Answer to Question #342469 in Statistics and Probability for Derek

Question

3. The Senior High School researchers claim that more than 20% of Senior High

School male students have tried smoking cigarette. After collecting 150 random

samples, they found that 60 of them have tried smoking cigarette.

Claim:

Parameter

Symbol for parameter

Ho and Ha complementary pair.

Hypotheses in words:

Ho

Ha:

Hypotheses in symbols

Ho

На:

4. In a certain town a school principal hypothesized that students enroll in schools

within 5 km from their homes. To check this claim you ask 38 students from

the said town. You found out that the average distance between the students

home and their schools is 5.6 km.

Claim:

Parameter

Symbol for parameter

Ho and Ha complementary pair.

Hypotheses in words:

Ho

Ha:

Hypotheses in symbols

Ho

На:

Answer 1

3. The population proportion $p$ is the parameter.

Claim $p>0.2.$

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

$H_0:$ the population proportion is no most 20%.

$H_a:$ the population proportion is greater than 20%.

$H_0:p\le0.2$

$H_a:p>0.2$

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

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

The z-statistic is computed as follows:

$$z=\dfrac{\hat{p}-p}{\sqrt{\dfrac{p(1-p)}{n}}}=\dfrac{\dfrac{60}{150}-0.2}{\sqrt{\dfrac{0.2(1-0.2)}{150}}}=6.1237$$

Since it is observed that $z =6.1237>1.6469=z_c,$ it is then concluded that the null hypothesis is rejected.

Using the P-value approach:

The p-value is $p=P(Z>6.1237)=0,$ and since $p = 0 < 0.05=\alpha,$ it is concluded that the null hypothesis is rejected.

Therefore, there is enough evidence to claim that the population proportion $p$ is greater than 0.2, at the $\alpha = 0.05$ significance level.

4.

The average distance between the students home and their schools $\mu$ is the parameter.

Claim $\mu=5km.$

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

$H_0:$ the average distance is 5 km.

$H_a:$ the average distance is not 5 km.

$H_0:\mu=5$

$H_a:\mu\not=5$

This corresponds to a two-tailed test, for which a t-test for one mean, with unknown population standard deviation, using the sample standard deviation, will be used.