Final answer:
To find the probability that the mean height for the sample is greater than 65 inches, we need to calculate the z-score for a sample mean of 65 inches. The area to the right of the z-score of 2.8884 is approximately 0.0019, or 0.19%.
Step-by-step explanation:
To find the probability that the mean height for the sample is greater than 65 inches, we need to calculate the z-score for a sample mean of 65 inches.
First, let's calculate the standard error of the sample mean using the formula:
Standard Error = Standard Deviation / Square Root of Sample Size
Given that the mean height of women in the country is 64.4 inches, we can assume that the standard deviation for the population is unknown. Therefore, we can use the standard deviation of the sample as an estimate.
Using the formula, the standard error is:
Standard Error = 1.8 / √75 ≈ 0.2073
Next, we need to calculate the z-score using the formula:
Z-score = (Sample Mean - Population Mean) / Standard Error
Using the given values, the z-score is:
Z-score = (65 - 64.4) / 0.2073 ≈ 2.8884
Finally, we can find the probability that the mean height for the sample is greater than 65 inches by looking up the z-score in the standard normal distribution table. The area to the right of the z-score of 2.8884 is approximately 0.0019, or 0.19%.