Respuesta :
Answer:
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).
Step-by-step 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:
[tex]H_0: \pi_1-\pi_2=0\\\\H_a:\pi_1-\pi_2< 0[/tex]
The significance level is estabilished in 0.01.
The sample 1 (low altitudes), of size n1=370 has a proportion of p1=0.116.
[tex]p_1=X_1/n_1=43/370=0.116[/tex]
The sample 2 (high altitudes), of size n2=80 has a proportion of p2=0.288.
[tex]p_2=X_2/n_2=23/80=0.288[/tex]
The difference between proportions is (p1-p2)=-0.171.
[tex]p_d=p_1-p_2=0.116-0.288=-0.171[/tex]
The pooled proportion, needed to calculate the standard error, is:
[tex]p=\dfrac{X_1+X_2}{n_1+n_2}=\dfrac{43+23.04}{370+80}=\dfrac{66}{450}=0.147[/tex]
The estimated standard error of the difference between means is computed using the formula:
[tex]s_{p1-p2}=\sqrt{\dfrac{p(1-p)}{n_1}+\dfrac{p(1-p)}{n_2}}=\sqrt{\dfrac{0.147*0.853}{370}+\dfrac{0.147*0.853}{80}}\\\\\\s_{p1-p2}=\sqrt{0.000338+0.001564}=\sqrt{0.001903}=0.044[/tex]
Then, we can calculate the z-statistic as:
[tex]z=\dfrac{p_d-(\pi_1-\pi_2)}{s_{p1-p2}}=\dfrac{-0.171-0}{0.044}=\dfrac{-0.171}{0.044}=-3.93[/tex]
This test is a left-tailed test, so the P-value for this test is calculated as (using a z-table):
[tex]\text{P-value}=P(z<-3.93)=0.00005[/tex]
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.
