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 64 purchasers of this table saw is taken. The probability that the sample mean will be between 77 and 89 minutes is ________.
To solve this problem, we need to use the central limit theorem and calculate the z-scores for 77 minutes and 89 minutes.
First, calculate the z-score for 77 minutes:
z = (77 - 80) / (40 / sqrt(64))
z = -3 / 5
z = -0.6
Next, calculate the z-score for 89 minutes:
z = (89 - 80) / (40 / sqrt(64))
z = 9 / 5
z = 1.8
Now, we look up the corresponding probabilities in the standard normal distribution table. The probability that the sample mean will be between 77 and 89 minutes is the difference between the probabilities for z = -0.6 and z = 1.8.
From the standard normal distribution table, we find:
P(z < -0.6) = 0.2743
P(z < 1.8) = 0.9641
Therefore, the probability that the sample mean will be between 77 and 89 minutes is:
P(-0.6 < z < 1.8) = P(z < 1.8) - P(z < -0.6)
= 0.9641 - 0.2743
= 0.6898
So, the probability that the sample mean will be between 77 and 89 minutes is approximately 0.6898, or 68.98%.