Question

The inside diameter of a randomly selected piston ring is a random variable with mean value 11 cm and standard deviation 0.06 cm.

(a) If

X

is the sample mean diameter for a random sample of n = 16 rings, where is the sampling distribution of

X

centered and what is the standard deviation of the

X

distribution? (Enter your standard deviation to five decimal places.)

center	cm
standard deviation	cm

(b) Answer the questions posed in part (a) for a sample size of n = 64 rings. (Enter your standard deviation to five decimal places.)

center	cm
standard deviation	cm

(c) For which of the two random samples, the one of part (a) or the one of part (b), is

X

more likely to be within 0.01 cm of 11 cm? Explain your reasoning.

X

is more likely to be within 0.01 cm of 11 cm in sample (b) because of the decreased variability with a larger sample size.

X

is more likely to be within 0.01 cm of 11 cm in sample (a) because of the increased variability with a smaller sample size.    

X

is more likely to be within 0.01 cm of 11 cm in sample (a) because of the decreased variability with a smaller sample size.

X

is more likely to be within 0.01 cm of 11 cm in sample (b) because of the increased variability with a larger sample size.

Answer 1

using the central Limit Theorem we have that

the mean of the samples will be the same mean of the poplation

and the standard deviation of the sample will be the SD of population divided by srqt (n)

x~N(=11,^2=0.06^2)

a)

center == 11 cm

SD = /sqrt(n)=0.06 / srqt(16) = 0.015 cm

b)

center == 11 cm

SD = 0.06 / srqt(64) = 0.0075

c)

is more likely to be within 0.01 cm of 11 cm in sample (b) because of the decreased variability with a larger sample size.