A researcher selects a sample of nine participants from a population with a mean of 8 and a standard deviation of 3.About 14% of the sample means in this sampling distribution should be between a sample mean of
A) 6 and 8.
B) 5 and 6.
C) 9 and 10.
D) 10 and 11.

Question

Quizplus · Accepted Answer

According to the Central Limit Theorem, the sampling distribution of the means will be approximately normal with a mean of 8 and a standard error of σ/√n = 3/√9 = 1. Therefore, we can use the z-score formula to find the proportion of sample means between two values.

For choice C: (10-8)/1 = 2, which corresponds to a probability of 0.4772. To find the probability between 9 and 10, we subtract the probability of less than 9 from the probability of less than 10: 0.4772 - 0.1587 = 0.3185 or about 31.85%. Therefore, about 14% of the sample means will fall between 9 and 10, which satisfies the condition.

For other choices, we can use the same method to find the probabilities:
A) (6-8)/1 = -2, P(z < -2) = 0.0228
B) (5-8)/1 = -3, P(z < -3) = 0.0013
D) (11-8)/1 = 3, P(z < 3) = 0.9987

As we can see, only choice C satisfies the condition of having about 14% of sample means between the given range.

A Researcher Selects a Sample of Nine Participants from a Population