Question

A simple random sample of front-seat occupants involved in car crashes is obtained. Among 2946occupants not wearing seat belts, 31 were killed. Among 7602 occupants wearing seat belts, 16were 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 (1) _________ the significance level of α = 0.01, so (2) _________ the null hypothesis. There (3) _________ 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 _________ < p1 − p2 < _________ .
(Round to three decimal places as needed.)

What is the conclusion based on the confidence interval?

Because the confidence interval limits (4) _________ 0, it appears that the two fatality rates are (5) _________.

Because the confidence interval limits include (6) __________ values, it appears that the fatality rate is (7) _________  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.

(1) less than
   greater than

(2) reject
   fail to reject

(3) is not
   is
(4) include
   do not include

(5) not equal.
   equal.

(6) only positive
   positive and negative
   only negative

(7) lower
   higher
   the same

Answer 1

D. H0: p1 = p2
   H1: p1 > p2

	pop 1	pop 2
x=	31	16
n =	2946	7602
p̂=x/n=	0.0105	0.0021
estimated prop. diff =p̂1-p̂2    =		0.0084
pooled prop p̂ =(x1+x2)/(n1+n2)=		0.0045
std error Se=√(p̂1*(1-p̂1)*(1/n1+1/n2) =		0.0014
test stat z=(p̂1-p̂2)/Se =		5.82
P value   =	0.000	(from excel:2*normsdist(-5.82)

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)

estimated diff. in proportion=p̂1-p̂2=

0.0084

Se =√(p̂1*(1-p̂1)/n1+p̂2*(1-p̂2)/n2) =

0.0020

for 98 % CI value of z=		2.326	from excel:normsinv((1+0.98)/2)
margin of error E=z*std error =		0.004541
lower bound=(p̂1-p̂2)-E=		0.0039
Upper bound=(p̂1-p̂2)+E=		0.0130
from above 98% confidence interval for difference in population proportion =(0.004<p1-p2<0.013)

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)

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