Question

Walking to school Refer to Exercise 36.

a. Explain why the sample result gives some evidence for the alternative hypothesis.

b. Calculate the standardized test statistic and $P$-value.

c. What conclusion would you make?

Short answer

a. Sample proportion of $0.17<0.13$.

b. $Z=1.19,P=0.1170$

c. No enough convincing evidence is present that true population proportion of all students at elementary school who walk is$>0.13$

Step-by-step solution

Given Information

It is given that $\alpha =0.05$

$H_0:p=0.13$

$H_1:p>0.13$

$n=100,x=17$

Explaining Sample Results

The sample proportion is:

$\hat{p}=\frac{x}{n}=\frac{17}{100}=0.17$

As $0.17>0.13$, sample result gives some evidence for alternate hypothesis as it agrees with it.

Hence,$H_1:p>0.17$

Standardized test statistic and P value

As sample proportion is $0.17$

Test statistic is $z=\frac{\hat{p}-p_0}{\sqrt{\frac{p_0\left(1-p_0\right)}{n}}}=\frac{0.17-0.13}{\sqrt{\frac{0.13(1-0.13)}{100}}}=1.19$

$P$value is $P=P(z>1.19)=1-P(Z<1.19)=1-0.8830$

$P=0.1170$

Conclusion

We get

$P=P(z>1.19)=1-P(Z<1.19)=1-0.8830=0.1170$

Now, $P>0.05\Rightarrow \text{Fail to reject}{H}_0$

Hence, there is no enough convincing evidence that true population of all students$>0.13$