Final answer:
To find the probability that the sample mean would be greater than 128.2 millimeters, we calculate the z-score and use a z-table or calculator to find the probability of a z-score less than that. The probability is approximately 0.9951, rounded to four decimal places.
Step-by-step explanation:
To find the mean and standard deviation for the sample, we first determine the standard deviation of the population using the variance given. The standard deviation is the square root of the variance, so in this case, it is √64, which equals 8 millimeters.
Next, we calculate the standard error of the mean by dividing the standard deviation of the population by the square root of the sample size. Since the sample size is 39, the standard error of the mean is 8 / √39, which is approximately 1.28 millimeters.
Finally, to find the probability that the sample mean would be greater than 128.2 millimeters, we calculate the z-score using the formula: (sample mean - population mean) / standard error of the mean.
Plugging in the values, we get a z-score of (128.2 - 132) / 1.28, which is approximately -2.656.
Using a z-table or a calculator, we can find that the probability of a z-score less than -2.656 is approximately 0.0049. Therefore, the probability that the sample mean would be greater than 128.2 millimeters is 1 - 0.0049, which is approximately 0.9951, rounded to four decimal places.