Question

Suppose x has a distribution with μ=17 and σ=15. (a) If a random sample of size n=41 is drawn, find μ x ˉ ′ ​ σ x ˉ ​ and P(17≤ x ˉ ≤19). (Round σ x ˉ ​ to two decimal places and the probability to four decimal places.) μ x ˉ ​ = σ x ˉ ​ = P(17≤ x ˉ ≤19)= ​ (b) If a random sample of size n=55 is drawn, find μ x ′ ​ ,σ x ˉ ​ and P(17≤ x ˉ ≤19 ). (Round σ x ˉ ​ to two decimal places and the probability to four decimal places.) μ x ˉ ​ = σ x ˉ ​ = P(17≤ x ˉ ≤19)= ​ (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. Therefore, the distribution about μ x ˉ ​ is

Answer 1

The question is focused on calculating the sample means and standard deviations for specific sample sizes and finding the probability that the sample mean falls within a particular range when drawn from a normally distributed population. It also highlights the effect of sample size on the standard deviation of the sampling distribution.

Step-by-step explanation:

The question pertains to the calculation of sample means and standard deviations for a population with a known mean (μ) and standard deviation (σ) and then finding probabilities for a given range using these sample statistics.

Specifically, it addresses how these probabilities change with increased sample sizes, in accordance with the Central Limit Theorem.

Part A

For a sample size of n = 41:

• μx = μ = 17
• σx= σ / √n = 15 / √41 = 15 / 6.403 = 2.34 (rounded to two decimal places)
• P(17≤ x ≤19) - This probability would require the use of the standard normal distribution and Z-scores.

Part B

For a sample size of n = 55:

• μx = μ = 17
• σx = σ / √n = 15 / √55 = 15 / 7.416 = 2.02 (rounded to two decimal places)
• P(17≤ x≤19) would again need the standard normal distribution for calculation.

Part C

The probability in part (b) is expected to be higher than in part (a) because larger samples tend to reduce the standard deviation of the sampling distribution (σx), hence the distribution of x narrows and becomes more concentrated around the mean (μx).