Question

In the EAI sampling problem, the population mean is $71,900 and the population standard deviation is $5000. For n = 30, there is a 0.4176 probability of obtaining a sample mean within +$500 of the population mean. Use z-table a. What is the probability that I is within $500 of the population mean if a sample of size 60 is used (to 4 decimals)? b. Answer part (a) for a sample of size 120 (to 4 decimals).

Answer 1

To find the probability of a sample mean being within $500 of the population mean for different sample sizes (60 and 120), the standard error of the mean needs to be calculated, which then helps determine the z-score, and ultimately, the probability can be found using a z-table.

Step-by-step explanation:

In the given EAI sampling problem, we’re tasked to find the probability of obtaining a sample mean within ±$500 of the population mean using different sample sizes. Knowing that the population mean (μ) is $71,900 and the population standard deviation (σ) is $5000, we use the formula for the standard error of the mean (SEM): SEM = σ / √n.

For part a, when n = 60:

1. Calculate SEM: SEM = 5000 / √60.
2. Find the z-score corresponding to a sample mean $500 away from μ: z = $500 / SEM.
3. Use the z-table to find the probability corresponding to that z-score.

For part b, when n = 120, repeat steps 1-3 with the new sample size.

As the sample size increases (n = 60 and n = 120), according to the Central Limit Theorem, the sampling distribution of the sample mean becomes more closely approximated to a normal distribution, making it more likely for a sample mean to fall within $500 of the population mean. The probability can be expected to increase with larger sample sizes.