Question

Construct the confidence interval for the population mean mu. c = 0.90​, x = 16.9​, s = 9.0​, and n = 45. A 90​% confidence interval for mu is:______.

Answer 1

The 90% confidence interval for population mean is
`14.7 < \mu < 19.1`

Explanation:

From the question we are told that

The sample mean is
`\= x = 16.9`

The confidence level is
`C = 0.90`

The sample size is
`n = 45`

The standard deviation

Now given that the confidence level is 0.90 the level of significance is mathematically evaluated as

`\alpha = 1-0.90`

`\alpha = 0.10`

Next we obtain the critical value of
`(\alpha )/(2)` from the standardized normal distribution table. The values is
`Z_{(\alpha )/(2) } = 1.645`

The reason we are obtaining critical values for
`(\alpha )/(2)` instead of that of
`\alpha` is because
`\alpha` represents the area under the normal curve where the confidence level 1 -
`\alpha` (90%) did not cover which include both the left and right tail while
`(\alpha )/(2)` is just considering the area of one tail which is what we required calculate the margin of error

Generally the margin of error is mathematically evaluated as

`MOE = Z_{(\alpha )/(2) } * (\sigma )/(√(n) )`

substituting values

`MOE = 1.645* ( 9 )/(√(45) )`

`MOE = 2.207`

The 90% confidence level interval is mathematically represented as

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

substituting values

`16.9 - 2.207 < \mu < 16.9 + 2.207`

`16.9 - 2.207 < \mu < 16.9 + 2.207`

`14.7 < \mu < 19.1`