You are looking at a population and are interested in the proportion \( p \) that has a certain characteristic. Unknown to you, this population proportion is \( p = 0.60 \). You have taken a random sample of size \( n = 120 \) from the population and found that the proportion of the sample that has the characteristic is \( \hat{p} = 0.53 \). Your sample is Sample 1 in the table below. (In the table, Sample 1 is written "S1", Sample 2 is written "S2", etc.)

(a) Based on Sample 1, graph the 75% and 95% confidence intervals for the population proportion. Use 1.150 for the critical value for the 75% confidence interval, and use 1.960 for the critical value for the 95% confidence interval.

- Enter the lower and upper limits on the graphs to show each confidence interval. Write your answers with two decimal places.
- For the points (\(\star\) and \(\star\)), enter the population proportion, 0.60.

(b) Press the "Generate Samples" button below to simulate taking 19 more samples of size \( n = 120 \) from the same population. Notice that the confidence intervals update automatically. Then complete parts (c) and (d) below the table.

(c) Notice that for 18 out of 20 samples (\(18/20 = 90\%\)), the 95% confidence interval contains the population proportion. Choose the correct statement.
- When constructing 95% confidence intervals for 20 samples of the same size from the population, exactly 95% of the samples must contain the population proportion.
- There must have been an error with the way our samples were chosen.
- When constructing 95% confidence intervals for 20 samples of the same size from the population, the percentage of the samples that contain the population proportion should be close to 95%, but it may not be exactly 95%.
- When constructing 95% confidence intervals for 20 samples of the same size from the population, at most 95% of the samples will contain the population proportion.

(d) Choose ALL that are true.
- To guarantee that a confidence interval will contain the population proportion, the level of confidence must match the sample proportion. For example, if \( p \) equals 0.45, then the 45% confidence interval will contain the population proportion.
- From the 75% confidence interval for Sample 6, we cannot say that there is a 75% probability that the population proportion is between 0.39 and 0.78.
- The 75% confidence interval for Sample 6 is narrower than the 95% confidence interval for Sample 6. This must be the case; when constructing a confidence interval for a sample, the greater the level of confidence, the wider the confidence interval.
- If there were a Sample 21 of size \( n = 160 \) with the same sample proportion as Sample 6, then the 75% confidence interval for Sample 21 would be wider than the 75% confidence interval for Sample 6.
- None of the choices above are true.