Question

In a random sample of 370 cars driven at low altitudes, 43 of them exceeded a standard of 10 grams of particulate pollution per gallon of fuel consumed. In an independent random sample of 80 cars driven at high altitudes, 23 of them exceeded the standard. Can you conclude that the proportion of high-altitude vehicles exceeding the standard is greater than the proportion of low-altitude vehicles exceeding the standard at an level of significance? Group of answer choices

Answer 1

There is enough evidence to support the claim that the proportion of high-altitude vehicles exceeding the standard is greater than the proportion of low-altitude vehicles exceeding the standard (P-value = 0.00005).

Explanation:

This is a hypothesis test for the difference between proportions.

The claim is that the proportion of high-altitude vehicles exceeding the standard is greater than the proportion of low-altitude vehicles exceeding the standard.

Then, the null and alternative hypothesis are:

`H_0: \pi_1-\pi_2=0\\\\H_a:\pi_1-\pi_2< 0`

The significance level is estabilished in 0.01.

The sample 1 (low altitudes), of size n1=370 has a proportion of p1=0.116.

`p_1=X_1/n_1=43/370=0.116`

The sample 2 (high altitudes), of size n2=80 has a proportion of p2=0.288.

`p_2=X_2/n_2=23/80=0.288`

The difference between proportions is (p1-p2)=-0.171.

`p_d=p_1-p_2=0.116-0.288=-0.171`

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

`p=(X_1+X_2)/(n_1+n_2)=(43+23.04)/(370+80)=(66)/(450)=0.147`

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

`s_(p1-p2)=\sqrt{(p(1-p))/(n_1)+(p(1-p))/(n_2)}=\sqrt{(0.147*0.853)/(370)+(0.147*0.853)/(80)}\\\\\\s_(p1-p2)=√(0.000338+0.001564)=√(0.001903)=0.044`

Then, we can calculate the z-statistic as:

`z=(p_d-(\pi_1-\pi_2))/(s_(p1-p2))=(-0.171-0)/(0.044)=(-0.171)/(0.044)=-3.93`

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

`\text{P-value}=P(z<-3.93)=0.00005`

As the P-value (0.00005) is smaller than the significance level (0.01), the effect is significant.

The null hypothesis is rejected.

There is enough evidence to support the claim that the proportion of high-altitude vehicles exceeding the standard is greater than the proportion of low-altitude vehicles exceeding the standard.