A simple random sample of? front-seat occupants involved in car crashes is obtai
ID: 2907510 • Letter: A
Question
A simple random sample of? front-seat occupants involved in car crashes is obtained. Among
27542754
occupants not wearing seat? belts,
3535
were killed. Among
78147814
occupants wearing seat? belts,
1818
were killed. Use a
0.010.01
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.
Upper H 0H0?:
p 1p1greater than or equals?p 2p2
Upper H 1H1?:
p 1p1not equals?p 2p2
B.
Upper H 0H0?:
p 1p1equals=p 2p2
Upper H 1H1?:
p 1p1less than<p 2p2
C.
Upper H 0H0?:
p 1p1equals=p 2p2
Upper H 1H1?:
p 1p1greater than>p 2p2
D.
Upper H 0H0?:
p 1p1not equals?p 2p2
Upper H 1H1?:
p 1p1equals=p 2p2
E.
Upper H 0H0?:
p 1p1equals=p 2p2
Upper H 1H1?:
p 1p1not equals?p 2p2
F.
Upper H 0H0?:
p 1p1less than or equals?p 2p2
Upper H 1H1?:
p 1p1not equals?p 2p2
Identify the test statistic.
zequals=nothing
?(Round to two decimal places as? needed.)
Identify the? P-value.
?P-valueequals=nothing
?(Round to three decimal places as? needed.)
What is the conclusion based on the hypothesis? test?
The? P-value is
less than
less than
greater than
the significance level of
alpha?equals=0.010.01?,
so
reject
fail to reject
reject
the null hypothesis. There
is
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
nothingless than<left parenthesis p 1 minus p 2 right parenthesisp1?p2less than<nothing.
?(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
do not include
?0, it appears that the two fatality rates are
not equal.
not equal.
equal.
Because the confidence interval limits include
only positive
only positive
only negative
positive and negative
?values, it appears that the fatality rate is
lower
higher
lower
the same
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 higher fatality rates than 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 the same fatality rates as not using seat belts.
D.
The results are inconclusive.
Click to select your answer(s).
Explanation / Answer
a) H0 : P1 = P2
H1 : P1 > P2
p1 = 35/2754 = 0.0127
p2 = 18/7814 = 0.0023
The pooled proportion(P) = (p1 * n1 + p2 * n2)/(n1 + n2)
= (0.0127 * 2754 + 0.0023 * 7814)/(2754 + 7814)
= 0.005
SE = sqrt(P(1 - P)(1/n1 + 1/n2))
= sqrt(0.005 * (1 - 0.005) * (1/2754 + 1/7814))
= 0.0016
The test statistic z = (p1 - p2)/SE
= (0.0127 - 0.0023)/0.0016
= 6.5
P-value = P(Z > 6.5)
= 1 - P(Z < 6.5)
= 1 - 1 = 0
The P-value is less than the significance level of alpha = 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) At 99% confidence interval the critical value is z0.005 = 2.58
The confidence interval is
(p1 - p2) +/- z0.005 * SE
= (0.0127 - 0.0023) +/- 2.58 * 0.0016
= 0.0104 +/- 0.0041
= 0.0063, 0.0145
= 0.006, 0.015
Because the confidence interval limits include only positive values, it appears that the fatality rate is higher for those not wearing seat belts.
c) Option - B) The results suggest that the use of seat belts is associated with lower fatality rates than not using seat belts.
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