Question

In the EAI sampling problem, the population mean is $71,800 and the population standard deviation is $5000. For n=20, there is a 0.4108 probability of obtaining a sample mean within ±$600 of the population mean. Use z-table. a. What is the probability that x ˉ is within $600 of the population mean if a sample of size 40 is used (to 4 decimals)? 7. Answer part (a) for a sample of size 80 (to 4 decimals).

Answer 1

To find the probability of the sample mean being within $600 of the population mean for samples of size 40 and 80, calculate the standard error of the mean for each sample size and use the Central Limit Theorem and z-table to determine the probabilities.

Step-by-step explanation:

The probability that the sample mean, x ˉ, is within $600 of the population mean can be found by using the Central Limit Theorem which tells us about the sampling distribution of the sample mean.

For a population with a mean (μ) of $71,800 and a standard deviation (σ) of $5000, and a sample size (n) of 40, we first need to calculate the standard error of the mean using σ/√n. Then, we use the z-table to find the probability that lies within ±$600 of the population mean. Similarly, this process is repeated for a sample size of 80 to find the corresponding probability.

For n = 40:

Calculate the standard error (SE) of the mean: SE = σ/√n = $5000/√40

Calculate the z-scores for ±$600 from the mean: Z = ±$600/SE

Look up these z-scores on the z-table to find the probabilities.

Subtract the lower probability from the higher probability to find the total probability.

Repeat the steps with n = 80 to find the corresponding probability.