-distribution-cis"><i class="fa fa-check"></i><b>18.2</b> Binomial distribution CIs</a></li>
</ul></li>
<li class="chapter" data-level="19" data-path="Chapter_19.html"><a href="Chapter_19.html"><i class="fa fa-check"></i><b>19</b> The t-interval</a></li>
<li class="chapter" data-level="20" data-path="Chapter_20.html"><a href="Chapter_20.html"><i class="fa fa-check"></i><b>20</b> <em>Practical</em>. z- and t-intervals</a>
<ul>
<li class="chapter" data-level="20.1" data-path="Chapter_20.html"><a href="Chapter_20.html#confidence-intervals-with-distraction"><i class="fa fa-check"></i><b>20.1</b> Confidence intervals with distrACTION</a></li>
<li class="chapter" data-level="20.2" data-path="Chapter_20.html"><a href="Chapter_20.html#confidence-intervals-from-z--and-t-scores"><i class="fa fa-check"></i><b>20.2</b> Confidence intervals from z- and t-scores</a></li>
<li class="chapter" data-level="20.3" data-path="Chapter_20.html"><a href="Chapter_20.html#confidence-intervals-for-different-sample-sizes"><i class="fa fa-check"></i><b>20.3</b> Confidence intervals for different sample sizes</a></li>
<li class="chapter" data-level="20.4" data-path="Chapter_20.html"><a href="Chapter_20.html#proportion-confidence-intervals"><i class="fa fa-check"></i><b>20.4</b> Proportion confidence intervals</a></li>
<li class="chapter" data-level="20.5" data-path="Chapter_20.html"><a href="Chapter_20.html#another-proportion-confidence-interval"><i class="fa fa-check"></i><b>20.5</b> Another proportion confidence interval</a></li>
</ul></li>
<li class="chapter" data-level="21" data-path="Chapter_21.html"><a href="Chapter_21.html"><i class="fa fa-check"></i><b>21</b> What is hypothesis testing?</a>
<ul>
<li class="chapter" data-level="21.1" data-path="Chapter_21.html"><a href="Chapter_21.html#how-ridiculous-is-our-hypothesis"><i class="fa fa-check"></i><b>21.1</b> How ridiculous is our hypothesis?</a></li>
<li class="chapter" data-level="21.2" data-path="Chapter_21.html"><a href="Chapter_21.html#statistical-hypothesis-testing"><i class="fa fa-check"></i><b>21.2</b> Statistical hypothesis testing</a></li>
<li class="chapter" data-level="21.3" data-path="Chapter_21.html"><a href="Chapter_21.html#p-values-false-positives-and-power"><i class="fa fa-check"></i><b>21.3</b> P-values, false positives, and power</a></li>
</ul></li>
<li class="chapter" data-level="22" data-path="Chapter_22.html"><a href="Chapter_22.html"><i class="fa fa-check"></i><b>22</b> The t-test</a>
<ul>
<li class="chapter" data-level="22.1" data-path="Chapter_22.html"><a href="Chapter_22.html#one-sample-t-test"><i class="fa fa-check"></i><b>22.1</b> One sample t-test</a></li>
<li class="chapter" data-level="22.2" data-path="Chapter_22.html"><a href="Chapter_22.html#independent-samples-t-test"><i class="fa fa-check"></i><b>22.2</b> Independent samples t-test</a></li>
<li class="chapter" data-level="22.3" data-path="Chapter_22.html"><a href="Chapter_22.html#paired-samples-t-test"><i class="fa fa-check"></i><b>22.3</b> Paired samples t-test</a></li>
<li class="chapter" data-level="22.4" data-path="Chapter_22.html"><a href="Chapter_22.html#assumptions-of-t-tests"><i class="fa fa-check"></i><b>22.4</b> Assumptions of t-tests</a></li>
<li class="chapter" data-level="22.5" data-path="Chapter_22.html"><a href="Chapter_22.html#non-parametric-alternatives"><i class="fa fa-check"></i><b>22.5</b> Non-parametric alternatives</a>
<ul>
<li class="chapter" data-level="22.5.1" data-path="Chapter_22.html"><a href="Chapter_22.html#wilcoxon-test"><i class="fa fa-check"></i><b>22.5.1</b> Wilcoxon test</a></li>
<li class="chapter" data-level="22.5.2" data-path="Chapter_22.html"><a href="Chapter_22.html#mann-whitney-u-test"><i class="fa fa-check"></i><b>22.5.2</b> Mann-Whitney U test</a></li>
</ul></li>
<li class="chapter" data-level="22.6" data-path="Chapter_22.html"><a href="Chapter_22.html#summary-3"><i class="fa fa-check"></i><b>22.6</b> Summary</a></li>
</ul></li>
<li class="chapter" data-level="23" data-path="Chapter_23.html"><a href="Chapter_23.html"><i class="fa fa-check"></i><b>23</b> <em>Practical</em>. Hypothesis testing and t-tests</a>
<ul>
<li class="chapter" data-level="23.1" data-path="Chapter_23.html"><a href="Chapter_23.html#one-sample-t-test-1"><i class="fa fa-check"></i><b>23.1</b> One sample t-test</a></li>
<li class="chapter" data-level="23.2" data-path="Chapter_23.html"><a href="Chapter_23.html#paired-t-test"><i class="fa fa-check"></i><b>23.2</b> Paired t-test</a></li>
<li class="chapter" data-level="23.3" data-path="Chapter_23.html"><a href="Chapter_23.html#wilcoxon-test-1"><i class="fa fa-check"></i><b>23.3</b> Wilcoxon test</a></li>
<li class="chapter" data-level="23.4" data-path="Chapter_23.html"><a href="Chapter_23.html#independent-samples-t-test-1"><i class="fa fa-check"></i><b>23.4</b> Independent samples t-test</a></li>
<li class="chapter" data-level="23.5" data-path="Chapter_23.html"><a href="Chapter_23.html#mann-whitney-u-test-1"><i class="fa fa-check"></i><b>23.5</b> Mann-Whitney U Test</a></li>
</ul></li>
<li class="chapter" data-level="24" data-path="Chapter_24.html"><a href="Chapter_24.html"><i class="fa fa-check"></i><b>24</b> Analysis of variance</a>
<ul>
<li class="chapter" data-level="24.1" data-path="Chapter_24.html"><a href="Chapter_24.html#f-distribution"><i class="fa fa-check"></i><b>24.1</b> F-distribution</a></li>
<li class="chapter" data-level="24.2" data-path="Chapter_24.html"><a href="Chapter_24.html#one-way-anova"><i class="fa fa-check"></i><b>24.2</b> One-way ANOVA</a>
<ul>
<li class="chapter" data-level="24.2.1" data-path="Chapter_24.html"><a href="Chapter_24.html#anova-mean-variance-among-groups"><i class="fa fa-check"></i><b>24.2.1</b> ANOVA mean variance among groups</a></li>
<li class="chapter" data-level="24.2.2" data-path="Chapter_24.html"><a href="Chapter_24.html#anova-mean-variance-within-groups"><i class="fa fa-check"></i><b>24.2.2</b> ANOVA mean variance within groups</a></li>
<li class="chapter" data-level="24.2.3" data-path="Chapter_24.html"><a href="Chapter_24.html#anova-f-statistic-calculation"><i class="fa fa-check"></i><b>24.2.3</b> ANOVA F-statistic calculation</a></li>
</ul></li>
<li class="chapter" data-level="24.3" data-path="Chapter_24.html"><a href="Chapter_24.html#assumptions-of-anova"><i class="fa fa-check"></i><b>24.3</b> Assumptions of ANOVA</a></li>
</ul></li>
<li class="chapter" data-level="25" data-path="Chapter_25.html"><a href="Chapter_25.html"><i class="fa fa-check"></i><b>25</b> Multiple comparisons</a></li>
<li class="chapter" data-level="26" data-path="Chapter_26.html"><a href="Chapter_26.html"><i class="fa fa-check"></i><b>26</b> Kruskal-Wallis H test</a></li>
<li class="chapter" data-level="27" data-path="Chapter_27.html"><a href="Chapter_27.html"><i class="fa fa-check"></i><b>27</b> Two-way ANOVA</a></li>
<li class="chapter" data-level="28" data-path="Chapter_28.html"><a href="Chapter_28.html"><i class="fa fa-check"></i><b>28</b> <em>Practical</em>. ANOVA and associated tests</a>
<ul>
<li class="chapter" data-level="28.1" data-path="Chapter_28.html"><a href="Chapter_28.html#one-way-anova-site"><i class="fa fa-check"></i><b>28.1</b> One-way ANOVA (site)</a></li>
<li class="chapter" data-level="28.2" data-path="Chapter_28.html"><a href="Chapter_28.html#one-way-anova-profile"><i class="fa fa-check"></i><b>28.2</b> One-way ANOVA (profile)</a></li>
<li class="chapter" data-level="28.3" data-path="Chapter_28.html"><a href="Chapter_28.html#multiple-comparisons"><i class="fa fa-check"></i><b>28.3</b> Multiple comparisons</a></li>
<li class="chapter" data-level="28.4" data-path="Chapter_28.html"><a href="Chapter_28.html#kruskal-wallis-h-test"><i class="fa fa-check"></i><b>28.4</b> Kruskal-Wallis H test</a></li>
<li class="chapter" data-level="28.5" data-path="Chapter_28.html"><a href="Chapter_28.html#two-way-anova"><i class="fa fa-check"></i><b>28.5</b> Two-way ANOVA</a></li>
</ul></li>
<li class="chapter" data-level="29" data-path="Chapter_29.html"><a href="Chapter_29.html"><i class="fa fa-check"></i><b>29</b> Frequency and count data</a>
<ul>
<li class="chapter" data-level="29.1" data-path="Chapter_29.html"><a href="Chapter_29.html#chi-square-distribution"><i class="fa fa-check"></i><b>29.1</b> Chi-square distribution</a></li>
<li class="chapter" data-level="29.2" data-path="Chapter_29.html"><a href="Chapter_29.html#chi-square-goodness-of-fit"><i class="fa fa-check"></i><b>29.2</b> Chi-square goodness of fit</a></li>
<li class="chapter" data-level="29.3" data-path="Chapter_29.html"><a href="Chapter_29.html#chi-square-test-of-association"><i class="fa fa-check"></i><b>29.3</b> Chi-square test of association</a></li>
</ul></li>
<li class="chapter" data-level="30" data-path="Chapter_30.html"><a href="Chapter_30.html"><i class="fa fa-check"></i><b>30</b> Correlation</a>
<ul>
<li class="chapter" data-level="30.1" data-path="Chapter_30.html"><a href="Chapter_30.html#scatterplots"><i class="fa fa-check"></i><b>30.1</b> Scatterplots</a></li>
<li class="chapter" data-level="30.2" data-path="Chapter_30.html"><a href="Chapter_30.html#correlation-coefficient"><i class="fa fa-check"></i><b>30.2</b> Correlation coefficient</a>
<ul>
<li class="chapter" data-level="30.2.1" data-path="Chapter_30.html"><a href="Chapter_30.html#pearson-product-moment-correlation-coefficient"><i class="fa fa-check"></i><b>30.2.1</b> Pearson product moment correlation coefficient</a></li>
<li class="chapter" data-level="30.2.2" data-path="Chapter_30.html"><a href="Chapter_30.html#spearmans-rank-correlation-coefficient"><i class="fa fa-check"></i><b>30.2.2</b> Spearman’s rank correlation coefficient</a></li>
</ul></li>
<li class="chapter" data-level="30.3" data-path="Chapter_30.html"><a href="Chapter_30.html#correlation-hypothesis-testing"><i class="fa fa-check"></i><b>30.3</b> Correlation hypothesis testing</a></li>
</ul></li>
<li class="chapter" data-level="31" data-path="Chapter_31.html"><a href="Chapter_31.html"><i class="fa fa-check"></i><b>31</b> <em>Practical</em>. Analysis of counts and correlations</a>
<ul>
<li class="chapter" data-level="31.1" data-path="Chapter_31.html"><a href="Chapter_31.html#survival-goodness-of-fit"><i class="fa fa-check"></i><b>31.1</b> Survival goodness of fit</a></li>
<li class="chapter" data-level="31.2" data-path="Chapter_31.html"><a href="Chapter_31.html#colony-goodness-of-fit"><i class="fa fa-check"></i><b>31.2</b> Colony goodness of fit</a></li>
<li class="chapter" data-level="31.3" data-path="Chapter_31.html"><a href="Chapter_31.html#chi-square-test-of-association-1"><i class="fa fa-check"></i><b>31.3</b> Chi-Square test of association</a></li>
<li class="chapter" data-level="31.4" data-path="Chapter_31.html"><a href="Chapter_31.html#pearson-product-moment-correlation-test"><i class="fa fa-check"></i><b>31.4</b> Pearson product moment correlation test</a></li>
<li class="chapter" data-level="31.5" data-path="Chapter_31.html"><a href="Chapter_31.html#spearmans-rank-correlation-test"><i class="fa fa-check"></i><b>31.5</b> Spearman’s rank correlation test</a></li>
<li class="chapter" data-level="31.6" data-path="Chapter_31.html"><a href="Chapter_31.html#untidy-goodness-of-fit"><i class="fa fa-check"></i><b>31.6</b> Untidy goodness of fit</a></li>
</ul></li>
<li class="chapter" data-level="32" data-path="Chapter_32.html"><a href="Chapter_32.html"><i class="fa fa-check"></i><b>32</b> Simple linear regression</a>
<ul>
<li class="chapter" data-level="32.1" data-path="Chapter_32.html"><a href="Chapter_32.html#visual-interpretation-of-regression"><i class="fa fa-check"></i><b>32.1</b> Visual interpretation of regression</a></li>
<li class="chapter" data-level="32.2" data-path="Chapter_32.html"><a href="Chapter_32.html#intercepts-slopes-and-residuals"><i class="fa fa-check"></i><b>32.2</b> Intercepts, slopes, and residuals</a></li>
<li class="chapter" data-level="32.3" data-path="Chapter_32.html"><a href="Chapter_32.html#regression-coefficients"><i class="fa fa-check"></i><b>32.3</b> Regression coefficients</a></li>
<li class="chapter" data-level="32.4" data-path="Chapter_32.html"><a href="Chapter_32.html#regression-line-calculation"><i class="fa fa-check"></i><b>32.4</b> Regression line calculation</a></li>
<li class="chapter" data-level="32.5" data-path="Chapter_32.html"><a href="Chapter_32.html#coefficient-of-determination"><i class="fa fa-check"></i><b>32.5</b> Coefficient of determination</a></li>
<li class="chapter" data-level="32.6" data-path="Chapter_32.html"><a href="Chapter_32.html#regression-assumptions"><i class="fa fa-check"></i><b>32.6</b> Regression assumptions</a></li>
<li class="chapter" data-level="32.7" data-path="Chapter_32.html"><a href="Chapter_32.html#regression-hypothesis-testing"><i class="fa fa-check"></i><b>32.7</b> Regression hypothesis testing</a>
<ul>
<li class="chapter" data-level="32.7.1" data-path="Chapter_32.html"><a href="Chapter_32.html#overall-model-significance"><i class="fa fa-check"></i><b>32.7.1</b> Overall model significance</a></li>
<li class="chapter" data-level="32.7.2" data-path="Chapter_32.html"><a href="Chapter_32.html#significance-of-the-intercept"><i class="fa fa-check"></i><b>32.7.2</b> Significance of the intercept</a></li>
<li class="chapter" data-level="32.7.3" data-path="Chapter_32.html"><a href="Chapter_32.html#significance-of-the-slope"><i class="fa fa-check"></i><b>32.7.3</b> Significance of the slope</a></li>
<li class="chapter" data-level="32.7.4" data-path="Chapter_32.html"><a href="Chapter_32.html#simple-regression-output"><i class="fa fa-check"></i><b>32.7.4</b> Simple regression output</a></li>
</ul></li>
<li class="chapter" data-level="32.8" data-path="Chapter_32.html"><a href="Chapter_32.html#prediction-with-linear-models"><i class="fa fa-check"></i><b>32.8</b> Prediction with linear models</a></li>
<li class="chapter" data-level="32.9" data-path="Chapter_32.html"><a href="Chapter_32.html#conclusion"><i class="fa fa-check"></i><b>32.9</b> Conclusion</a></li>
</ul></li>
<li class="chapter" data-level="33" data-path="Chapter_33.html"><a href="Chapter_33.html"><i class="fa fa-check"></i><b>33</b> Multiple regression</a>
<ul>
<li class="chapter" data-level="33.1" data-path="Chapter_33.html"><a href="Chapter_33.html#adjusted-coefficient-of-determination"><i class="fa fa-check"></i><b>33.1</b> Adjusted coefficient of determination</a></li>
</ul></li>
<li class="chapter" data-level="34" data-path="Chapter_34.html"><a href="Chapter_34.html"><i class="fa fa-check"></i><b>34</b> <em>Practical</em>. Using regression</a>
<ul>
<li class="chapter" data-level="34.1" data-path="Chapter_34.html"><a href="Chapter_34.html#predicting-pyrogenic-carbon-from-soil-depth"><i class="fa fa-check"></i><b>34.1</b> Predicting pyrogenic carbon from soil depth</a></li>
<li class="chapter" data-level="34.2" data-path="Chapter_34.html"><a href="Chapter_34.html#predicting-pyrogenic-carbon-from-fire-frequency"><i class="fa fa-check"></i><b>34.2</b> Predicting pyrogenic carbon from fire frequency</a></li>
<li class="chapter" data-level="34.3" data-path="Chapter_34.html"><a href="Chapter_34.html#multiple-regression-depth-and-fire-frequency"><i class="fa fa-check"></i><b>34.3</b> Multiple regression depth and fire frequency</a></li>
<li class="chapter" data-level="34.4" data-path="Chapter_34.html"><a href="Chapter_34.html#large-multiple-regression"><i class="fa fa-check"></i><b>34.4</b> Large multiple regression</a></li>
<li class="chapter" data-level="34.5" data-path="Chapter_34.html"><a href="Chapter_34.html#predicting-temperature-from-fire-frequency"><i class="fa fa-check"></i><b>34.5</b> Predicting temperature from fire frequency</a></li>
</ul></li>
<li class="chapter" data-level="35" data-path="Chapter_35.html"><a href="Chapter_35.html"><i class="fa fa-check"></i><b>35</b> Randomisation</a>
<ul>
<li class="chapter" data-level="35.1" data-path="Chapter_35.html"><a href="Chapter_35.html#summary-of-parametric-hypothesis-testing"><i class="fa fa-check"></i><b>35.1</b> Summary of parametric hypothesis testing</a></li>
<li class="chapter" data-level="35.2" data-path="Chapter_35.html"><a href="Chapter_35.html#randomisation-approach"><i class="fa fa-check"></i><b>35.2</b> Randomisation approach</a></li>
<li class="chapter" data-level="35.3" data-path="Chapter_35.html"><a href="Chapter_35.html#randomisation-for-hypothesis-testing"><i class="fa fa-check"></i><b>35.3</b> Randomisation for hypothesis testing</a></li>
<li class="chapter" data-level="35.4" data-path="Chapter_35.html"><a href="Chapter_35.html#randomisation-assumptions"><i class="fa fa-check"></i><b>35.4</b> Randomisation assumptions</a></li>
<li class="chapter" data-level="35.5" data-path="Chapter_35.html"><a href="Chapter_35.html#bootstrapping"><i class="fa fa-check"></i><b>35.5</b> Bootstrapping</a></li>
<li class="chapter" data-level="35.6" data-path="Chapter_35.html"><a href="Chapter_35.html#randomisation-conclusions"><i class="fa fa-check"></i><b>35.6</b> Randomisation conclusions</a></li>
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<li class="appendix"><span><b>Appendix</b></span></li>
<li class="chapter" data-level="A" data-path="appendexA.html"><a href="appendexA.html"><i class="fa fa-check"></i><b>A</b> Answers to chapter exercises</a>
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<li class="chapter" data-level="A.1" data-path="appendexA.html"><a href="appendexA.html#chapter-3"><i class="fa fa-check"></i><b>A.1</b> Chapter 3</a>
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<li class="chapter" data-level="A.1.1" data-path="appendexA.html"><a href="appendexA.html#exercise-3.1"><i class="fa fa-check"></i><b>A.1.1</b> Exercise 3.1:</a></li>
<li class="chapter" data-level="A.1.2" data-path="appendexA.html"><a href="appendexA.html#exercise-3.2"><i class="fa fa-check"></i><b>A.1.2</b> Exercise 3.2</a></li>
<li class="chapter" data-level="A.1.3" data-path="appendexA.html"><a href="appendexA.html#exercise-3.3"><i class="fa fa-check"></i><b>A.1.3</b> Exercise 3.3</a></li>
<li class="chapter" data-level="A.1.4" data-path="appendexA.html"><a href="appendexA.html#exercise-3.4"><i class="fa fa-check"></i><b>A.1.4</b> Exercise 3.4</a></li>
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<li class="chapter" data-level="A.2" data-path="appendexA.html"><a href="appendexA.html#chapter-8"><i class="fa fa-check"></i><b>A.2</b> Chapter 8</a>
<ul>
<li class="chapter" data-level="A.2.1" data-path="appendexA.html"><a href="appendexA.html#exercise-8.1"><i class="fa fa-check"></i><b>A.2.1</b> Exercise 8.1</a></li>
<li class="chapter" data-level="A.2.2" data-path="appendexA.html"><a href="appendexA.html#exercise-8.2"><i class="fa fa-check"></i><b>A.2.2</b> Exercise 8.2</a></li>
<li class="chapter" data-level="A.2.3" data-path="appendexA.html"><a href="appendexA.html#exercise-8.3"><i class="fa fa-check"></i><b>A.2.3</b> Exercise 8.3</a></li>
</ul></li>
<li class="chapter" data-level="A.3" data-path="appendexA.html"><a href="appendexA.html#chapter-14"><i class="fa fa-check"></i><b>A.3</b> Chapter 14</a>
<ul>
<li class="chapter" data-level="A.3.1" data-path="appendexA.html"><a href="appendexA.html#exercise-14.1"><i class="fa fa-check"></i><b>A.3.1</b> Exercise 14.1</a></li>
<li class="chapter" data-level="A.3.2" data-path="appendexA.html"><a href="appendexA.html#exercise-14.2"><i class="fa fa-check"></i><b>A.3.2</b> Exercise 14.2</a></li>
<li class="chapter" data-level="A.3.3" data-path="appendexA.html"><a href="appendexA.html#exercise-14.3"><i class="fa fa-check"></i><b>A.3.3</b> Exercise 14.3</a></li>
<li class="chapter" data-level="A.3.4" data-path="appendexA.html"><a href="appendexA.html#exercise-14.4"><i class="fa fa-check"></i><b>A.3.4</b> Exercise 14.4</a></li>
<li class="chapter" data-level="A.3.5" data-path="appendexA.html"><a href="appendexA.html#exercise-14.5"><i class="fa fa-check"></i><b>A.3.5</b> Exercise 14.5</a></li>
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<li class="chapter" data-level="A.4" data-path="appendexA.html"><a href="appendexA.html#chapter-17"><i class="fa fa-check"></i><b>A.4</b> Chapter 17</a>
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<li class="chapter" data-level="A.4.1" data-path="appendexA.html"><a href="appendexA.html#exercise-17.1"><i class="fa fa-check"></i><b>A.4.1</b> Exercise 17.1</a></li>
<li class="chapter" data-level="A.4.2" data-path="appendexA.html"><a href="appendexA.html#exercise-17.2"><i class="fa fa-check"></i><b>A.4.2</b> Exercise 17.2</a></li>
<li class="chapter" data-level="A.4.3" data-path="appendexA.html"><a href="appendexA.html#exercise-17.3"><i class="fa fa-check"></i><b>A.4.3</b> Exercise 17.3</a></li>
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<li class="chapter" data-level="A.5" data-path="appendexA.html"><a href="appendexA.html#chapter-20"><i class="fa fa-check"></i><b>A.5</b> Chapter 20</a>
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<li class="chapter" data-level="A.5.1" data-path="appendexA.html"><a href="appendexA.html#exercise-20.1"><i class="fa fa-check"></i><b>A.5.1</b> Exercise 20.1</a></li>
<li class="chapter" data-level="A.5.2" data-path="appendexA.html"><a href="appendexA.html#exercise-20.2"><i class="fa fa-check"></i><b>A.5.2</b> Exercise 20.2</a></li>
<li class="chapter" data-level="A.5.3" data-path="appendexA.html"><a href="appendexA.html#exercise-20.3"><i class="fa fa-check"></i><b>A.5.3</b> Exercise 20.3</a></li>
<li class="chapter" data-level="A.5.4" data-path="appendexA.html"><a href="appendexA.html#exercise-20.4"><i class="fa fa-check"></i><b>A.5.4</b> Exercise 20.4</a></li>
<li class="chapter" data-level="A.5.5" data-path="appendexA.html"><a href="appendexA.html#exercise-20.5"><i class="fa fa-check"></i><b>A.5.5</b> Exercise 20.5</a></li>
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<li class="chapter" data-level="A.6" data-path="appendexA.html"><a href="appendexA.html#chapter-23"><i class="fa fa-check"></i><b>A.6</b> Chapter 23</a>
<ul>
<li class="chapter" data-level="A.6.1" data-path="appendexA.html"><a href="appendexA.html#exercise-23.1"><i class="fa fa-check"></i><b>A.6.1</b> Exercise 23.1</a></li>
<li class="chapter" data-level="A.6.2" data-path="appendexA.html"><a href="appendexA.html#exercise-23.2"><i class="fa fa-check"></i><b>A.6.2</b> Exercise 23.2</a></li>
<li class="chapter" data-level="A.6.3" data-path="appendexA.html"><a href="appendexA.html#exercise-23.3"><i class="fa fa-check"></i><b>A.6.3</b> Exercise 23.3</a></li>
<li class="chapter" data-level="A.6.4" data-path="appendexA.html"><a href="appendexA.html#exercise-23.4"><i class="fa fa-check"></i><b>A.6.4</b> Exercise 23.4</a></li>
<li class="chapter" data-level="A.6.5" data-path="appendexA.html"><a href="appendexA.html#exercise-23.5"><i class="fa fa-check"></i><b>A.6.5</b> Exercise 23.5</a></li>
</ul></li>
<li class="chapter" data-level="A.7" data-path="appendexA.html"><a href="appendexA.html#chapter-28"><i class="fa fa-check"></i><b>A.7</b> Chapter 28</a>
<ul>
<li class="chapter" data-level="A.7.1" data-path="appendexA.html"><a href="appendexA.html#exercise-28.1"><i class="fa fa-check"></i><b>A.7.1</b> Exercise 28.1</a></li>
<li class="chapter" data-level="A.7.2" data-path="appendexA.html"><a href="appendexA.html#exercise-28.2"><i class="fa fa-check"></i><b>A.7.2</b> Exercise 28.2</a></li>
<li class="chapter" data-level="A.7.3" data-path="appendexA.html"><a href="appendexA.html#exercise-28.3"><i class="fa fa-check"></i><b>A.7.3</b> Exercise 28.3</a></li>
<li class="chapter" data-level="A.7.4" data-path="appendexA.html"><a href="appendexA.html#exercise-28.4"><i class="fa fa-check"></i><b>A.7.4</b> Exercise 28.4</a></li>
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<li class="chapter" data-level="A.8" data-path="appendexA.html"><a href="appendexA.html#chapter-31"><i class="fa fa-check"></i><b>A.8</b> Chapter 31</a>
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<li class="chapter" data-level="A.8.1" data-path="appendexA.html"><a href="appendexA.html#exercise-31.1"><i class="fa fa-check"></i><b>A.8.1</b> Exercise 31.1</a></li>
<li class="chapter" data-level="A.8.2" data-path="appendexA.html"><a href="appendexA.html#exercise-31.2"><i class="fa fa-check"></i><b>A.8.2</b> Exercise 31.2</a></li>
<li class="chapter" data-level="A.8.3" data-path="appendexA.html"><a href="appendexA.html#exercise-31.3"><i class="fa fa-check"></i><b>A.8.3</b> Exercise 31.3</a></li>
<li class="chapter" data-level="A.8.4" data-path="appendexA.html"><a href="appendexA.html#exercise-31.4"><i class="fa fa-check"></i><b>A.8.4</b> Exercise 31.4</a></li>
<li class="chapter" data-level="A.8.5" data-path="appendexA.html"><a href="appendexA.html#exercise-31.5"><i class="fa fa-check"></i><b>A.8.5</b> Exercise 31.5</a></li>
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<li class="chapter" data-level="A.9" data-path="appendexA.html"><a href="appendexA.html#chapter-34"><i class="fa fa-check"></i><b>A.9</b> Chapter 34</a>
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<li class="chapter" data-level="A.9.1" data-path="appendexA.html"><a href="appendexA.html#exercise-34.1"><i class="fa fa-check"></i><b>A.9.1</b> Exercise 34.1</a></li>
<li class="chapter" data-level="A.9.2" data-path="appendexA.html"><a href="appendexA.html#exercise-34.2"><i class="fa fa-check"></i><b>A.9.2</b> Exercise 34.2</a></li>
<li class="chapter" data-level="A.9.3" data-path="appendexA.html"><a href="appendexA.html#exercise-34.3"><i class="fa fa-check"></i><b>A.9.3</b> Exercise 34.3</a></li>
<li class="chapter" data-level="A.9.4" data-path="appendexA.html"><a href="appendexA.html#exercise-34.4"><i class="fa fa-check"></i><b>A.9.4</b> Exercise 34.4</a></li>
<li class="chapter" data-level="A.9.5" data-path="appendexA.html"><a href="appendexA.html#exercise-33.5"><i class="fa fa-check"></i><b>A.9.5</b> Exercise 33.5</a></li>
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<li class="chapter" data-level="B" data-path="uncertainty_derivation.html"><a href="uncertainty_derivation.html"><i class="fa fa-check"></i><b>B</b> Uncertainty derivation</a>
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<li class="chapter" data-level="B.1" data-path="uncertainty_derivation.html"><a href="uncertainty_derivation.html#propagation-of-error-for-addition-and-subtraction"><i class="fa fa-check"></i><b>B.1</b> Propagation of error for addition and subtraction</a></li>
<li class="chapter" data-level="B.2" data-path="uncertainty_derivation.html"><a href="uncertainty_derivation.html#propagation-of-error-for-multiplication-and-division"><i class="fa fa-check"></i><b>B.2</b> Propagation of error for multiplication and division</a></li>
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<li class="chapter" data-level="" data-path="references.html"><a href="references.html"><i class="fa fa-check"></i>References</a></li>
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<i class="fa fa-circle-o-notch fa-spin"></i><a href="./">Fundamental statistical concepts and techniques in the biological and environmental sciences: With jamovi</a>
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<h1><span class="header-section-number">Chapter 20</span> <em>Practical</em>. z- and t-intervals<a href="Chapter_20.html#Chapter_20" class="anchor-section" aria-label="Anchor link to header"></a></h1>
<p>This chapter focuses on applying the concepts from <a href="Chapter_18.html#Chapter_18">Chapter 18</a> and <a href="Chapter_19.html#Chapter_19">Chapter 19</a> in jamovi <span class="citation">(<a href="#ref-Jamovi2022" role="doc-biblioref">The jamovi project, 2024</a>)</span>.
Specifically, we will practice calculating confidence intervals (CIs).
There will be four exercises focused on calculating CIs in jamovi.
To complete the first two exercises, you will need the distrACTION module in jamovi.
If you need to download it again, the instructions to do this are in the second exercise of <a href="Chapter_17.html#Chapter_17">Chapter 17</a> (briefly, go to the Modules option and select ‘jamovi library’, then scroll down until you find the ‘distraACTION’ module).</p>
<p>The data for this chapter are inspired by ongoing work in the Woodland Creation and Ecological Networks (WrEN) project <span class="citation">(<a href="#ref-Fuentes-Montemayor2022" role="doc-biblioref">Fuentes-Montemayor, Park, et al., 2022</a>; <a href="#ref-Fuentes-Montemayor2022a" role="doc-biblioref">Fuentes-Montemayor, Watts, et al., 2022</a>)</span>.
The WrEN project is led by a collaboration between University of Stirling researchers Dr Elisa Fuentes-Montemayor and Prof Kirsty Park, and at Forest Research, Prof Kevin Watts (<a href="https://www.wren-project.com/">https://www.wren-project.com/</a>).
It focuses on questions about what kinds of conservation actions should be prioritised to restore degraded ecological networks.
The WrEN project encompasses a huge amount of work and data collection from hundreds of surveyed secondary or ancient woodland sites.
Here we will focus on observations of tree diameter at breast height (DBH) and grazing to calculate confidence intervals.</p>
<div id="confidence-intervals-with-distraction" class="section level2 hasAnchor" number="20.1">
<h2><span class="header-section-number">20.1</span> Confidence intervals with distrACTION<a href="Chapter_20.html#confidence-intervals-with-distraction" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>First, it is important to download the distrACTION module if it has not been downloaded already.
If the distrACTION module has already been downloaded (see <a href="Chapter_17.html#Chapter_17">Chapter 17</a>), it should appear in the toolbar of jamovi.
Once the distrACTION module has been made available, download the WrEN trees dataset<a href="#fn38" class="footnote-ref" id="fnref38"><sup>38</sup></a> and open it in a spreadsheet.
Notice that the dataset is not in a tidy format.
There are four different sites represented by different columns in the dataset.
The numbers under each column are measurements of tree diameter at breast height (DBH) in centimetres.
Before doing anything else, it is therefore necessary to put the WrEN dataset into a tidy format.
The tidy dataset should include two columns: one for site and the other for DBH.</p>
<p>Once the WrEN trees dataset has been reorganised into a tidy format, save it as a CSV file and open it in jamovi.
In jamovi, go to Exploration and Descriptives in the toolbar and build a histogram that shows the distribution of DBH.
Do these data appear to be roughly normal?
Why or why not?</p>
<pre><code>
</code></pre>
<p>Next, calculate the grand mean and standard deviation of tree DBH (i.e., the mean and standard deviation of trees across all sites).</p>
<p>Grand mean: _____________________</p>
<p>Grand standard deviation: _____________________</p>
<p>We will use this mean and standard deviation to compute quantiles and obtain 95% z-scores.
First, click on the distrACTION icon in the toolbar.
From the distrACTION pull-down menu, select ‘Normal Distribution’.
To the left, you should see boxes to input parameter values for the mean and standard deviation (SD).
Below the ‘Parameters’ options, you should also see different functions for computing probability or quantiles.
To the right, you should see a standard normal distribution (i.e., a normal distribution with a mean of 0 and a standard deviation of 1).</p>
<p>For this exercise, we will assume that the population of DBH from which our sample came is normally distributed.
In other words, if we somehow had access to <em>all possible</em> DBH measurements in the woodland sites (not just the 120 trees sampled), we assume that DBH would be normally distributed.
To find the probability of sampling a tree within a given interval of DBH (e.g., greater than 30), we therefore need to build this distribution with the correct mean and standard deviation.
We do not know the <em>true</em> mean (<span class="math inline">\(\mu\)</span>) and standard deviation (<span class="math inline">\(\sigma\)</span>) of the population, but our best estimate of these values are the mean (<span class="math inline">\(\bar{x}\)</span>) and standard deviation (<span class="math inline">\(s\)</span>) of the sample, as reported above (i.e., the grand mean and standard deviation).
Using the Mean and SD parameter input boxes in distrACTION, we can build a normal distribution with the same mean and standard deviation as our sample.
Do this now by inputting the calculated Grand mean and Grand standard deviation from above in the appropriate boxes.
Note that the normal distribution on the right has the same shape, but the table of parameters has been updated to reflect the mean and standard deviation.</p>
<p>In <a href="Chapter_17.html#Chapter_17">Chapter 17</a>, we calculated the probability of sampling a value within a given interval of the normal distribution.
If we wanted to do the same exercise here, we might find the probability of sampling a DBH < 30 using the Compute probability function (the answer is p = 0.264).
Instead, we are now going to do the opposite using the Compute quantile(s) function.
We might want to know, for example, what 75% of DBH values will be less than (i.e., what is the cutoff DBH, below which DBH values will be lower than this cutoff with a probability of 0.75).
To find this, uncheck the ‘Compute probability’ box and check the ‘Compute quantile(s)’ box.
Make sure that the ‘cumulative quantile’ radio button is selected, then set p = 0.75 (Figure 20.1).</p>
<div class="figure"><span style="display:block;" id="fig:unnamed-chunk-80"></span>
<img src="img/jamovi_distrACTION_p75.png" alt="Jamovi window showing with parameters for a normal distribution with a mean of 36.9 and a standard deviatio of 11, and a normal distribution shows a vertical dotted line on an x-axis value of 44.3, below which 75% of values fall." width="100%" />
<p class="caption">
Figure 20.1: Jamovi interface for the ‘distrACTION’ module, in which quantiles have been computed to find the diameter at breast height (DBH) below which 75% of DBHs will be given a normal distribution with a mean of 36.9 and standard deviation of 11. Data for these parameter values were inspired by the Woodland Creation and Ecological Networks (WrEN) project.
</p>
</div>
<p>From Figure 20.1, we can see that the cumulative 0.75 quantile is 44.3, so if DBH is normally distributed with the mean and standard deviation calculated above, 75% of DBH values in a population will be below 44.3 cm.
Using the same principles, what is the cumulative 0.4 quantile for the DBH data?</p>
<p>Quantile: _____________________</p>
<p>We can also use the Compute quantile(s) option in jamovi to compute interval quantiles.
For example, if we want to know the DBH values within which 95% of the probability density is contained, we can set p = 0.95, then select the radio button ‘central interval quantiles’.
Do this for the DBH data.
From the Results table on the right, what interval of DBH values will contain 95% of the probability density around the mean?</p>
<p>Interval: _____________________</p>
<p>Remember that we are looking at the full sample distribution of DBH.
That is, getting intervals for the probability of sampling DBH values around the mean, <em>not</em> confidence intervals around the mean as introduced in <a href="Chapter_18.html#Chapter_18">Chapter 18</a>.
How would we get confidence intervals around the mean?
That is, what if we want to say that we have 95% confidence that the <em>mean</em> lies between two values?
We would need to use the standard deviation <em>of the sample mean</em> <span class="math inline">\(\bar{x}\)</span> around the true mean <span class="math inline">\(\mu\)</span>, rather than the sample standard deviation.
Recall from <a href="Chapter_12.html#the-standard-error">Section 12.6</a> that the standard error is the standard deviation of <span class="math inline">\(\bar{x}\)</span> values around <span class="math inline">\(\mu\)</span>.
We can therefore use the standard error to calculate confidence intervals (CIs) around the mean value of DBH.
From the ‘Descriptives’ panel in jamovi (recall that this is under the ‘Exploration’ button), find the standard error of DBH,</p>
<p>Std. error of Mean: ___________________</p>
<p>Now, go back to the distrACTION Normal Distribution and put the DBH mean into the parameters box as before.
But this time, put the standard error calculated above into the box for SD.
Next, choose the ‘Compute quantile(s)’ option and set p = 0.95 to calculate a 95% confidence interval.
Based on the Results table, what can you infer are the lower and upper 95% CIs around the mean?</p>
<p>Lower 95% CI: ________________</p>
<p>Upper 95% CI: ________________</p>
<p>Remember that this assumed that the sample means (<span class="math inline">\(\bar{x}\)</span>) are normally distributed around the true mean (<span class="math inline">\(\mu\)</span>).
But as we saw in <a href="Chapter_19.html#Chapter_19">Chapter 19</a>, when we assume that our sample standard deviation (<span class="math inline">\(s\)</span>) is the same as the population standard deviation (<span class="math inline">\(\sigma\)</span>), then the shape of the normal distribution will be at least a bit off.
Instead, we can get a more accurate estimate of CIs using a t-distribution.
Jamovi usually does this automatically when calculating CIs outside of the distrACTION module.
To get 95% CIs, go back to the Descriptives panel in jamovi, then choose DBH as a variable.
Scroll down to the Statistics options and check ‘Confidence interval for Mean’ under the <strong>Mean Dispersion</strong> options, and make sure that the number in the box is 95 for 95% confidence.
Confidence intervals will appear in the Descriptives table on the right.
From this Descriptives table now, write the lower and upper 95% CIs below.</p>
<p>Lower 95% CI: ________________</p>
<p>Upper 95% CI: ________________</p>
<p>You might have been expecting a bit more of a difference, but remember, for sufficiently large sample sizes (around N = 30), the normal and t-distributions are very similar (see <a href="Chapter_19.html#Chapter_19">Chapter 19</a>).
We really do not expect much of a difference until sample sizes become small, which we will see in Exercise 20.3.</p>
</div>
<div id="confidence-intervals-from-z--and-t-scores" class="section level2 hasAnchor" number="20.2">
<h2><span class="header-section-number">20.2</span> Confidence intervals from z- and t-scores<a href="Chapter_20.html#confidence-intervals-from-z--and-t-scores" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>While jamovi can be very useful for calculating CIs from a dataset, you might also need to calculate CIs from just a set of summary statistics (e.g., mean, standard error, and sample size).
This activity will demonstrate how to calculate CIs from z- and t-scores.
Recall the formula for lower and upper CIs from <a href="Chapter_18.html#normal-distribution-cis">Section 18.1</a>,</p>
<p><span class="math display">\[LCI = \bar{x} - (z \times SE),\]</span></p>
<p><span class="math display">\[UCI = \bar{x} + (z \times SE).\]</span></p>
<p>We could calculate 95% CIs for DBH with just the sample mean (<span class="math inline">\(\bar{x}\)</span>), z-score (<span class="math inline">\(z\)</span>), and standard error (SE).
We have already calculated <span class="math inline">\(\bar{x}\)</span> and SE for the DBH in Exercise 20.1 above, so we just need to figure out z.
Recall that z-scores are <em>standard normal deviates</em>, that is, deviations from the mean given a standard normal distribution, in which the mean equals 0 and standard deviation equals 1.
For example, <span class="math inline">\(z = -1\)</span> is 1 standard deviation below the mean of a standard normal distribution, and <span class="math inline">\(z = 2\)</span> is 2 standard deviations above the mean of a standard normal distribution.
What values of z contain 95% of the probability density of a standard normal distribution?
We can use the distrACTION module again to find this out.
Select ‘Normal Distribution’ from the pull-down menu of the distrACTION module.
Notice that by default, a standard normal distribution is already set (Mean = 0 and SD = 1).
All that we need to do now is compute quantiles for p = 0.95.
From these quantiles, what is the proper z-score to use in the equations for LCI and UCI above?</p>
<p>z-score: ________________</p>
<p>Now, use the values of <span class="math inline">\(\bar{x}\)</span>, <span class="math inline">\(z\)</span>, and SE for DBH in the equations above to calculate lower and upper 95% CIs again.</p>
<p>Lower 95% CI: ________________</p>
<p>Upper 95% CI: ________________</p>
<p>Are these CIs the same as what you calculated in Exercise 20.1?</p>
<pre><code>
</code></pre>
<p>Lastly, instead of using the z-score, we can do the same with a t-score.
We can find the appropriate t-score from the t-distribution in the distrACTION module.
To get the t-score, click on the distrACTION module button and choose ‘T-Distribution’ from the pull-down menu.
To get quantiles with the t-distribution, we need to know the degrees of freedom (<span class="math inline">\(df\)</span>) of the sample.
<a href="Chapter_19.html#Chapter_19">Chapter 19</a> explains how to calculate <span class="math inline">\(df\)</span> from the sample size <span class="math inline">\(N\)</span>.
What are the appropriate <span class="math inline">\(df\)</span> for DBH?</p>
<p><span class="math inline">\(df\)</span>: _________________</p>
<p>Put the df in the Parameters box.
Ignore the box for lambda (<span class="math inline">\(\lambda\)</span>); this is not needed.
Under the <strong>Function</strong> options, choose ‘Compute quantile(s)’ as before to calculate Quantiles.
From the Results table, what is the proper t-score to use in the equations for LCI and UCI?</p>
<p>t-score: _______________</p>
<p>Again, use the values of <span class="math inline">\(\bar{x}\)</span>, <span class="math inline">\(t\)</span>, and <span class="math inline">\(SE\)</span> for DBH in the equations above to calculate lower and upper 95% CIs.</p>
<p>Lower 95% CI: ________________</p>
<p>Upper 95% CI: ________________</p>
<p>How similar are the estimates for lower and upper CIs when using z- versus t-scores.
Reflect on any similarities or differences that you see in all of these different ways of calculating CIs.</p>
<pre><code>
</code></pre>
</div>
<div id="confidence-intervals-for-different-sample-sizes" class="section level2 hasAnchor" number="20.3">
<h2><span class="header-section-number">20.3</span> Confidence intervals for different sample sizes<a href="Chapter_20.html#confidence-intervals-for-different-sample-sizes" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>In Exercises 20.1 and 20.2, the sample size of DBH was fairly large (<span class="math inline">\(N = 120\)</span>).
Now, we will calculate CIs for the mean DBH of each of the four different sites using both z- and t-scores.
These sites have much different sample sizes.
From the Descriptives tool in jamovi, write the sample sizes for DBH split by site below.</p>
<p>Site 1182: <span class="math inline">\(N =\)</span> _________</p>
<p>Site 1223: <span class="math inline">\(N =\)</span> _________</p>
<p>Site 3008: <span class="math inline">\(N =\)</span> _________</p>
<p>Site 10922: <span class="math inline">\(N =\)</span> _________</p>
<p>For which of these sites would you predict CIs calculated from z-scores versus t-scores to differ the most?</p>
<p>Site: ______________</p>
<p>The next part of this exercise is self-guided.
In Exercises 20.1 and 20.2, you used different approaches for calculating 95% CIs from the normal and t-distributions.
Now, fill in Table 20.1 reporting 95% CIs calculated using each distribution from the four sites using any method you prefer.</p>
<table>
<caption><strong>TABLE 20.1</strong> 95% confidence intervals calculated for tree diameter at breast height (DBH) in centimetres. Data for these parameter values were inspired by the Woodland Creation and Ecological Networks (WrEN) project.</caption>
<thead>
<tr class="header">
<th>Site</th>
<th>N</th>
<th>95% CIs (Normal)</th>
<th>95% CIs (t-distribution)</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td>1182</td>
<td></td>
<td></td>
<td></td>
</tr>
<tr class="even">
<td>1223</td>
<td></td>
<td></td>
<td></td>
</tr>
<tr class="odd">
<td>3008</td>
<td></td>
<td></td>
<td></td>
</tr>
<tr class="even">
<td>10922</td>
<td></td>
<td></td>
<td></td>
</tr>
</tbody>
</table>
<p>Next, do the same in Table 20.2, but now calculate 99% CIs instead of 95% CIs.</p>
<table>
<caption><strong>TABLE 20.2</strong> 99% confidence intervals calculated for tree diameter at breast height (DBH) in centimetres. Data for these parameter values were inspired by the Woodland Creation and Ecological Networks (WrEN) project.</caption>
<thead>
<tr class="header">
<th>Site</th>
<th>N</th>
<th>99% CIs (Normal)</th>
<th>99% CIs (t-distribution)</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td>1182</td>
<td></td>
<td></td>
<td></td>
</tr>
<tr class="even">
<td>1223</td>
<td></td>
<td></td>
<td></td>
</tr>
<tr class="odd">
<td>3008</td>
<td></td>
<td></td>
<td></td>
</tr>
<tr class="even">
<td>10922</td>
<td></td>
<td></td>
<td></td>
</tr>
</tbody>
</table>
<p>What do you notice about the difference between CIs calculated from the normal distribution versus the t-distribution across the different sites?</p>
<pre><code>
</code></pre>
<p>In your own words, what do these CIs <em>actually mean</em>?</p>
<pre><code>
</code></pre>
<p>We will now move on to calculating CIs for proportions.</p>
</div>
<div id="proportion-confidence-intervals" class="section level2 hasAnchor" number="20.4">
<h2><span class="header-section-number">20.4</span> Proportion confidence intervals<a href="Chapter_20.html#proportion-confidence-intervals" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>We will now try calculating CIs for proportional data using the WrEN Sites dataset<a href="#fn39" class="footnote-ref" id="fnref39"><sup>39</sup></a>.</p>
<p>Notice that there are more sites included than there were in the dataset used in Exercises 20.1–20.3, and that some of these sites are grazed while others are not (column ‘Grazing’).
From the Descriptives options, find the number of sites grazed versus not grazed (hint, remember from <a href="Chapter_17.html#Chapter_17">Chapter 17</a> to put ‘Grazing’ in the variable box and click the ‘Frequency tables’ checkbox).</p>
<p>Grazed: ____________</p>
<p>Not Grazed: _____________</p>
<p>From these counts above, what is the estimate (<span class="math inline">\(p\)</span>, or more technically <span class="math inline">\(\hat{p}\)</span>, with the hat indicating that it is an estimate) of the proportion of sites that are grazed?</p>
<p><span class="math inline">\(p =\)</span> __________</p>
<p><a href="Chapter_18.html#Chapter_18">Chapter 18</a> explained how to calculate lower and upper CIs for binomial distributions (i.e., proportion data).
It showed how to calculate Wald CIs, but also noted that Clopper-Pearson CIs are generally more accurate.
First, we will calculate Wald CIs by hand, then use jamovi to calculate Clopper-Pearson CIs.
To calculate the Wald CIs, we can use equations similar to the ones used for LCI and UCI from Exercise 20.2 above,</p>
<p><span class="math display">\[LCI = p - z \times \mathrm{SE}(p),\]</span></p>
<p><span class="math display">\[UCI = p + z \times \mathrm{SE}(p),\]</span></p>
<p>We have already calculated <span class="math inline">\(p\)</span>, and we can find z-scores for CIs in the same way that we did in Exercise 20.2 (i.e., the z-scores associated with 95% CIs do not change just because we are working with proportions).
All that is left to calculate LCI and UCI are the standard errors of the proportions.
Remember from <a href="Chapter_18.html#Chapter_18">Chapter 18</a> that these are calculated differently from a standard error of continuous values such as diameter breast height.
The formula for standard error of a proportion is,</p>
<p><span class="math display">\[\mathrm{SE}(p) = \sqrt{\frac{p\left(1 - p\right)}{N}}.\]</span></p>
<p>We can estimate <span class="math inline">\(p\)</span> as the total number of grazed sites divided by <span class="math inline">\(N\)</span>, where <span class="math inline">\(N\)</span> is the total sample size.
Using the above equation, what is the standard error of p?</p>
<p>SE(p) = ____________</p>
<p>Using this standard error, what are the Wald lower and upper 95% CIs around <span class="math inline">\(p\)</span>?</p>
<p>Wald <span class="math inline">\(LCI_{95\%} =\)</span> ______________</p>
<p>Wald <span class="math inline">\(UCI_{95\%} =\)</span> ______________</p>
<p>Next, find the lower and upper 99% CIs around <span class="math inline">\(p\)</span> and report them below (hint: the only difference here from the calculation of the 95% CIs is the z-score).</p>
<p>Wald <span class="math inline">\(LCI_{99\%} =\)</span> ______________</p>
<p>Wald <span class="math inline">\(UCI_{99\%} =\)</span> ______________</p>
<p>Do you notice anything unusual about the lower 99% CI?</p>
<pre><code>
</code></pre>
<p>Now we can use jamovi to find the Clopper-Pearson 95 and 99% CIs.
Jamovi does this for us, so no calculation is required.
To calculate Clopper-Pearson CIs, Find the ‘Frequencies’ button on the toolbar in the ‘Analyses’ tab.
Click on ‘Frequencies’, then choose ‘2 Outcomes’ from the pull-down menu.
You will see a box on the left called ‘Proportion Test (2 Outcomes)’.
From here, move ‘Grazing’ from the left box to the right box.
Under <strong>Additional Statistics</strong>, check the box for ‘Confidence intervals’, and make sure that the interval is 95.
A table called ‘Proportion Test (2 Outcomes)’ will appear to the right.
Find the row with the Grazing Level ‘Yes’, then report what you see for <span class="math inline">\(p\)</span>, and the lower and upper CIs below.</p>
<p><span class="math inline">\(p =\)</span> __________</p>
<p>Clopper-Pearson <span class="math inline">\(LCI_{95\%} =\)</span> ______________</p>
<p>Clopper-Pearson <span class="math inline">\(UCI_{95\%} =\)</span> ______________</p>
<p>To calculate 99% CIs, change the number in the Interval box from 95 to 99.
Report the 99% CIs below.</p>
<p>Clopper-Pearson <span class="math inline">\(LCI_{99\%} =\)</span> ______________</p>
<p>Clopper-Pearson <span class="math inline">\(UCI_{99\%} =\)</span> ______________</p>
<p>What do you notice about the difference between the Wald CIs and the Clopper-Pearson CIs?</p>
<pre><code>
</code></pre>
</div>
<div id="another-proportion-confidence-interval" class="section level2 hasAnchor" number="20.5">
<h2><span class="header-section-number">20.5</span> Another proportion confidence interval<a href="Chapter_20.html#another-proportion-confidence-interval" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>Next, find the 80, 95, and 99% CIs for the proportion of sites that are classified as Ancient woodland using the Clopper-Pearson method for calculating binomial CIs.
First consider an 80% CI.</p>
<p><span class="math inline">\(LCI_{80\%} =\)</span> ______________</p>
<p><span class="math inline">\(UCI_{80\%} =\)</span> ______________</p>
<p>Next, calculate 95% CIs for the proportion of sites classified as Ancient woodland.</p>
<p><span class="math inline">\(LCI_{95\%} =\)</span> ______________</p>
<p><span class="math inline">\(UCI_{95\%} =\)</span> ______________</p>
<p>Finally, calculate 99% CIs for the proportion of sites classified as Ancient woodland.</p>
<p><span class="math inline">\(LCI_{99\%} =\)</span> ______________</p>
<p><span class="math inline">\(UCI_{99\%} =\)</span> ______________</p>
<p>Reflect again on what these values actually mean.
For example, what does it mean to have 95% confidence that the proportion of sites classified as Ancient woodland are between two values?
Are there any situations in which this might be useful, from a scientific or conservation standpoint?
There is no right or wrong answer here, but CIs are very challenging to understand conceptually, so having now done the calculations to get them, it is a good idea to think again about what they mean.</p>
<pre><code>
</code></pre>
</div>
</div>
<h3>References<a href="references.html#references" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<div id="refs" class="references csl-bib-body hanging-indent" line-spacing="2">
<div id="ref-Fuentes-Montemayor2022" class="csl-entry">
Fuentes-Montemayor, E., Park, K. J., Cordts, K., & Watts, K. (2022). <span class="nocase">The long-term development of temperate woodland creation sites: from tree saplings to mature woodlands</span>. <em>Forestry</em>, <em>95</em>, 28–37. <a href="https://doi.org/10.1093/forestry/cpab027">https://doi.org/10.1093/forestry/cpab027</a>
</div>
<div id="ref-Fuentes-Montemayor2022a" class="csl-entry">
Fuentes-Montemayor, E., Watts, K., Sansum, P., Scott, W., & Park, K. J. (2022). <span class="nocase">Moth community responses to woodland creation: The influence of woodland age, patch characteristics and landscape attributes</span>. <em>Diversity and Distributions</em>, <em>28</em>(9), 1993–2007. <a href="https://doi.org/10.1111/ddi.13599">https://doi.org/10.1111/ddi.13599</a>
</div>
<div id="ref-Jamovi2022" class="csl-entry">
The jamovi project. (2024). <em>Jamovi (version 2.5)</em>. <a href="https://www.jamovi.org">https://www.jamovi.org</a>
</div>
</div>
<div class="footnotes">
<hr />
<ol start="38">
<li id="fn38"><p><a href="https://bradduthie.github.io/stats/data/wren_trees.xlsx">https://bradduthie.github.io/stats/data/wren_trees.xlsx</a><a href="Chapter_20.html#fnref38" class="footnote-back">↩︎</a></p></li>
<li id="fn39"><p><a href="https://bradduthie.github.io/stats/data/wren_sites.csv">https://bradduthie.github.io/stats/data/wren_sites.csv</a><a href="Chapter_20.html#fnref39" class="footnote-back">↩︎</a></p></li>
</ol>
</div>
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