Question

You are given the sample mean and the population standard deviation. Use this information to construct the​ 90% and​ 95% confidence intervals for the population mean. Interpret the results and compare the widths of the confidence intervals. From a random sample of business​ days, the mean closing price of a certain stock was ​$. Assume the population standard deviation is ​$. The​ 90% confidence interval is ​( nothing​, nothing​). ​(Round to two decimal places as​ needed.) The​ 95% confidence interval is ​( nothing​, nothing​). ​(Round to two decimal places as​ needed.) Which interval is​ wider? Choose the correct answer below

Answer 1

Complete Question

The complete question is shown on the first uploaded image

Answer:

The 90% confidence interval is
`[108.165 ,112.895]`

The 95% confidence interval is
`[107.7123 ,113.3477]`

The correct option is D

Explanation:

From the question we are told that

The sample size is n = 48

The sample mean is
`\= x = \$ 110.53`

The standard deviation is
`\sigma = \$ 9.96`

Considering first question

Given that the confidence level is 90% then the level of significance is mathematically represented as

`\alpha = (100 - 90)\%`

`\alpha = 0.10`

The critical value of
`(\alpha )/(2)` from the normal distribution table is

`Z_{(\alpha )/(2) } = 1.645`

Generally the margin of error is mathematically represented as

`E = ZZ_{ (x)/(y) } * (\sigma)/( √(n) )`

`E = 1.645 * (9.96)/( √( 48) )`

`E = 2.365`

The 90% confidence interval is

`\= x - E < \mu < \= x + E`

=>
`110.53 - 2.365 < \mu < 110.53 + 2.365`

=>
`108.165 < \mu < 112.895`

Considering second question

Given that the confidence level is 95% then the level of significance is mathematically represented as

`\alpha = (100 - 95)\%`

`\alpha = 0.05`

The critical value of
`(\alpha )/(2)` from the normal distribution table is

`Z_{(\alpha )/(2) } = 1.96`

Generally the margin of error is mathematically represented as

`E = Z_{ (x)/(y) } * (\sigma)/( √(n) )`

`E = 1.96 * (9.96)/( √( 48) )`

`E = 2.8177`

The 95% confidence interval is

`\= x - E < \mu < \= x + E`

=>
`110.53 - 2.8177 < \mu < 110.53 + 2.8177`

=>
`107.7123 < \mu < 113.3477`