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:   $(188.24,\ 211.76 )$.

Step-by-step 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 :_

$\overline{x}\ \pm\ z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}$

$=200\ \pm\ (1.96)\dfrac{48}{\sqrt{64}}\\\\=200\pm11.76=(200-11.76, 200+11.76)=(188.24,\ 211.76 )$

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