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, we will use the sample proportion and sample size provided. First, let's calculate the standard error of the sample proportion:

Standard Error (SE) = sqrt((sample proportion * (1 - sample proportion)) / sample size)

In this case:
Sample Proportion (p̂) = 20% or 0.2
Sample Size (n) = 200

SE = sqrt((0.2 * (1 - 0.2)) / 200)
= sqrt(0.16 / 200)
= sqrt(0.0008)
≈ 0.0283

Next, we can calculate the margin of error by multiplying the standard error by the appropriate critical value. For a 90% confidence level, the critical value is 1.645.

Margin of Error = Critical Value * Standard Error
= 1.645 * 0.0283
≈ 0.0467

Now, we can construct the confidence interval by adding and subtracting the margin of error from the sample proportion:

Lower Limit = Sample Proportion - Margin of Error
= 0.2 - 0.0467
= 0.1533

Upper Limit = Sample Proportion + Margin of Error
= 0.2 + 0.0467
= 0.2467

Therefore, the 90% confidence interval for the population proportion of students who believe the mascot should be changed is approximately 0.1533 to 0.2467.

Now, let's comment on the credibility of the local journalist's comment that "over 50% of the students would like a new mascot." Based on the confidence interval, we can see that the lower limit is 0.1533. Since this lower limit is below 50%, it suggests that there is a significant possibility that less than 50% of the students actually want a new mascot. Therefore, the journalist's comment seems to lack credibility based on the provided sample findings and associated confidence interval.