Question

A manufacturer of large appliances must decide which of two​ machines, A and​ B, they want to purchase to perform a specific task in the production process. The goal is to buy the machine that has smaller mean time required to perform the task. The plant supervisor selects 15 machine operators at​ random, and each operator performs the task on each of the two machines. The production times are paired for each worker. A paired​ t-test is to be performed to determine if there is evidence that the population mean time using machine A is less than the population mean time using machine B. The summary statistics for the differences in the times required for the task in minutes​ (machine A​ - machine​ B) for the 15 randomly selected workers are given below.
n=15; xÌ… = -10.9 and s=20.3

What must be true about the population of differences in the times required for the task between machine A and machine B for conclusions from the paired t-test to be valid for the population of differences among all workers?

a. Because of the small sample size of differences in times required between machine A and machine B, the distribution of sample means of the differences cannot be normal.
b. Because there were a total of 30 obervations (15 times from machine A and 15 times from machine B), the distribution of sample means of the differences will be approximately normal by the Central Limit Theorem.
c. Because the sample size is "large" enough, the distribution of differences for all workers will be normal.
d. Because of the small sample size of differences in times required between machine A and machine B, the distribution of differences for all workers must be normal.

Answer 1

Answer:

d. Because of the small sample size of differences in times required between machine A and machine B, the distribution of differences for all workers must be normal.

Step-by-step explanation:

A paired t- test conclusion is said to be valid if one of the assumptions that must be satisfied is that: the distribution of the differences must be normal in most cases for which the sample size is small.

From the given information:

the sample size n = 15 ;which is far less than 30

Therefore;we require the distribution of differences in times required between machine A and machine B for all workers to be normal.

From the first option; it is incorrect because even if the sample size is small; the distribution of sample means of the differences will be normal but in the first option ; it is stated that the differences cannot be normal. That makes the first option to be incorrect.

From the second option; is not correct because the sample size (for differences) is 15 and therefore that is a minimal sample which makes the Central Limit Theorem to be invalid and not applicable here.

From the third option; we all know that the sample size is small and not large since it is lesser than 30.