Question

A simple random sample of​ front-seat occupants involved in car crashes is obtained. Among 2901 occupants not wearing seat​ belts,27 were killed. Among 7877 occupants wearing seat​ belts,16 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?

A. H0​: p1=p2

H1​:p1≠p2

B.H0​: p1=p2

H1​: p1> p2

C.H0​: p1≤p2

H1​:p1≠p2

D.H0​:p1≠p2

H1​:p1=p2

E.H0​:p1=p2

H1​:p1<p2

F.H0​:p1≥p2

H1​:p1≠p2

Identify the test statistic.

z=

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

Identify the​ P-value.

​P-value=

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

What is the conclusion based on the hypothesis​ test?

The​ P-value is

greater than

less than

the significance level of

alphaαequals=0.01​, so

reject

fail to reject

the null hypothesis. There

is not

is

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

< (p 1 - p 2)<

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

What is the conclusion based on the confidence​ interval?

Because the confidence interval limits

do not include

include

​0, it appears that the two fatality rates are

equal.

not equal.

Because the confidence interval limits include

only positive

only negative

positive and negative

​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 lower fatality rates than not using seat belts.

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

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

D.The results are inconclusive.

Answer 1

B.H0​: p1=p2

H1​: p1> p2

===============

sample #1   ----->              
first sample size,     n1=   2901          
number of successes, sample 1 =     x1=   27          
proportion success of sample 1 , p̂1=   x1/n1=   0.0093          
                  
sample #2   ----->              
second sample size,     n2 =    7877          
number of successes, sample 2 =     x2 =    16          
proportion success of sample 1 , p̂ 2=   x2/n2 =    0.0020          
                  
difference in sample proportions, p̂1 - p̂2 =     0.0093   -   0.0020   =   0.0073
                  
pooled proportion , p =   (x1+x2)/(n1+n2)=   0.003989608          
                  
std error ,SE =    =SQRT(p*(1-p)*(1/n1+ 1/n2)=   0.00137          
Z-statistic = (p̂1 - p̂2)/SE = (   0.007   /   0.0014   ) =   5.31
                  
p-value =        0.0000
decision :    p-value<α,Reject null hypothesis   

The​ P-value is

less than

the significance level of

alphaαequals=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)

level of significance, α =   0.01              
Z critical value =   Z α/2 =    2.576   [excel function: =normsinv(α/2)      
                  
Std error , SE =    SQRT(p̂1 * (1 - p̂1)/n1 + p̂2 * (1-p̂2)/n2) =     0.00185          
margin of error , E = Z*SE =    2.576   *   0.0019   =   0.00477
                  
confidence interval is                   
lower limit = (p̂1 - p̂2) - E =    0.007   -   0.0048   =   0.0025014
upper limit = (p̂1 - p̂2) + E =    0.007   +   0.0048   =   0.0120504
                  
so, confidence interval is (   0.003   < p1 - p2 <   0.012   )  

Because the confidence interval limits include

only positive

​values, it appears that the fatality rate is

higher

for those not wearing seat belts.

c)

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

please revert for doubts and
please UPVOTE the solution