This is a hypothesis test for the difference in proportions between two independent groups. Here are the steps to solve the problem:
a. State the appropriate Null and Alternative Hypothesis. Do the Hypothesis test with significance level a=0.1.
Null Hypothesis (H0): The proportion of high altitude vehicles exceeding the standard is less than or equal to the proportion of low altitude vehicles exceeding the standard.
Alternative Hypothesis (Ha): The proportion of high altitude vehicles exceeding the standard is greater than the proportion of low altitude vehicles exceeding the standard.
We can use a two-sample z-test to test this hypothesis. The test statistic is given by:
z = (p1 - p2) / sqrt(p(1-p)(1/n1 + 1/n2))
where:
p = (x1 + x2) / (n1 + n2)
p1 = x1 / n1
p2 = x2 / n2
x1 = 46 (number of cars exceeding the standard in low altitude sample)
n1 = 340 (sample size for low altitude sample)
x2 = 21 (number of cars exceeding the standard in high altitude sample)
n2 = 85 (sample size for high altitude sample)
Using these values, we get: p = (46 + 21) / (340 + 85) = 0.129 p1 = 46 / 340 = 0.135 p2 = 21 / 85 = 0.247
z = (0.135 - 0.247) / sqrt(0.129(1 - 0.129)(1/340 + 1/85)) = -3.07
The critical value for a one-tailed test with alpha=0.1 and degrees of freedom=423 is -1.28.
Since our test statistic (-3.07) is less than our critical value (-1.28), we reject the null hypothesis and conclude that there is sufficient evidence to suggest that the proportion of high altitude vehicles exceeding the standard is greater than the proportion of low altitude vehicles exceeding the standard.
b. Construct a 95% confidence interval for the difference in proportions.
The formula for calculating a confidence interval for the difference in proportions is:
(p1 - p2) ± z*sqrt((p1(1-p1)/n1) + (p2(1-p2)/n2))
where:
z is the critical value from the standard normal distribution for a given level of confidence
p1 and p2 are the sample proportions
n1 and n2 are the sample sizes
Using our values from above, we get:
z*sqrt((p1(1-p1)/n1) + (p2(1-p2)/n2)) = 0.098
So our confidence interval is:
(0.135 - 0.247) ± 0.098 = (-0.21, -0.05)
Therefore, we are 95% confident that the true difference in proportions between high and low altitude vehicles that exceed the standard is between -0.21 and -0.05.
c. Calculate the Margin of Error for 95% confidence level.
The margin of error for a confidence interval is half of its width, so in this case it would be:
(0.21 - 0.05) / 2 = 0.08
Therefore, our margin of error is ±0.08.