4. Working for a car company, you have been assigned to find the average miles p

Question

4. Working for a car company, you have been assigned to find the average miles per gallon (mpg) for a certain model of car. You take a random sample of 49 cars of the assigned model. The observed sample mean is 26.7 mpg. Based on previous evidence and a QQ plot, you have reason to believe that the gas mileage is normally distributed with standard deviation (sd) 6.2 mpg. (a) Construct and interpret a 95% confidence interval for the mean mpg, , for the certain model of car. (b) What would happen to the interval if you increased the confidence level from 95% to 99%? Explain your reasoning. (Do not compute the new interval.) (c) Suppose you are asked to repeat this process with data gathered from all across the country. You end up constructing a total of 60 separate 95% confidence intervals at different factories for the estimated average mpg of the model of car. Of these intervals, how many of them do you expect would fail to contain the true value of ? (d) Now suppose based on previous evidence, it is not clear that the gas mileage is normally distributed. But you know the distribution is approximately symmetric with the same mean and sd. Then does the confidence interval for , found in (a), remain valid? Give reasons.

Explanation / Answer

(A) 95% confidence interval for the mean mpg = x +- Z95% (/sqrt(n)

Where = 6.2 mpg

sample mean x = 26.7 ,pg

95% confidence interval = 26.7 +- 1.96 * (6.2 / sqrt(49)) = 26.7 +- 1.736

= (24.96, 28.44)

so, we can interpret that population mean in miles per gallon for a certain model has 95% probability of being in between 24.96 mpg and 28.44 mpg.

(B) if confidence interval is increased to 99%, the width of confidence interval will increase. It will increase as it will increase the Z - value for 99% confidencelevel.

(C) There are 60 Seperate 95% confidence interval. As each of the confidence interval has 95% chance to contain the true population mean so that measns there is 5% chance that the given confidence ineterval will not be able to contain the true population mean. So,

The Expected number of intervals which would fail to contain the true value of mean = 60 * 0.05 = 3

so we will expect that on an average 3 would fail to contain the true value of population mean.

(d) Yes, even the gas mileage is not normaly distributed but as per the Central Limit Theorem and Law of large numbers entails that for any given distribution if a large sample is taken the distribution of mean of that samples will form a normal distibution. So, the confidenceinterval for population will remain valid.

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