Question

Suppose x has a distribution with μ=16 and σ=15. (a) If a random sample of size n=36 is drawn, find the following. Find μx∗−​ μx−​= Find σxˉ∗​ σxˉ−​= Find P(16≤xˉ≤18). (Round your answer to four decimal places.) P(16≤xˉ≤18)= (b) If a random sample of size n=64 is drawn, find the following. Find μx−−​ μx−​= Find σxˉ​ σx−​= Find P(16≤xˉ≤18). (Round your answer to four decimal places.) P(16≤x≤18)= (c) Why should you expect the probability of part (b) to be higher than that of part (a)? (Hint: Consider the standard deviations in parts (a) and (b).) The standard deviation of part (b) is part (a) because of the sample size. Therefore, the distribution about μx−​is

Answer 1

This answer explains how to find the difference between the sample mean and population mean for different sample sizes, and how to calculate the probability of the sample mean falling within a range.

Step-by-step explanation:

In part (a), the question asks for the difference between the sample mean and the population mean, which is denoted as μx* - μx-. Since the population mean (μ) is given as 16 and the sample size (n) is 36, the sampling distribution of the sample mean (μx*) will have a mean equal to the population mean (16) and a standard deviation equal to the population standard deviation divided by the square root of the sample size (15/√36).

In part (b), the question asks for the same values as part (a), but with a different sample size (n=64). The difference between the sample mean and the population mean (μx∗−​ μx∗−​) will have a mean equal to the population mean (16) and a standard deviation equal to the population standard deviation divided by the square root of the sample size (15/√64).

In part (c), the probability of the sample mean falling between 16 and 18 is calculated. This can be calculated using the z-score formula and the standard deviation of the sample means calculated in part (b). The z-score is equal to (x - μx*) / (σx*) where x is the value of the sample mean (16 or 18). The probability P(16≤xˉ≤18) can be calculated using the standard normal distribution table or a calculator.