Question

4- (14%) A random sample of 10 yields a mean and standard deviation, respectively, of 80 and 7.9. Does this sample confirm that the population mean is greater than 78 with 95% confidence level?

Answer 1

Given : $n=10,\bar{X}=80,s=7.9,\mu_0=78,\alpha=0.05$

The null and alternative hypothesis is ,

$H_0:\mu=78;H_1:\mu>78$

Now , df=degrees of freedom=n-1=10-1=9

The critical value is ,

$t_{df,\alpha}=t_{9,0.05}=1.833$ ; From t-table

The decision rule is ,

Reject Ho , if t-stat>1.833 , otherwise fail to reject Ho.

The test statistic is ,

$t_{stat}=\frac{\bar{X}-\mu_0}{s/\sqrt{n}}=\frac{80-78}{7.9/\sqrt{10}}=0.8006$

Decision : Here , t-stat=0.8006<1.833

Therefore , fail to reject Ho.

Conclusion : The sample does not confirm that the population mean is greater than 78