Question

The thickness (in millimeters) of the coating applied to hard drives is one characteristic that determines the usefulness of the product. When no unusual circumstances are present, the thickness (x) has a normal distribution with a mean of 2 mm and a standard deviation of 0.04 mm. Suppose that the process will be monitored by selecting a random sample of 25 drives from each shift's production and determining x, the mean coating thickness for the sample.

Required:
Describe the sampling distribution of X (for a sample of size 16).

Answer 1

The sampling distribution of X (for a sample of size 16) is a normal distribution with a mean of 2 mm and a standard deviation of 0.01 mm.

Step-by-step explanation:

The sampling distribution of X (for a sample of size 16) can be described as a normal distribution with a mean equal to the population mean, which is 2 mm. The standard deviation of the sampling distribution is calculated by dividing the population standard deviation by the square root of the sample size. In this case, the population standard deviation is 0.04 mm and the sample size is 16. So, the standard deviation of the sampling distribution is 0.04 / √16 = 0.01 mm.

Answer 2

The sampling distribution of X, for a sample of size 16, will be approximately normal with mean 2 mm and standard deviation 0.01 mm.

Step-by-step explanation:

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean
`\mu` and standard deviation
`\sigma`, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
`\mu` and standard deviation
`s = (\sigma)/(√(n))`.

For the population:

Mean 2 mm, standard deviation 0.04 mm

Describe the sampling distribution of X (for a sample of size 16).

Sample of size 16 means that
`n = 16`

By the Central Limit Theorem, approximately normal with mean 2 and standard deviation
`s = (0.04)/(√(16)) = 0.01`