Question

A random sample of 64 items is selected from a population of 400 items. The sample mean is 200. The population standard deviation is 48. From this data, a 95% confidence interval to estimate the population mean can be computed as

Answer 1

Answer:
`(188.24,\ 211.76 )`.

Explanation:

Given : Sample size : n= 64 , the sample is a large sample (n>30), so we can apply z-test.

Sample mean =
`\overline{x}=200`

Standard deviation :
`\sigma=48`

Level of confidence:
`1-\alpha=0.95`

`\Rightarrow\ \alpha=0.05`

Then, critical z-value =
`z_(\alpha/2)=1.96`

The confidence interval to estimate the population mean is given by :_

`\overline{x}\ \pm\ z_(\alpha/2)(\sigma)/(√(n))`

`=200\ \pm\ (1.96)(48)/(√(64))\\\\=200\pm11.76=(200-11.76, 200+11.76)=(188.24,\ 211.76 )`

Hence, the 95% confidence interval to estimate the population mean can be computed as
`(188.24,\ 211.76 )`.