Question

an online travel agent that specializes in cruises wanted to compare the cost of cruising to different destinations on various cruise lines. the following random sample data show the average cost of a seven - day cruise in a standard room to two locations. assume the population variances for the cruise fares for these destinations are equal. complete parts a through c
location 1 location 2
sample mean $982 $920
sample standard deviation $132 $127
sample size 8 11
b. what conclusions can be made about the difference in rates between these destinations?
with 90% confidence, it can be said that the difference in rates between these destinations falls within the endpoints of the confidence interval. the fact that this interval includes zero provides evidence that there is not a significant difference between these two rates.
c. what assumptions need to be made in order to perform this procedure?
○ a. if both sample sizes are not each at least 30, then one must assume that the underlying populations are normally distributed to perform this procedure. also, one must assume that the samples are independent.
○ b. since the sample sizes are large enough, no assumptions are needed.
○ c. although the sample size in the second sample is large enough, it must be assumed that the distribution for the first sample is normally distributed.
○ d. although the sample size in the first sample is large enough, it must be assumed that the distribution for the second sample is normally distributed.

Explanation:

When performing a two - sample t - test (which is likely the procedure here, given the context of comparing two means), if the sample sizes (\(n_1 = 8\) and \(n_2=11\)) are not large (a common rule of thumb is \(n\geq30\) for the Central Limit Theorem to make the sampling distribution of the sample mean approximately normal without assuming population normality), we need to assume that the underlying populations are normally distributed. Also, for a two - sample t - test, the samples should be independent.

Answer:

A. If both sample sizes are not each at least 30, then one must assume that the underlying populations are normally distributed to perform this procedure. Also, one must assume that the samples are independent.