Question

For a random sample of 50 measurements on the breaking strength of cotton threads, the mean breaking strength was found to be 210 grams and the standard deviation 18 grams. Obtain a confidence interval for the true mean breaking strength of cotton threads of this type, with confidence coefficient 0.99.

Answer 1

The 99% confidence interval is given by (203.178;216.822)

Explanation:

1) Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

`\bar X=210` represent the sample mean

`\mu` population mean (variable of interest)

s=18 represent the sample standard deviation

n=50 represent the sample size

2) Confidence interval

The confidence interval for the mean is given by the following formula:

`\bar X \pm t_(\alpha/2)(s)/(√(n))` (1)

In order to calculate the critical value
`t_(\alpha/2)` we need to find first the degrees of freedom, given by:

`df=n-1=50-1=49`

Since the Confidence is 0.99 or 99%, the value of
`\alpha=0.01` and
`\alpha/2 =0.005`, and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.005,49)".And we see that
`t_(\alpha/2)=2.68`

Now we have everything in order to replace into formula (1):

`210-2.68(18)/(√(50))=203.178`

`210+2.68(18)/(√(50))=216.822`

So on this case the 99% confidence interval would be given by (203.178;216.822)