Question

The manager of a computer retails store is concerned that his suppliers have been giving him laptop computers with lower than average quality. His research shows that replacement times for the model laptop of concern are normally distributed with a mean of 3.9 years and a standard deviation of 0.5 years. He then randomly selects records on 39 laptops sold in the past and finds that the mean replacement time is 3.8 years.

Assuming that the laptop replacement times have a mean of 3.9 years and a standard deviation of 0.5 years, find the probability that 39 randomly selected laptops will have a mean replacement time of 3.8 years or less.

P(M < 3.8 years) =

Answer 1

To find the probability that 39 randomly selected laptops will have a mean replacement time of 3.8 years or less, we can use the Central Limit Theorem. The probability is approximately 0.1020, or 10.20%.

Step-by-step explanation:

To find the probability that 39 randomly selected laptops will have a mean replacement time of 3.8 years or less, we can use the Central Limit Theorem. According to the Central Limit Theorem, as the sample size increases, the sampling distribution of the sample mean approaches a normal distribution even if the population distribution is not normal.

Given that the population mean replacement time is 3.9 years and the standard deviation is 0.5 years, we can calculate the standard error of the mean using the formula:

Standard Error = Standard Deviation / √Sample Size

Standard Error = 0.5 / √39 ≈ 0.079

Next, we can use a z-score table or a calculator to find the z-score corresponding to the sample mean of 3.8 years. The formula for z-score is:

z = (X - Mean) / Standard Error

z = (3.8 - 3.9) / 0.079 ≈ -1.2658

Using the z-score, we can find the probability that the sample mean is 3.8 years or less by looking up the z-score in the z-table. The area under the normal curve to the left of -1.2658 is approximately 0.1020.

Therefore, the probability that 39 randomly selected laptops will have a mean replacement time of 3.8 years or less is approximately 0.1020, or 10.20%.