Question

The inside diameter of a randomly selected piston ring is a random variable with mean value 12 cm and standard deviation .04 cm.

a. If Xˉ is the sample mean diameter for a random sample of n=16 rings, where is the sampling distribution of Xˉcentered, and what is the standard deviation of the Xˉdistribution?
b. Answer the questions posed in part (a) for a sample size of n=64 rings.
c. For which of the two random samples, the one of part (a) or the one of part (b), is Xˉmore likely to be within .01 cm of 12 cm? Explain your reasoning.

Answer 1

For a sample size of 16, the sampling distribution of Ä” is centered at the population mean with a standard error of 0.01 cm. For a sample size of 64, the standard error decreases to 0.005 cm, thus making Ä” more likely to be within 0.01 cm of the population mean of 12 cm.

Step-by-step explanation:

The question deals with the concept of sampling distributions and the properties of these distributions when sampling from a normally distributed population. We are given a population with a mean (μ) of 12 cm and a standard deviation (σ) of 0.04 cm.

Part a

For a sample size of n = 16, the sampling distribution of the sample mean Ä” will be centered at the population mean which is 12 cm, as the sample mean is an unbiased estimator of the population mean. The standard deviation of the sample mean, often called the standard error (SE), is calculated using the formula SE = σ/√n. So for this part, SE = 0.04/√16 = 0.01 cm.

Part b

For a sample size of n = 64, again the sampling distribution of Ä” is centered at 12 cm. The standard deviation of this distribution is SE = 0.04/√64 = 0.005 cm.

Part c

For the sample of size n = 64, Ä” is more likely to be within .01 cm of 12 cm, because the standard deviation of the sample mean for n = 64 is smaller (0.005 cm) than for n = 16 (0.01 cm), indicating that the sample mean for larger samples will vary less from the population mean and hence more likely to be closer to the true population mean.