Question

Construct a confidence interval for the proportion of students who purchased used textbooks in the first-year and second-year groups based on a survey.

Answer 1

The confidence interval for the proportion of students who purchased used textbooks in the first-year group is estimated to be between
`\( \hat{p}_1 - E_1 \) and \( \hat{p}_1 + E_1 \)`, and for the second-year group, it is estimated to be between
`\( \hat{p}_2 - E_2 \) and \( \hat{p}_2 + E_2 \)`, where
`\( \hat{p}_1 \) and \( \hat{p}_2 \)` are the sample proportions, and
`\( E_1 \) and \( E_2 \` are the respective margin of errors.

Step-by-step explanation:

To construct the confidence interval for the proportion of students who purchased used textbooks in the first-year and second-year groups, we use the formula:

`\[ \text{Confidence Interval} = \hat{p} \pm E \]`

Where
`\( \hat{p} \)` is the sample proportion and
`\( E \)` is the margin of error. The margin of error is calculated as:

`\[ E = Z * \sqrt{\frac{\hat{p} * (1 - \hat{p})}{n}} \]`

Here
`, \( Z \)` is the Z-score corresponding to the desired confidence level,
`\( \hat{p} \)` is the sample proportion, and
`\( n \)` is the sample size. For each group, this formula is applied to find the confidence interval.

It's crucial to note that the Z-score varies based on the chosen confidence level. For example, for a 95% confidence level, the Z-score is approximately 1.96. The sample proportion
`(\( \hat{p} \))` is the observed proportion of students who purchased used textbooks in each group, and
`\( n \)` is the sample size.

In summary, the confidence interval provides a range within which we are reasonably confident that the true proportion of students who purchased used textbooks lies. This statistical approach allows us to make inferences about the entire population of first-year and second-year students based on the survey results.