Question 3 (1 point Automobile insurance companies take many factors into consid

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Question 3 (1 point Automobile insurance companies take many factors into consideration when setting rates, including age, marital status, and miles driven per year. In order to determine the effect of gender, 350 males and 30 female drivers were surveyed. Each was asked how many miles he or she had driven in the past year. The data collected are summarized in the table below Male Drivers Female Drivers Xi = 10,541 X2 =9,457 s, -3,45.2 s, = 3,438 Construct a 98% confidence interval estimate of the average difference in miles driven per year for males and females Make sure that your interval estimate is in the correct format, as discussed in class. Interpret the practical meaning of the resulting interval estimate, in plain English (i.e., in the context of the problem) Question 4 (3 points). A pharmaceutical company has produced a flu vaccine. In a test of its effectiveness, 1,000 people were randomly selected; of these, 500 were injected with the vaccine, and the other 500 went untreated. The number of people in each group who contacted the flu during the next three months is summarized in the table below NUMBER OF PEOPLE CONDITION Developed the flu Did not develop the flu Treated with Vaccine 65 435 Untreated 125 375 Does this study provide significant evidence that the vaccine is effective in preventing the flu? In other words, do proportionally fewer people develop the flu when treated with vaccine? Conduct a complete and appropriate hypothesis test using -.05, the p-value approach, and 5-step procedure discussed in class Step 1: State the null and alternative hypotheses Step 2: State the decision rule for rejecting the null hypothesis Step 3: State the p-value of the hypothesis test Step 4: Evaluate the null hypothesis Step 5: State the practical conclusion(s) to be drawn from the hypothesis test, in the context of the problem, in plain Englisih 4b Construct a 95% confidence interval estimate of the difference in the proportion of people who do and do not develop the flu when treated with vaccine. Make sure that your interval estimate is in the correct format, as discussed in class. Interpret the practical meaning of the resulting interval estimate, in plain English (i.e., in the context of the problem)

Explanation / Answer

(1-alpha)*100% confidence interval for population mean difference =

=sample mean difference ±z(alpha/2)*SE(difference)

98% confidence interval =(10541-9457)±z(0.02/2)*654.25=(10541-9457)±1.96*654.25=1084±1518=(-434,2602)

SE(difference)=sqrt(s12/n1 + s22/n2)=sqrt(3452*3452/350 + 3438*3438/30)=654.25

To interpret a confidence interval the sample information is random - but there is a pattern to its behavior if we look at all possible samples. Each possible sample gives us a different sample and a different interval. But, even though the results vary from sample-to-sample, we are "confident" because the margin-of-error would be satisfied for 98% of all samples .

(4a) proprotion of treated with vaccine and developed flue=p1=x1/n1=65/500=0.13,

var(p1)=(p1(1-p1)/n1)=(0.13*(1-0.13)/500)=0.000226

proprotion of treated with vaccine and did not developed flue p2=x2/n2=435/500=0.87,

var(p2)=(p2(1-p2)/n2)=(0.87*(1-0.87)/500)=0.000226

SE(p1-p2)=sqrt(var(p1)+var(p2))=sqrt(0.000226+0.000226)=0.0213

null hypothesisH0:p1=p2

alternate hypothesis H1:p1<p2

here we use z-test and critical z(0.05)=1.645

z=(p1-p2)/SE(p1-p2)=(0.13-0.87)/0.0213=-34.74

since the calculated absolute z=34.74 is more than the critical z=1.645, so we reject null hypothesis and conclude that vaccine is effective. here p-value =0.000 ( less than 0.001)

(1-alpha)*100% confidence interval for difference of proporiton =(p1-p2) ±z(alpha/2)*SE(p1-p2)

95% confidence interval =(0.13-0.87)±z(0.05/2)*SE(p)=-0.74.±1.96*0.0213=-0.74±0.0417=(-0.7817, 0.7383)

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