Question

At a certain college, it is estimated that approximately 19% of the students ride bicycles to school. Would you consider this to be valid estimate if, in a random sample of 85 college students, 20 are found to ride bicycles to class

Answer 1

The following null and alternative hypotheses for the population proportion needs to be tested:

$H_0:p=0.19$

$H_a:p\not=0.19$

This corresponds to a two-tailed test, for which a z-test for one population proportion will be used.

Based on the information provided, the significance level is $\alpha = 0.05,$ and the critical value for a two-tailed test is $z_c =1.96.$

The rejection region for this two-tailed test is $R = \{z: |z| > 1.96\}.$

The z-statistic is computed as follows:

$$z=\dfrac{\hat{p}-p_0}{\sqrt{\dfrac{p_0(1-p_0)}{n}}}=\dfrac{\dfrac{20}{85}-0.19}{\sqrt{\dfrac{0.19(1-0.19)}{85}}}\approx1.0645$$

Since it is observed that $|z|=1.0645< 1.96=z_c,$ it is then concluded that the null hypothesis is not rejected.

Using the P-value approach:

The p-value is $p=2P(Z>1.0645)=0.287102,$ and since $p=0.287102>0.05=\alpha,$ it is concluded that the null hypothesis is not rejected.

Therefore, there is not enough evidence to claim that the population proportion $p$ is different than $0.19,$ at the $\alpha = 0.05$ significance level.