Question

The number of hours spent studying by students on a large campus in the week before the final exams follows a normal distribution with standard deviation of 8.4 hours. A random sample of these students is taken to estimate the population mean number of hours studying.

a. How large a sample is needed to ensure that the probability that the sample mean differs from the population mean by more than 2.0 hours is less than 0.05?

b. Without doing the calculations, state whether a larger or smaller sample size compared to the sample size in part (a) would be required to guarantee that the probability of the sample mean differing from the population mean by more than 2.0 hours is less than 0.10. Explain your answer.

c. Without doing the calculations, state whether a larger or smaller sample size compared to the sample size in part (a) would be required to guarantee that the probability of the sample mean differing from the population mean by more than 1.5 hours is less than 0.05. Explain your answer.

Answer 1

The central limit theorem states that the sample mean, follows approximately the normal distribution with mean μ and standard deviation √, where µ and are the mean and standard deviation of the population from where the sample was selected. If the population is normal, then the sample size doesn't matter, but if the population is non-normal, the Central limit theorem can be applied to the sampling distribution only if the sample size is large enough (n≥30)

i.e. ~N(µ,/√n)