Question

A simple random sample of size n is drawn from a population that is normally distributed. The sample​ mean, x overbar is found to be 112, and the sample standard​ deviation, s, is found to be 10.

​(a) Construct a 95% confidence interval about μ if the sample​ size, n, is 26.

​(b) Construct a 95​% confidence interval about μ if the sample​ size, n, is 17.

​(c) Construct an 80​% confidence interval about μ if the sample​ size, n, is 26.

​(d) Could we have computed the confidence intervals in parts​ (a)-(c) if the population had not been normally​ distributed?

Click the icon to view the table of areas under the​ t-distribution.

Answer 1

(a)

We need to construct the 95% confidence interval for the population mean μ. The following information is provided:

Sample Mean =	112
Sample Standard Deviation (s) =	10
Sample Size (n) =	26

The critical value for α=0.05 and df=n−1=25 degrees of freedom is . The corresponding confidence interval is computed as shown below:

Therefore, based on the data provided, the 95% confidence interval for the population mean is 107.961<μ<116.039, which indicates that we are 95% confident that the true population mean μ is contained by the interval (107.961,116.039).

(b)

The critical value for α=0.05 and df=n−1=16 degrees of freedom is . The corresponding confidence interval is computed as shown below:

(c)

We need to construct the 80% confidence interval for the population mean μ. The following information is provided:

Sample Mean =	112
Sample Standard Deviation (s) =	10
Sample Size (n) =	26

The critical value for α=0.2 and df=n−1=25 degrees of freedom is . The corresponding confidence interval is computed as shown below:

d)

For non-normal data, we can use bootstrap method in order to create confidence intervals.

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