Statistics Homework: Confidence Intervals The age of students in a math class at PSU are as follows:

18 18 18 18 18 18 18 18 18 18 18 18 18 18 18
19 19 19 19 19 19 19 19 19 19 19 19 19 19 19
19 19 19 19 19 19 20 20 20 20 21 21 21 21 21
21 21 21 22 22 23 24 24 25 26 26 34 35 35 36

1. Find a 99% confidence interval for the mean age of PSU math students. Do the median and mode fall into this range? If we want the margin of error to be less than 1 year, how many total students should we sample?

2. The average age of PSU students is 26. Does this value fall within the confidence interval? Assuming that this is the true mean age and a standard deviation of σ=5, what is the probability of seeing a mean age (of a sample of n=60) less than 24? Does this provide evidence that the true mean of PSU math students is less than 26? Can we generalize this result to all PSU students?

3. Find a 90% confidence interval for the proportion of math students who are 21 or over. If we want the margin of error to be within 5 percentage points, how many total students must we sample?

4. If we assume that the data is normal and treat the mean and standard deviation of the sample as if they were the population values, find the probability that a random math student is able to legally consume recreational drugs, or rather, calculate P(age>21). Does this value fall in the confidence interval from problem 3?

5. Based on your answers from problem 1 (regarding the median and mode) and problem 4, do we have evidence that the original data is not normal? Considering the CLT and its requirements, does this mean that we are not justified in using the normal distribution for our confidence intervals?