Question

A large box contains 10,000 ball bearings. A random sample of 120 is chosen. The sample mean diameter is 10 mm, and the standard deviation is 0.24 mm. True or false: a. A 95% confidence interval for the mean diameter of the 120 bearings in the sample is 10 ± (1.96)(0.24)/√ 120. b. A 95% confidence interval for the mean diameter of the 10,000 bearings in the box is 10 ± (1.96)(0.24)/√120. c. A 95% confidence interval for the mean diameter of the 10,000 bearings in the box is 10 ± (1.96)(0.24)/√10,000.

Answer 1

Option a) A 95% confidence interval for the mean diameter of the 120 bearings in the sample is
`10 \pm 1.96\displaystyle(0.24)/(√(120))`.

Explanation:

We are given the following information in the question:

Sample size, n = 120

Sample mean = 10 mm

Standard Deviation = 0.24 mm

Formula:

`\mu \pm z_(critical)(\sigma)/(√(n))`

`z_(critical)\text{ at}~\alpha_(0.05) = 1.96`

`10 \pm 1.96\displaystyle(0.24)/(√(120))`

Hence, the correct interpretation for the confidence interval is given by option a).

A 95% confidence interval for the mean diameter of the 120 bearings in the sample is
`10 \pm 1.96\displaystyle(0.24)/(√(120))`.

We have to consider the factor of sampling of 120 ball bearings from a population of 10,000 ball bearings.