Question

a manufacturer of power tools claims that the mean amount of time required to assemble their top-of-the-line table saw is 80 minutes with a standard deviation of 40 minutes. suppose a random sample of 50 of this table saw is taken. what is the probability that the sample mean will be between 77 and 89 minutes ? (use 4 decimals)

Answer 1

0.6832

Explanation:

To find the probability that the sample mean will be between 77 and 89 minutes, we can use the Central Limit Theorem and assume that the distribution of sample means follows a normal distribution.

The mean of the sample mean distribution is the same as the population mean, which is 80 minutes. The standard deviation of the sample mean distribution, also known as the standard error, is calculated by dividing the population standard deviation by the square root of the sample size.

Standard error = Population standard deviation / √sample size

Standard error = 40 / √50

Standard error ≈ 5.6569

To calculate the probability, we need to standardize the values of 77 and 89 using the z-score formula.

Z-score for 77: (77 - 80) / 5.6569 ≈ -0.5303

Z-score for 89: (89 - 80) / 5.6569 ≈ 1.5907

We can then use a standard normal distribution table or a calculator to find the probabilities associated with these z-scores.

Using the table or calculator, the probability that the sample mean will be between 77 and 89 minutes is approximately 0.6832.