Question

Condé Nast Traveler conducts an annual survey in which readers rate their favorite cruise ship. All ships are rated on a 100-point scale, with higher values indicating better service. A sample of 36 ships that carry fewer than 500 passengers resulted in an average rating of 85.33 , and a sample of 43 ships that carry 500 or more passengers provided an average rating of 81.3. Assume that the population standard deviation is 4.55 for ships that carry fewer than 500 passengers and 3.95 for ships that carry 500 or more passengers.

A.) What is the point estimate of the difference between the population mean rating for ships that carry fewer than 500 passengers and the population mean rating for ships that carry 500 or more passengers?

B.) At 95% confidence, what is the margin of error?

C.) What is a 95% confidence interval estimate of the difference between the population mean ratings for the two sizes of ships?

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Answer 1

(A) Point estimate of the difference between the population mean rating for ships that carry fewer than 500 passengers and the population mean rating for ships that carry 500 or more passengers is the sample mean difference between these two ships.

sample mean for ships fewer than 500 passengers = 85.33

sample mean for ships 500 or more passengers = 81.3

So, point estimate = sample mean difference = 85.33-81.3 = 4.03

(B) Formula for margin of error is given as

where z = 1.96 for 95% confidence interval (using z table)

n1(sample size for ship with less than 500 passenger) =36 and n2(sample size for ship with 500 or more passenger) = 43

is population standard for first sample = 4.55 and is population standard for second sample = 3.95

setting the values in the above formula, we get

(rounded to decimals) or 1.898 (rounded to 3 decimals)

(C) Formula for 95% confidence interval when we have the estimate for sample mean and margin of error is given as

where and ME(margin of error) = 1.90

setting the values, we get

Therefore, required 95% confidence interval for the difference between population means is (2.13, 5.93)