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A confidence interval gives a range of values within which we expect the true population parameter to lie. This range is calculated from the sample data. For instance, in Jane's survey, the 95% confidence interval indicates where we believe the true proportion of students who eat cauliflower can be found, based on her sample of 20 students. The confidence intervals are especially useful because they provide an estimated range rather than a single number, giving us a better sense of the uncertainty associated with the sample data.
Confidence intervals typically depend on the sample size, the sample proportion, and the desired level of confidence (like 95%). As the sample size increases, the confidence interval tends to become narrower, meaning we get a more precise estimate of the population parameter.

The sample proportion, denoted as \(\hat{p}\), is the ratio of the number of favorable outcomes to the total number of observations in the sample. In Jane's case, she surveyed 20 students and found that 2 of them eat cauliflower. Therefore, the sample proportion is \(\hat{p} = \frac{2}{20} = 0.10\).
This sample proportion is a point estimate of the true population proportion. In other words, it provides a single best guess of the proportion of all students on Jane's campus who eat cauliflower based on her sample. However, due to sampling variability, this estimate will have some degree of uncertainty, which is why we use confidence intervals.

The standard error measures the variability of a sample proportion as we repeatedly sample from the same population. It's crucial for constructing confidence intervals. Agresti and Coull's method adjusts the sample proportion and sample size to determine the standard error, making their method more accurate, especially with small sample sizes.
For Jane's data, the adjusted sample proportion is calculated as \(\tilde{p} = \frac{x + 2}{n + 4}\) and the adjusted sample size as \(\tilde{n} = n + 4\). Thus, \(\tilde{p} = \frac{4}{24} = 0.167\) and \(\tilde{n} = 24\). The standard error formula is:
\[\text{SE} = \sqrt{\frac{\tilde{p}(1 - \tilde{p})}{\tilde{n}}}\]
By substituting the adjusted values, we obtain:
\[\text{SE} = \sqrt{\frac{0.167(1 - 0.167)}{24}} \approx 0.075\],
indicating a measure of the sampling distribution's variability.

A 95% confidence interval provides a range within which we are 95% confident the true population parameter lies. For a 95% confidence level, the Z-value is approximately 1.96. To calculate the 95% confidence interval for Jane's adjusted proportion, we use the formula:
\[ \tilde{p} \pm Z \times SE \]
Substituting the values:
\[ 0.167 \pm 1.96 \times 0.075 = 0.167 \pm 0.147 \]
This results in an interval of \[\left[0.020, 0.314\right]\].
This means we are 95% confident that the true proportion of students who eat cauliflower at Jane's campus is between 2% and 31.4%. This range accounts for the inherent uncertainty in the sample data while providing a useful estimation of the population parameter.