Question

Suppose the true rate at which MSU students self-identify as 'in-state' students is π = 0.7 π=0.7. The true rate at which Harvard students self-identify as 'in-state' students is only π = 0.01 π=0.01. For which group of students would you need a larger sample to use the normal density curve as an approximation of π ^ π ^ 's sampling distribution? Explain in 1-2 sentences.

Answer 1

A larger sample size is needed for Harvard students to use the normal approximation of the sampling distribution because np and n(1-p) must both be greater than 5, which is less likely to occur with a low proportion (π = 0.01).

Step-by-step explanation:

The group of students for which you would need a larger sample to use the normal density curve as an approximation of π's sampling distribution is the Harvard students. This is because the central limit theorem suggests that for the normal approximation to be reasonable, np and n(1-p) should both be greater than 5. With π = 0.7 for MSU students, even a small sample size would typically satisfy this condition since both np and n(1-p) would be greater than 5. However, with π = 0.01 for Harvard students, a much larger sample size is necessary to ensure that both np and n(1-p) are greater than 5, due to the low rate of 'in-state' identification.