Final answer:
The question involves using the central limit theorem and standard error to calculate the likelihood that a sample mean of steel bolts will differ from the population mean by more than 3 mm.
Step-by-step explanation:
The student is asking about the probability that the sample mean of diameters of steel bolts will differ from the population mean by more than 3 mm, given that the population mean diameter is 132 mm with a variance of 64. To answer this, we can use the central limit theorem, which states that the sampling distribution of the sample mean will be normally distributed if the sample size is large enough. In this case, with a sample size of 39, the sample mean would approximately follow a normal distribution.
Firstly, we calculate the standard error of the mean (SEM), which is the standard deviation of the sampling distribution of the sample mean:
- SEM = √SD/n = √64/39.
- Then we find the z-score for a deviation of 3 mm from the mean:
- Z = (X - mean) / SEM = (3 mm) / SEM.
- Lastly, we use the z-score to find the probability that the sample mean would differ by more than 3 mm from the population mean using the standard normal distribution table.
The probability that the sample mean differs from the population mean by greater than 3 mm is equal to the sum of the probabilities of the sample mean being 3 mm less than or 3 mm greater than the population mean (two-tailed test).