Certain states require your vehicle to be inspected for safety on a yearly basis

Question

Certain states require your vehicle to be inspected for safety on a yearly basis. Police randomly stop drivers from time to time to ensure that the car's safety inspection is up to date. Last year, an average of 8% of cars stopped were ticketed for having expired stickers. The state then funded a marketing campaign to increase awareness of the safety inspection requirement in hopes of lowering the amount of tickets issued. Last month the police issued tickets for expired inspections to 45 of 614 cars they stopped. Is this evidence to suggest that the marketing campaign was effective? Assume all conditions have been met.

Part 1: State the null and alternative hypotheses in symbols and words. Assume conditions have been met.

Part 2: Perform the appropriate hypothesis test "by hand" using the equation editor. Assume an alpha level of 0.05 for your test. Be sure to use the appropriate probability notation. Then find the corresponding confidence interval for the proportion of cars whose safety inspections have expired.

Part 3: State your conclusion making sure to site whether we reject or fail to reject and what that means in context. Then support your conclusion with the results from your confidence interval. Be sure to note whether the hypothesized value falls in our interval and what that means.

Explanation / Answer

here P=0.08 , n=614

SE(P)=sqrt(P(1-P)/n)=sqrt(0.08*(1-0.08)/614)=0.011

p=45/614=0.0733

x=issued tickets

E(x)=np=614*0.08=49.12

var(x)=np(1-p)=614*0.08*(1-0.08)=45.19

SE(x)=6.72

(1) null hypothesis H0:P=0.08

alternate hypothesis H1:P<0.08

(2) here we use z-test and z=(p-P)/SE(P)=(0.0733-0.08)/0.011=-0.61

critical z(0.05)=1.645

(1-alpha)*100% confidence interval for p=p± z(alpha/2)*SE(p)

95% confidence interval for sample mean=0.0733±z(0.05/2)*0.011=0.0733±1.96*0.011=0.0733±0.0216=(0.0517,0.0949)

(3) since critical z is more than absolute value of calculated z=0.61, so we fail to reject null hypothesis and conclude that there is no improvement.

Related Questions

Navigate

• Browse (All)
• Browse C
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.