Question

Assume that the height of adult females in the United States is approximately normally distributed with a mean of 64 inches ando standard deviation of 2.75 inches. A sample of 13 such women is selected at random. Find the probability that the mean height of the sample is within 0.6 inches of the population mean height.

Answer 1

To find the probability that the mean height of the sample is within 0.6 inches of the population mean height, we need to calculate the z-scores for the lower and upper bounds of the interval and find the area under the standard normal curve between those z-scores.

Step-by-step explanation:

To find the probability that the mean height of the sample is within 0.6 inches of the population mean height, we need to calculate the z-scores for the lower and upper bounds of the interval and find the area under the standard normal curve between those z-scores.

First, we calculate the z-score for the lower bound:

z = (lower bound - population mean) / (population standard deviation / sqrt(sample size))

Substituting in the values:

z = (64 - 64) / (2.75 / sqrt(13)) = 0

The z-score for the lower bound is 0.

Next, we calculate the z-score for the upper bound:

z = (upper bound - population mean) / (population standard deviation / sqrt(sample size))

Substituting in the values:

z = (64 + 0.6 - 64) / (2.75 / sqrt(13)) = 0.4108

The z-score for the upper bound is 0.4108

Now, we look up the areas corresponding to these z-scores in the standard normal distribution table or use a calculator to find the cumulative probability. Subtracting the lower bound area from the upper bound area will give us the probability that the mean height of the sample is within 0.6 inches of the population mean height.