Question

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

$0.757$	$0.723$	$0.754$	$0.737$	$0.757$	$0.741$	$0.722$	$0.741$	$0.743$	$0.742$
$0.740$	$0.758$	$0.724$	$0.739$	$0.736$	$0.735$	$0.760$	$0.750$	$0.759$	$0.754$
$0.744$	$0.758$	$0.765$	$0.756$	$0.738$	$0.742$	$0.758$	$0.757$	$0.724$	$0.757$
$0.744$	$0.738$	$0.763$	$0.756$	$0.760$	$0.768$	$0.761$	$0.742$	$0.734$	$0.754$
$0.758$	$0.735$	$0.740$	$0.743$	$0.737$	$0.737$	$0.725$	$0.761$	$0.758$	$0.756$

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?

Short answer

From the given data,

a) Mean diameter =$0.74682$

The standard deviation for the sample = $0.012322$

b) the probability that $50$randomly selected screws will be within stated tolerance level is approximately$22.27\%$

Step-by-step solution

Given Information ( part a)

Find the mean diameter and standard deviation for the sample

Use supplied data (Part a)

Use supplied data of screw right company data to solve subparts.

To calculate mean diameter and standard deviation for the sample, use MS-Excel software, the process is given as below:

Firstly write the provided data into an MS-Excel spreadsheet, the screenshot is conveyed as below.

calculate the mean diameter and standard deviation (Part a)

To calculate the mean diameter and standard deviation for the sample, use the following formula as shared below in the screenshot.

Step 4:acquired mean diameter and standard deviation (Part a)

The screenshot of the acquired mean diameter and standard deviation for the sample is conveyed below.

Given Information (part b)

Find the probability that $50$randomly selected screws will be within the stated tolerance levels. Is the company’s diameter claim plausible?

calculate the probability  (part b)

To calculate the probability that $50$randomly selected screws will be within stated tolerance levels, use the Ti-$83$calculator, for this, click on $2$^nI, then DISTR, and then scroll down to the normal CDF option and enter the supplied details. After this, click on ENTER button of the calculator to have the expected result. The screenshot is conveyed as below:

Explanation (part b)

Thus, the probability that $50$randomly selected screws will be within stated tolerance levels is around $0.2227$. The probability that $50$ randomly selected screws will be within stated tolerance levels is approximately $22.27$ %. So the company's diameter claims are not plausible.