Answer to Question #342470 in Statistics and Probability for Derek

Question

5. A teacher wants to test his assumption that less than 30% of the Senior High

School students liked research class. After randomly collecting 150 samples,

he found out that only 40 students like their research class.

Claim:

Parameter

Symbol for parameter

Ho and Ha complementary pair.

Hypotheses in words:

Ho

Ha:

Hypotheses in symbols

Ho

На:

Answer 1

The population proportion $p$ is the parameter.

Claim $p<0.3.$

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

$H_0:$ the population proportion is at least 30%.

$H_a:$ the population proportion is less than 30%.

$H_0:p\ge0.3$

$H_a:p<0.3$

This corresponds to a left-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 left-tailed test is $z_c =-1.6469.$

The rejection region for this left-tailed test is $R = \{z: z<-1.6469\}.$

The z-statistic is computed as follows:

$$z=\dfrac{\hat{p}-p}{\sqrt{\dfrac{p(1-p)}{n}}}=\dfrac{\dfrac{40}{150}-0.3}{\sqrt{\dfrac{0.3(1-0.3)}{150}}}=-0.891$$

Since it is observed that $z =-0.891>-1.6469=z_c,$ it is then concluded that the null hypothesis is not rejected.

Using the P-value approach:

The p-value is $p=P(Z<-0.891)=0.186465,$ and since $p = 0.186465 > 0.05=\alpha,$ it is concluded that the null hypothesis is not rejected.

Therefore, there is not enough evidence to claim that the population proportion $p$ is less than 0.3, at the $\alpha = 0.05$ significance level.