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.

Question

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.

Answer 1

To construct a confidence interval for the population proportion, we can follow these steps:

1. Identify the sample proportion: In this case, the sample proportion is the percentage of students who believe the mascot should be changed. Given that 20% of the 200 students surveyed hold this belief, the sample proportion is 0.20 (or 20%).

2. Determine the level of confidence: The problem states that we need to construct a 90% confidence interval. This means that we want to be 90% confident that the true proportion of students who believe the mascot should be changed falls within our interval.

3. Calculate the margin of error: The margin of error is the range within which we expect the true population proportion to fall. It depends on the level of confidence and the standard deviation (which is approximated by using the sample proportion). The formula for the margin of error is:

Margin of Error = Critical Value * Standard Error

The critical value is obtained from a standard normal distribution table or a calculator based on the chosen level of confidence. For a 90% confidence interval, the critical value is approximately 1.645.

The standard error can be calculated using the formula:

Standard Error = √[(sample proportion * (1 - sample proportion)) / sample size]

Plugging in the values, we have:

Standard Error = √[(0.20 * (1 - 0.20)) / 200] ≈ 0.02

Now, we can calculate the margin of error:

Margin of Error = 1.645 * 0.02 ≈ 0.033

4. Construct the confidence interval: The confidence interval is calculated by subtracting and adding the margin of error from the sample proportion. In this case, the population proportion is estimated to be between the sample proportion minus the margin of error and the sample proportion plus the margin of error. Thus, the confidence interval can be calculated as:

Confidence Interval = Sample Proportion ± Margin of Error

Confidence Interval = 0.20 ± 0.033

This gives us a confidence interval of (0.167, 0.233) or in percentage terms, (16.7%, 23.3%).

Now, considering the confidence interval obtained, we can comment on the credibility of the local journalist's statement. The journalist claimed that "over 50% of the students would like a new mascot." From the confidence interval, we see that the proportion of students who believe the mascot should be changed could be as low as 16.7% and as high as 23.3%. Since this range does not include 50%, it suggests that the journalist's comment is not credible based on the provided sample findings and associated confidence interval.