Question

The mean height of 50 male students who showed above-average participation in college athletics was 173.3 cm with s.d. 6.4 cm, while 50 male students who showed no interest in such participation had a mean of 171.5 cm and s.d. 7.1 cm. Test the hypothesis that male students who participate in college athletics are taller than other students at 5% level of significance using sample s.d. as an estimate of population s.d. By how much the sample size of each group should be increased in order that the observed difference of 1.8 cm in the mean heights is significant at the same level of significance.

Answer 1

Given :

n1=50, $\overline{x_{1}}=173.3$ , s1=6.4

n2=50 $\overline{x_{2}}=171.5$ ,S2=7.1

Step 1: State the hypothesis

H0: $\mu _{1}=\mu _{2}$

H1: $\mu _{1}> \mu _{2}$

Step 2:

Name of the test:

since population sd known, use 2 sample t test

Step 3 : Test statstic:

$Z= \frac{\overline{x1}-\overline{x2}}{\sqrt{\left ( s1^{2}/n1-(s2^{2}/n2) \right )}}$

$Z= \frac{173.3-171.5}{\sqrt{\left ( 6.4^{2}/50-(7.1^{2}/50) \right )}}$

Z=1.332

Step 4: Since it is observed that t = 1.332 , CV = 1.661

t=1.332≤CV ​=1.661, and p value = 0.0931 >0.05,

it is then concluded that the null hypothesis is not rejected.

Step 5: It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population $\mu _{1}> \mu _{2}$ ​, at the 0.05 significance level.

That is, p-values tend to become smaller as sample size increases, unless H0 is true.

Hence when sample size of each data is 80, p values becomes smaller and hence result is significant.

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