Question

Random samples of students were compared to see whether or not there was a difference in the proportion favoring the university's proposed switch from MWF (three-day) classes to MW and TR (two-day) classes. In the resident population 80 out of 200 favored the switch. In the commuter population, 120 out of 200 favored the switch. Conduct a hypothesis test at the .05 significance level to see if there is a difference in the proportion of residents and commuters who prefer the switch.

Answer 1

Answer:

There is enough evidence to support the claim that there is a significant difference in the proportion of residents and commuters who prefer the switch.

Step-by-step explanation:

This is a hypothesis test for the difference between proportions.

The claim is that there is a significant difference in the proportion of residents and commuters who prefer the switch.

Then, the null and alternative hypothesis are:

$H_0: \pi_1-\pi_2=0\\\\H_a:\pi_1-\pi_2\neq 0$

The significance level is 0.05.

The sample 1 (residents), of size n1=200 has a proportion of p1=0.4.

$p_1=X_1/n_1=80/200=0.4$

The sample 2 (conmuters), of size n2=200 has a proportion of p2=0.6.

$p_2=X_2/n_2=120/200=0.6$

The difference between proportions is (p1-p2)=-0.2.

$p_d=p_1-p_2=0.4-0.6=-0.2$

The pooled proportion, needed to calculate the standard error, is:

$p=\dfrac{X_1+X_2}{n_1+n_2}=\dfrac{80+120}{200+200}=\dfrac{200}{400}=0.5$

The estimated standard error of the difference between means is computed using the formula:

$s_{p1-p2}=\sqrt{\dfrac{p(1-p)}{n_1}+\dfrac{p(1-p)}{n_2}}=\sqrt{\dfrac{0.5*0.5}{200}+\dfrac{0.5*0.5}{200}}\\\\\\s_{p1-p2}=\sqrt{0.0013+0.0013}=\sqrt{0.0025}=0.05$

Then, we can calculate the z-statistic as:

$z=\dfrac{p_d-(\pi_1-\pi_2)}{s_{p1-p2}}=\dfrac{-0.2-0}{0.05}=\dfrac{-0.2}{0.05}=-4$

This test is a two-tailed test, so the P-value for this test is calculated as (using a z-table):

$P-value=2\cdot P(z<-4)=0.00008$

As the P-value (0.00008) is smaller than the significance level (0.05), the effect is significant.

The null hypothesis is rejected.

There is enough evidence to support the claim that there is a significant difference in the proportion of residents and commuters who prefer the switch.