In a random sample of 340 cars driven at low altitudes 46 of

In a random sample of 340 cars driven at low altitudes, 46 of them exceeded a standard of 10 grams of particulate pollution per gallon of fuel consumed. In an independent random sample of 85 cars driven at high altitudes, 21 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?

Solution

Let p1 be the proportion for high-altitude vehicles

Let p2 be the proportion for low-altitude vehicles

The test hypothesis:

Ho: p1=p2 (i.e. null hypothesis)

Ha: p1>p2 (i.e. alternative hypothesis)

p1=21/85 = 0.2470588

p2=46/340 = 0.1352941

The test statistic is

Z=(p1-p2)/sqrt(p1*(1-p1)/n1+p2*(1-p2)/n2)

=(0.2470588-0.1352941)/sqrt(0.2470588*(1-0.2470588)/85+0.1352941*(1-0.1352941)/340)

=2.22

Assume that the significant level a=0.05

It is a right-tailed test.

The critical value is Z(0.05) = 1.645 (from standard normal table)

The rejection region is if Z>1.645, we reject the null hypothesis.

Since Z=2.22 is larger than 1.645, we reject the null hypothesis.

So we can conclude that the proportion of high-altitude vehicles exceeding the standard is greater than the proportion of low-altitude vehicles exceeding the standard