Question

A simple random sample of​ front-seat occupants involved in car crashes is obtained. Among

2853 occupants not wearing seat​ belts, 36 were killed. Among 7754 occupants wearing seat​ belts, 18 were killed. Use a 0.01 significance level to test the claim that seat belts are effective in reducing fatalities. Complete parts​ (a) through​ (c) below.

a. Test the claim using a hypothesis test.

Consider the first sample to be the sample of occupants not wearing seat belts and the second sample to be the sample of occupants wearing seat belts. What are the null and alternative hypotheses for the hypothesis​ test?

Identify the test statistic.

zequals=___

​(Round to two decimal places as​ needed.)

Identify the​ P-value.

​P-value equals=____

​(Round to three decimal places as​ needed.)

What is the conclusion based on the hypothesis​ test?

The​ P-value is ______(less than or greater)________ than the significance level of α=0.01​,

So ______(Reject or fail to reject)_____ the null hypothesis. There _____( is or is not)_____   

sufficient evidence to support the claim that the fatality rate is higher for those not wearing seat belts.

b. Test the claim by constructing an appropriate confidence interval.

The appropriate confidence interval is _____?

​(Round to three decimal places as​ needed.)

What is the conclusion based on the confidence​ interval?

Because the confidence interval limits _______(do not include or include)_______   

​0, it appears that the two fatality rates are _____(equal or not equal)_____. Because the confidence interval limits include______(only negative, positive and negative, only positive)_______    values, it appears that the fatality rate is ______(higher, the same,    lower)_______ for those not wearing seat belts.

c. What do the results suggest about the effectiveness of seat​ belts?

A.

The results suggest that the use of seat belts is associated with the same fatality rates as not using seat belts.

B.

The results suggest that the use of seat belts is associated with lower fatality rates than not using seat belts.

C.

The results suggest that the use of seat belts is associated with higher fatality rates than not using seat belts.

D. The results are inconclusive.

Answer 1

a)Consider the first sample to be the sample of occupants not wearing seat belts and the second sample to be the sample of occupants wearing seat belts.

Ho:   p1 - p2 =   0
Ha:   p1 - p2 >   0
      
first sample size,     n1=   2853
number of successes, sample 1 =     x1=   36
proportion success of sample 1 , p̂1=   x1/n1=   0.012618297
      
second sample size,     n2 =    7754
number of successes, sample 2 =     x2 =    18
proportion success of sample 1 , p̂ 2=   x2/n2 =    0.002321383
      
difference in sample proportions,    p̂1 - p̂2 =     0.010296914
pooled proportion , p =   (x1+x2)/(n1+n2)=   0.005090978
      
std error ,SE =    =SQRT(p*(1-p)*(1/n1+ 1/n2)   0.001558385
      
Z-statistic =    (p̂1 - p̂2)/SE =     6.61
      
p-value =        0.000
decision :    p-value<α,Reject null hypothesis

The​ P-value is ______(less )________ than the significance level of α=0.01​,

So ______(Reject )_____ the null hypothesis. There _____( is )_____   

sufficient evidence to support the claim that the fatality rate is higher for those not wearing seat belts.

b)

proportion success of sample 1 , p̂1=   x1/n1=   0.012618297
proportion success of sample 1 , p̂ 2=   x2/n2 =    0.002321383

Std error , SE =    SQRT(p̂1 * (1 - p̂1)/n1 + p̂2 * (1-p̂2)/n2) =     0.0022
Z critical value =   Z (α/2)=Z(0.01/2) =    2.5758
      
margin of error , E =   Z*SE =    0.0056
      
confidence interval is       
lower limit =    (p̂1 - p̂2) - E =    0.005
upper limit =   (p̂1 - p̂2) + E =    0.016

The appropriate confidence interval is _____(0.005,0.016)

Because the confidence interval limits _______(do not include )_______   

​0, it appears that the two fatality rates are _____(not equal)_____. Because the confidence interval limits include__(only positive)_______ values, it appears that the fatality rate is ______(higher )_______ for those not wearing seat belts.

c)

The results suggest that the use of seat belts is associated with higher fatality rates than not using seat belts.