In response to media inquiries and concerns expressed by groups opposed to violence, the president of a university with over 25,000 students has agreed to survey a simply random sample of her students to find out whether the student body thinks the school’s “Plundering Pirate” mascot should be changed to one that is less aggressive in name and appearance. Of the 200 students selected for participation in the survey, only 20% believe the school should select a new and more kindly mascot. Construct a 90% confidence interval for the population proportion of students who believe the mascot should be changed. Based on the sample findings and associated confidence interval, comment on the credibility of a local journalist’s comment that “over 50% of the students would like a new mascot.

To construct a confidence interval for the population proportion of students who believe the mascot should be changed, we can follow these steps:

Step 1: Identify the relevant information:
- Sample size (n): 200 students
- Number of students in the sample who believe the mascot should be changed (x): 20% of 200 = 0.2 * 200 = 40 students
- Confidence level (CL): 90%
- Standard deviation (σ): We will use the formula for estimating the standard deviation, assuming we don't have prior knowledge about the population proportion. The formula is: σ = sqrt((p * (1 - p)) / n), where p is the sample proportion. In this case, p = 0.2.

Step 2: Calculate the standard error:
Standard error (SE) = sqrt((p * (1 - p)) / n)
SE = sqrt((0.2 * (1 - 0.2)) / 200) = sqrt(0.16 / 200) ≈ 0.020

Step 3: Determine the critical value:
Since the confidence level is 90%, we need to find the Z-score corresponding to a 95% confidence level (one-tailed test). The critical value can be found using a Z-table or a statistical calculator. For a 90% confidence level, the critical value is approximately 1.645.

Step 4: Calculate the margin of error:
Margin of error (ME) = critical value * standard error
ME = 1.645 * 0.020 ≈ 0.033

Step 5: Calculate the lower and upper bounds of the confidence interval:
Lower bound = sample proportion - margin of error
Upper bound = sample proportion + margin of error

Lower bound = 0.2 - 0.033 = 0.167
Upper bound = 0.2 + 0.033 = 0.233

Step 6: Interpret the confidence interval:
The 90% confidence interval for the population proportion of students who believe the mascot should be changed is approximately 16.7% to 23.3%.

Based on this confidence interval, we cannot support the local journalist's comment that "over 50% of the students would like a new mascot." The lower bound of the confidence interval is significantly below 50%, suggesting that less than 50% of the students in the population actually believe the mascot should be changed.