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<!DOCTYPE html>
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<title>Chapter 18 Confidence 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 18 Confidence intervals | Fundamental statistical concepts and techniques in the biological and environmental sciences: With jamovi" />
<meta name="twitter:description" content="This is an introductory statistics textbook for students in the biological and environmental sciences with examples using jamovi statistical software." />
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<meta name="author" content="A. Bradley Duthie" />
<meta name="date" content="2024-08-06" />
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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>
</ul></li>
<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>
<ul>
<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>
<ul>
<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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.9.1" 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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<div id="Chapter_18" class="section level1 hasAnchor" number="18">
<h1><span class="header-section-number">Chapter 18</span> Confidence intervals<a href="Chapter_18.html#Chapter_18" class="anchor-section" aria-label="Anchor link to header"></a></h1>
<p>In <a href="Chapter_16.html#Chapter_16">Chapter 16</a>, we saw how it is possible to calculate the probability of sampling values from a specific interval of the normal distribution (e.g., the probability of sampling a value within 1 standard deviation of the mean).
In this chapter, we will see how to apply this knowledge to calculating intervals that express confidence in the mean value of a population.</p>
<p>Remember that we almost never really know the true mean value of a <em>population</em>, <span class="math inline">\(\mu\)</span>.
Our best estimate of <span class="math inline">\(\mu\)</span> is the mean that we have calculated from a <em>sample</em>, <span class="math inline">\(\bar{x}\)</span> (see <a href="Chapter_4.html#Chapter_4">Chapter 4</a> for a review of the difference between populations and samples).
But how good of an estimate is <span class="math inline">\(\bar{x}\)</span> of <span class="math inline">\(\mu\)</span>, really?
Since we cannot know <span class="math inline">\(\mu\)</span>, one way of answering this question is to find an interval that expresses a degree of confidence about the value of <span class="math inline">\(\mu\)</span>.
The idea is to calculate two numbers that we can say with some degree of confidence that <span class="math inline">\(\mu\)</span> is between (i.e., a lower confidence interval and an upper confidence interval).
The wider this interval is, the more confident that we can be that the true mean <span class="math inline">\(\mu\)</span> is somewhere within it.
The narrower the interval is, the less confident we can be that our confidence intervals (CIs) contain <span class="math inline">\(\mu\)</span>.</p>
<p>Confidence intervals are notoriously easy to misunderstand.
I will explain this verbally first, focusing on the general ideas rather than the technical details.
Then I will present the calculations before coming back to their interpretation again.
The idea follows a similar logic to the standard error from <a href="Chapter_12.html#Chapter_12">Chapter 12</a>.</p>
<p>Suppose that we want to know the mean body mass of all domestic cats (Figure 18.1).
We cannot weigh every living cat in the world, but maybe we can find enough to get a sample of 20.
From these 20 cats, we want to find some interval of masses (e.g., 3.9–4.3 kg) within which the <em>true</em> mean mass of the population is contained.
The only way to be 100% certain that our proposed interval <em>definitely</em> contains the true mean would be to make the interval absurdly large.
Instead, we might more sensibly ask what the interval would need to be to contain the mean with 95% confidence.
What does ‘with 95% confidence’ actually mean?
It means that when we do the calculation to get the interval, the true mean should be somewhere within the interval 95% of the time that a sample is collected.</p>
<div class="figure"><span style="display:block;" id="fig:unnamed-chunk-76"></span>
<img src="img/housecats.png" alt="Two cats sitting close together on a windowsill, a large orange one in the front and small black and brown one in the back" width="100%" />
<p class="caption">
Figure 18.1: Two domestic cats sitting side by side with much different body masses.
</p>
</div>
<p>In other words, if we were to go back out and collect another sample of 20 cats, and then another, and another (and so forth), calculating 95% CIs each time, then in 95% of our samples the true mean will be within our CIs (meaning that 5% of the time it will be outside the CIs).
Note that this is slightly different than saying that there is a 95% probability that the true mean is between our CIs.<a href="#fn25" class="footnote-ref" id="fnref25"><sup>25</sup></a>
Instead, the idea is that if we were to repeatedly resample from a population and calculate CIs each time, then 95% of the time the true mean would be within our CIs <span class="citation">(<a href="#ref-Sokal1995" role="doc-biblioref">Sokal & Rohlf, 1995</a>)</span>.
If this idea does not make sense at first, that is okay.
The calculation is actually relatively straightforward, and we will come back to the statistical concept again afterwards to interpret it.
First, we will look at CIs assuming a normal distribution, then the separate case of a binomial distribution.</p>
<div id="normal-distribution-cis" class="section level2 hasAnchor" number="18.1">
<h2><span class="header-section-number">18.1</span> Normal distribution CIs<a href="Chapter_18.html#normal-distribution-cis" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>Remember from the Central Limit Theorem in <a href="Chapter_16.html#Chapter_16">Chapter 16</a> that as our sample size <span class="math inline">\(N\)</span> increases, the distribution of our sample mean <span class="math inline">\(\bar{x}\)</span> will start looking more and more like a normal distribution.
Also from <a href="Chapter_16.html#Chapter_16">Chapter 16</a>, we know that we can calculate the probability associated with any interval of values in a normal distribution.
For example, we saw that about 68.2% of the probability density of a normal distribution is contained within one standard deviation of the mean.
We can use this knowledge from <a href="Chapter_16.html#Chapter_16">Chapter 16</a> to set confidence intervals for any percentage of values around the sample mean (<span class="math inline">\(\bar{x}\)</span>) using a standard error (SE) and z-score (<span class="math inline">\(z\)</span>).
Confidence intervals include two numbers.
The <strong>lower confidence interval</strong> (LCI) is below the mean, and the <strong>upper confidence interval</strong> (UCI) is above the mean.
Here is how they are calculated,</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>Note that the equations are the same, except that for the LCI, we are subtracting <span class="math inline">\(z \times SE\)</span>, and for the UCI we are adding it.
The specific value of <span class="math inline">\(z\)</span> determines the confidence interval that we are calculating.
For example, about 95% of the probability density of a standard normal distribution lies between <span class="math inline">\(z = -1.96\)</span> and <span class="math inline">\(z = 1.96\)</span> (Figure 18.2).
Hence, if we use <span class="math inline">\(z = 1.96\)</span> to calculate LCI and UCI, we would be getting 95% confidence intervals around our mean (an interactive application<a href="#fn26" class="footnote-ref" id="fnref26"><sup>26</sup></a> helps visualise the relationship between probability intervals and z-scores more generally).</p>
<div class="figure"><span style="display:block;" id="fig:unnamed-chunk-77"></span>
<img src="bookdown-demo_files/figure-html/unnamed-chunk-77-1.png" alt="A plot of a bell curve in which the middle 95% of the distribution is shaded in grey. A value of zero centres the x-axis, which is labelled 'z-score'." width="672" />
<p class="caption">
Figure 18.2: Standard normal probability distribution showing 95% of probability density surrounding the mean.
</p>
</div>
<p>Now suppose that we want to calculate 95% CIs around the sample mean of our <span class="math inline">\(N = 20\)</span> domestic cats from earlier.
We find that the mean body mass of cats in our sample is <span class="math inline">\(\bar{x} = 4.1\)</span> kg, and that the standard deviation is <span class="math inline">\(s = 0.6\)</span> kg (suppose that we are willing to assume, for now, that <span class="math inline">\(s = \sigma\)</span>, that is, we know the true standard deviation of the population).
Remember from <a href="Chapter_12.html#Chapter_12">Chapter 12</a> that the sample standard error can be calculated as <span class="math inline">\(s / \sqrt{N}\)</span>.
Our lower 95% confidence interval is therefore,</p>
<p><span class="math display">\[LCI_{95\%} = 4.1 - \left(1.96 \times \frac{0.6}{\sqrt{20}}\right) = 3.837\]</span></p>
<p>Our upper 95% confidence interval is,</p>
<p><span class="math display">\[UCI_{95\%} = 4.1 + \left(1.96 \times \frac{0.6}{\sqrt{20}}\right) = 4.363\]</span></p>
<p>Our 95% CIs are therefore <span class="math inline">\(LCI = 3.837\)</span> and <span class="math inline">\(UCI = 4.363\)</span>.
We can now come back to the statistical concept of what this actually means.
If we were to go out and repeatedly collect new samples of 20 cats, and do the above calculations each time, then 95% of the time our true mean cat body mass would be somewhere between the LCI and UCI.</p>
<p>The 95% confidence intervals are the most commonly used in the biological and environmental sciences.
In other words, we accept that about 5% of the time (1 in 20 times), our confidence intervals will not contain the true mean that we are trying to estimate.
Suppose, however, that we wanted to be a bit more cautious.
We could calculate 99% CIs, that is, CIs that contain the true mean in 99% of samples.
To do this, we just need to find the z-score that corresponds with 99% of the probability density of the standard normal distribution.
This value is about <span class="math inline">\(z = 2.58\)</span>, which we could find with an interactive application, a <a href="https://www.z-table.com/">z table</a>, some maths, or a quick online search<a href="#fn27" class="footnote-ref" id="fnref27"><sup>27</sup></a>.
Consequently, the lower 99% confidence interval for our example of cat body masses would be,</p>
<p><span class="math display">\[LCI_{99\%} = 4.1 - \left(2.58 \times \frac{0.6}{\sqrt{20}}\right) = 3.754\]</span></p>
<p>Our upper 99% confidence interval is,</p>
<p><span class="math display">\[UCI_{99\%} = 4.1 + \left(2.58 \times \frac{0.6}{\sqrt{20}}\right) = 4.446\]</span></p>
<p>Notice that the confidence intervals became wider around the sample mean.
The 99% CI is now 3.754–4.446, while the 95% CI was 3.837–4.363.
This is because if we want to be more confident about our interval containing the true mean, we need to make a bigger interval.</p>
<p>We could make CIs using any percentage that we want, but in practice it is very rare to see anything other than 90% (<span class="math inline">\(z = 1.65\)</span>), 95% (<span class="math inline">\(z = 1.96\)</span>), or 99% (<span class="math inline">\(z = 2.58\)</span>).
It is useful to see what these different intervals look like when calculated from actual data (an interactive application<a href="#fn28" class="footnote-ref" id="fnref28"><sup>28</sup></a> illustrates CIs on a histogram).
Unfortunately, the CI calculations from this section are a bit of an idealised situation.
We assumed that the sample means are normally distributed around the population mean.
While we know that this <em>should</em> be the case as our sample size increases, it is not quite true when our sample is small.
In practice, what this means is that our z-scores are usually not going to be the best values to use when calculating CIs, although they are often good enough when a sample size is large<a href="#fn29" class="footnote-ref" id="fnref29"><sup>29</sup></a>.
We will see what to do about this in <a href="Chapter_19.html#Chapter_19">Chapter 19</a>, but first we turn to the special case of how to calculate CIs from binomial proportions.</p>
</div>
<div id="binomial-distribution-cis" class="section level2 hasAnchor" number="18.2">
<h2><span class="header-section-number">18.2</span> Binomial distribution CIs<a href="Chapter_18.html#binomial-distribution-cis" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>For a binomial distribution, our data are counts of successes and failures (see <a href="Chapter_15.html#Chapter_15">Chapter 15</a>).
For example, we might flip a coin 40 times and observe 22 heads and 18 tails.
Suppose that we do not know in advance the coin is fair, so we cannot be sure that the probability of it landing on heads is <span class="math inline">\(p = 0.5\)</span>.
From our collected data, our estimated probability of landing on heads is, <span class="math inline">\(\hat{p} = 22/40 = 0.55\)</span>.<a href="#fn30" class="footnote-ref" id="fnref30"><sup>30</sup></a>
But how would we calculate the CIs around this estimate?
There are multiple ways to calculate CIs around proportions.
One common method relies on a normal approximation, that is, approximating the discrete counts of the binomial distribution using the continuous normal distribution.
For example, we can note that the variance of <span class="math inline">\(p\)</span> for a binomial distribution is <span class="math inline">\(\sigma^{2} = p\left(1 - p\right)\)</span> <span class="citation">(<a href="#ref-Box1978" role="doc-biblioref">Box et al., 1978</a>; <a href="#ref-Sokal1995" role="doc-biblioref">Sokal & Rohlf, 1995</a>)</span>.<a href="#fn31" class="footnote-ref" id="fnref31"><sup>31</sup></a>
This means that the standard deviation of <span class="math inline">\(p\)</span> is <span class="math inline">\(\sigma = \sqrt{p\left(1 - p\right)}\)</span>, and <span class="math inline">\(p\)</span> has a standard error,</p>
<p><span class="math display">\[\mathrm{SE}(p) = \sqrt{\frac{p\left(1 - p\right)}{N}}.\]</span></p>
<p>We could then use this standard error in the same equation from earlier for calculating CIs.
For example, if we wanted to calculate the lower 95% CI for <span class="math inline">\(\hat{p} = 0.55\)</span>,</p>
<p><span class="math display">\[LCI_{95\%} = 0.55 - 1.96 \sqrt{\frac{0.55\left(1 - 0.55\right)}{40}} = 0.396\]</span></p>
<p>Similarly, to calculate the upper 95% CI,</p>
<p><span class="math display">\[UCI_{95\%} = 0.55 + 1.96 \sqrt{\frac{0.55\left(1 - 0.55\right)}{40}} = 0.704.\]</span></p>
<p>Our conclusion is that, based on our sample, 95% of the time we flip a coin 40 times, the true mean <span class="math inline">\(p\)</span> will be somewhere between 0.396 and 0.704.
These are quite wide CIs, which suggests that our flip of <span class="math inline">\(\hat{p} = 0.55\)</span> would not be particularly remarkable even if the coin was fair (<span class="math inline">\(p = 0.5\)</span>).<a href="#fn32" class="footnote-ref" id="fnref32"><sup>32</sup></a>
This CI is called the Wald interval.
It is easy to calculate, and is instructive for showing another way that CIs can be calculated.
But it does not actually do a very good job of accurately producing 95% CIs around a proportion, especially when our proportion is very low or high <span class="citation">(<a href="#ref-Andersson2023" role="doc-biblioref">Andersson, 2023</a>; <a href="#ref-Schilling2014" role="doc-biblioref">Schilling & Doi, 2014</a>)</span>.</p>
<p>In fact, there are many different methods proposed to calculate binomial CIs, and a lot of statistical research has been done to compare and contrast these methods <span class="citation">(<a href="#ref-Reed2007" role="doc-biblioref">Reed, 2007</a>; <a href="#ref-Schilling2014" role="doc-biblioref">Schilling & Doi, 2014</a>; <a href="#ref-Thulin2014" role="doc-biblioref">Thulin, 2014</a>)</span>.
Jamovi uses Clopper-Pearson CIs <span class="citation">(<a href="#ref-Clopper1934" role="doc-biblioref">Clopper & Pearson E. S., 1934</a>)</span>.
Instead of relying on a normal approximation, the Clopper-Pearson method uses the binomial probability distribution (see <a href="Chapter_15.html#Chapter_15">Chapter 15</a>) to build CIs.
This is what is known as an ‘exact method’.
Clopper-Pearson CIs tend to err on the side of caution (they are often made wider than necessary), but they are a much better option than Wald CIs <span class="citation">(<a href="#ref-Reed2007" role="doc-biblioref">Reed, 2007</a>)</span>.</p>
</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-Andersson2023" class="csl-entry">
Andersson, P. G. (2023). The <span>W</span>ald confidence interval for a binomial p as an illuminating <span>“bad”</span> example. <em>American Statistician</em>, <em>77</em>(4), 443–448. <a href="https://doi.org/10.1080/00031305.2023.2183257">https://doi.org/10.1080/00031305.2023.2183257</a>
</div>
<div id="ref-Box1978" class="csl-entry">
Box, G. E. P., Hunter, W. G., & Hunter, S. J. (1978). <em><span class="nocase">Statistics for Experimenters: An Introduction to Design, Data Analysis, and Model Building</span></em>. John Wiley & Sons, New York, USA.
</div>
<div id="ref-Clopper1934" class="csl-entry">
Clopper, C. J., & Pearson E. S. (1934). <span class="nocase">The use of confidence or fiducial limits illustrated in the case of the binomial</span>. <em>Biometrika</em>, <em>26</em>(4), 404–413.
</div>
<div id="ref-Ellison2004" class="csl-entry">
Ellison, A. M. (2004). <span class="nocase">Bayesian inference in ecology</span>. <em>Ecology Letters</em>, <em>7</em>(6), 509–520. <a href="https://doi.org/10.1111/j.1461-0248.2004.00603.x">https://doi.org/10.1111/j.1461-0248.2004.00603.x</a>
</div>
<div id="ref-Reed2007" class="csl-entry">
Reed, J. F. (2007). <span class="nocase">Better binomial confidence intervals</span>. <em>Journal of Modern Applied Statistical Methods</em>, <em>6</em>(1), 153–161. <a href="https://doi.org/10.22237/jmasm/1177992840">https://doi.org/10.22237/jmasm/1177992840</a>
</div>
<div id="ref-Schilling2014" class="csl-entry">
Schilling, M. F., & Doi, J. A. (2014). A coverage probability approach to finding an optimal binomial confidence procedure. <em>American Statistician</em>, <em>68</em>(3), 133–145. <a href="https://doi.org/10.1080/00031305.2014.899274">https://doi.org/10.1080/00031305.2014.899274</a>
</div>
<div id="ref-Sokal1995" class="csl-entry">
Sokal, R. R., & Rohlf, F. J. (1995). <em><span>Biometry</span></em> (3rd ed., p. 887). W. H. Freeman & Company, New York, USA.
</div>
<div id="ref-Thulin2014" class="csl-entry">
Thulin, M. (2014). <span class="nocase">The cost of using exact confidence intervals for a binomial proportion</span>. <em>Electronic Journal of Statistics</em>, <em>8</em>(1), 817–840. <a href="https://doi.org/10.1214/14-EJS909">https://doi.org/10.1214/14-EJS909</a>
</div>
</div>
<div class="footnotes">
<hr />
<ol start="25">
<li id="fn25"><p>The reason that these two ideas are different has to do with the way that probability is defined in the frequentist approach to statistics (see <a href="Chapter_15.html#Chapter_15">Chapter 15</a>). With this approach, there is no way to get the probability of the true mean being within an interval, strictly speaking. Other approaches to probability, such as Bayesian probability, do allow you to build intervals in which the true mean is contained with some probability. These are called ‘credible intervals’ rather than ‘confidence intervals’ <span class="citation">(e.g., <a href="#ref-Ellison2004" role="doc-biblioref">Ellison, 2004</a>)</span>. The downside to credible intervals (or not, depending on your philosophy of statistics) is that Bayesian probability is at least partly subjective, i.e., based in some way on the subjective opinion of the individual researcher.<a href="Chapter_18.html#fnref25" class="footnote-back">↩︎</a></p></li>
<li id="fn26"><p><a href="https://bradduthie.github.io/stats/app/zandp/" class="uri">https://bradduthie.github.io/stats/app/zandp/</a><a href="Chapter_18.html#fnref26" class="footnote-back">↩︎</a></p></li>
<li id="fn27"><p>While it is always important to be careful when searching, typing ‘z-score 99% confidence interval’ will almost always get the intended result.<a href="Chapter_18.html#fnref27" class="footnote-back">↩︎</a></p></li>
<li id="fn28"><p><a href="https://bradduthie.github.io/stats/app/CI_hist_app/" class="uri">https://bradduthie.github.io/stats/app/CI_hist_app/</a><a href="Chapter_18.html#fnref28" class="footnote-back">↩︎</a></p></li>
<li id="fn29"><p>What defines a ‘small’ or a ‘large’ sample is a bit arbitrary. A popular suggestion <span class="citation">(e.g., <a href="#ref-Sokal1995" role="doc-biblioref">Sokal & Rohlf, 1995, p. 145</a>)</span> is that any <span class="math inline">\(N < 30\)</span> is too small to use z-scores, but any cut-off <span class="math inline">\(N\)</span> is going to be somewhat arbitrary. Technically, the z-score is not completely accurate until <span class="math inline">\(N \to \infty\)</span>, but for all intents and purposes, it is usually only trivially inaccurate for sample sizes in the hundreds. Fortunately, you do not need to worry about any of this when calculating CIs from continuous data in jamovi because jamovi applies a correction for you, which we will look at in <a href="Chapter_19.html#Chapter_19">Chapter 19</a>.<a href="Chapter_18.html#fnref29" class="footnote-back">↩︎</a></p></li>
<li id="fn30"><p>The hat over the p, (<span class="math inline">\(\hat{p}\)</span>) is just being used here to indicate the <em>estimate</em> of <span class="math inline">\(P(heads)\)</span>, rather than the <em>true</em> <span class="math inline">\(P(heads)\)</span>.<a href="Chapter_18.html#fnref30" class="footnote-back">↩︎</a></p></li>
<li id="fn31"><p>Note, the variance of total <em>successes</em> is simply <span class="math inline">\(np\left(1 - p\right)\)</span>, i.e., just multiply the variance of <span class="math inline">\(p\)</span> by <span class="math inline">\(n\)</span>.<a href="Chapter_18.html#fnref31" class="footnote-back">↩︎</a></p></li>
<li id="fn32"><p>You might ask, why are we doing all of this for the binomial distribution? The central limit theorem is supposed to work for the mean of any distribution, so should that not include the distribution of <span class="math inline">\(p\)</span> too? Can we not just indicate success (heads) with a 1 and failures (tails) with a 0, then estimate the standard error of 22 values of 1 and 18 values of 0? Well, yes! That actually does work and gives an estimate of 0.079663, which is very close to the <span class="math inline">\(\sqrt{\hat{p}(1-\hat{p})/N} = 0.078661\)</span>. The problem arises when the sample size is low, or when <span class="math inline">\(p\)</span> is close to 0 or 1, and we are trying to map the z-score to probability density.<a href="Chapter_18.html#fnref32" class="footnote-back">↩︎</a></p></li>
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