Use technology to solve the following problem: The mean annual income for people in a certain city (in thousands of dollars) is 40, with a standard deviation of 38. A pollster draws a sample of 35 People to interview. Find the 50th percentile of the sample mean.
A) 37.1 thousand dollars
B) 43.1 thousand dollars
C) 38.2 thousand dollars
D) 40.0 thousand dollars

Question

Quizplus · Accepted Answer

We can use the Central Limit Theorem to solve this problem. The sample size is n=35, which is large enough to apply the Central Limit Theorem. The mean of the sample means (also known as the sampling distribution) is equal to the population mean, which is 40. The standard deviation of the sampling distribution (also known as the standard error) is equal to the population standard deviation divided by the square root of the sample size:

standard error = 38 / sqrt(35) ≈ 6.42

To find the 50th percentile of the sample mean, we need to find the z-score that corresponds to the 50th percentile, and then use that z-score to find the raw score (the sample mean). We can use the standard normal distribution table or a calculator to find the z-score:

z-score = invNorm(0.5) ≈ 0

The 50th percentile corresponds to a z-score of 0, which means the sample mean is at the population mean:

sample mean = population mean + (z-score * standard error)
sample mean = 40 + (0 * 6.42)
sample mean = 40

Therefore, the 50th percentile of the sample mean is 40 thousand dollars. Choice D is the correct answer.

Use Technology to Solve the Following Problem: the Mean Annual