Question

A campus legend tells the story of two friends who lied to their professor by blaming a flat tire for their having missed an exam. On the make-up exam, the professor sent them to separate rooms and asked: "Which tire was it?" If you were faced with this situation, which of the four tires would you say it was? 1. Record the counts of responses for everyone in your class in the table below. This has been completed you by your instructor. Tire Left Front Left Rear Right Front Right Rear Count 122 115 154 109 2. It has been suggested that when asked to say which tire on a car is likely to have gone flat, people choose the right front tire more often than they would if the four tires were equally likely to be chosen. Define the parameter of interest in the context of the problem (assume your population of interest is all 500 students taking a course this semester). 3. If the four tires were, in fact, equally likely to be chosen, how often would the right front tire be selected in the long run?

Answer 1

2) The parameter of interest here is "p" which denotes the proportion time people choose the right front tire.

3) If four tire were, in fact, equally likely to be chosen, then in the long run "p" which denotes the proportion time people choose the right front tire will be equal to 0.25.

4) Here the hypothesis is:

$\small H_0=p=0.25\ vs\ H_a:p> 0.25$

5) The sample proportion of students who chose the right front tire is:

$\small p=\frac{154}{500}=0.308$

6) Now under the null hypothesis $\small \hat{p}$ follows a normal distribution with mean 0.25 and standard deviation $\small \sqrt{\frac{0.25*0.75}{500}}$ . Thus, the probability of observing $\small \hat{p}$ st least as large as p obtained is:

$\small P(\hat{p}>0.308)=P(\frac{\hat{p}-0.25}{SE}>\frac{0.308-0.25}{SE})=P(Z>2.995)=0.0014,\\ here\ SE=\sqrt{\frac{0.25*0.75}{500}}$

8) As the p-value is very small. We reject the null hypothesis. Hence, we conclude that there is sufficient evidence to support the claim that the proportion of students choosing the right front tire is more likely.