. The average family size in the Philippines was reported as 4.25. A random of families in a particular street resulted in the following family sizes: 5, 6, 7, 4, 3, 2, and 9. At 𝛼=0.10, does the average family size more than the national average?

$n=7\\\bar x={\sum x\over n}={36\over7}=5.14285714\\ s^2={\sum x^2-{(\sum(x))^2\over n}\over n-1}={220-185.142857\over6}=5.8095\\s=\sqrt{s^2}=\sqrt{5.8095}=2.4103$

The hypotheses are,

$H_0:\mu=4.25\\vs\\H_1:\mu\gt4.25$

The test statistic is,

$t={\bar x-\mu\over {s\over\sqrt{n}}}={5.14-4.25\over{2.4103\over\sqrt{7}}}={0.8929\over0.911}=0.98$

The critical value is,

$t_{\alpha,n-1}=t_{0.1,6}=1.439756$

The null hypothesis is rejected if, $t\gt t_{0.1,6}$

Since $t=0.98\lt t_{0.1,6}=1.439756$, the null hypothesis is not rejected. Therefore, we can conclude that there is no sufficient evidence to show that the average family size is more than the national average at 1% level of significance.