Using the Central Limit Theorem · Exercise 43

For each problem, wherever possible, provide graphs and use a calculator.

The Screw Right Company claims their $\frac{3}{4}$inch screws are within ±0.23 of the claimed mean diameter of 0.750 inches with a standard deviation of 0.115 inches. The following data were recorded.

$$\begin{array}{cccccccccc}\text{0.757}& \text{0.723}& \text{0.754}& \text{0.737}& \text{0.757}& \text{0.741}& \text{0.722}& \text{0.741}& \text{0.743}& \text{0.742}\\ \text{0.740}& \text{0.758}& \text{0.724}& \text{0.739}& \text{0.736}& \text{0.735}& \text{0.760}& \text{0.750}& \text{0.759}& \text{0.754}\\ \text{0.744}& \text{0.758}& \text{0.765}& \text{0.756}& \text{0.738}& \text{0.742}& \text{0.758}& \text{0.757}& \text{0.724}& \text{0.757}\\ \text{0.744}& \text{0.738}& \text{0.763}& \text{0.756}& \text{0.760}& \text{0.768}& \text{0.761}& \text{0.742}& \text{0.734}& \text{0.754}\\ \text{0.758}& \text{0.735}& \text{0.740}& \text{0.743}& \text{0.737}& \text{0.737}& \text{0.725}& \text{0.761}& \text{0.758}& \text{0.756}\end{array}$$
The screws were randomly selected from the local home repair store.

(a) Find the mean diameter and standard deviation for the sample.
(b) Find the probability that 50 randomly selected screws will be within the stated tolerance levels. Is the company’s diameter claim plausible?

Its source prints no answer for this problem, so there's nothing to check against.

OpenStax, Statistics · §7.3 Using the Central Limit Theorem · Exercise 43