Question

A manufacturer knows that their items have a normally distributed length, with a mean of 16.4 inches, and standard deviation of 5.4 inches. If 9 items are chosen at random, what is the probability that their mean length is less than 13.4 inches?

Answer 1

To find the probability that the mean length is less than 13.4 inches, we use the standard deviation of the population mean and perform a Z-score calculation. The probability is approximately 4.62%.

Step-by-step explanation:

To find the probability that the mean length is less than 13.4 inches, we need to use the standard deviation of the population mean, which is calculated by dividing the standard deviation of the population by the square root of the sample size. In this case, the standard deviation of the population mean is calculated as 5.4 / sqrt(9) = 1.8 inches.

We can then standardize the sample mean by subtracting the population mean and dividing by the standard deviation of the population mean: Z = (13.4 - 16.4) / 1.8 = -1.67.

Using a standard normal distribution table, we can find the probability that the Z-score is less than -1.67, which is approximately 0.0462, or 4.62%.