Question

A simple random sample of​ front-seat occupants involved in car crashes is obtained. Among 2805 occupants not wearing seat​ belts, 27 were killed. Among 7745 occupants wearing seat​ belts, 10 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.

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?

z=?

p-value=?

What is the conclusion based on the hypothesis​ test?

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

What is the conclusion based on the confidence​ interval?

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

Answer 1

Let p₁ = The Proportion of deaths where occupants who did not wear seat belts = 27 / 2805 = 0.0096

Let p₂ = The Proportion of deaths where occupants who wore seat belts = 10 / 7745 = 0.0013

Let = Overall proportion = (27+10)/(2285+7745) = 0.0035

1 - = 0.0065

= 0.01

(a) The Hypothesis:

H0: p₁ = p₂ : The proportion of deaths where occupants who did not wear seat belts is equal to the proportion of deaths where occupants who wore seat belts.

Ha: p₁ > p₂ :The proportion of deaths where occupants who did not wear seat belts is greater than the proportion of deaths where occupants who wore seat belts.

This is a Right tailed Test.

The Test Statistic:

The p Value:    The p value (Right tail) for Z = 6.41, is; p value = 0.0000

Since p value is < , we reject H0.

There is sufficient evidence to conclude that the proportion of deaths where occupants who did not wear seat belts is greater than the proportion of deaths where occupants who wore seat belts.

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The 99% Confidence interval

= 0.0096, 1 - = 0.9904, n₁ = 2805,

= 0.0013, 1 - = 0.9987, n₂ = 7745,

= 0.01

The Zcritical (2 tail) for = 0.01, is 2.576

The Confidence Interval is given by (- ) ME, where
(- ) = 0.0096 – 0.0013 = 0.0083

The Lower Limit = 0.0083 - 0.0049 = 0.0034

The Upper Limit = 0.0083 + 0.0049 = 0.0132

The Confidence Interval is (0.0034 , 0.0132)

Since both the values are positive, i.e 0 does not lie in the confidence interval, the results are statistically significant. We Reject H0.

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The results suggest that wearing seatbelts is effective in decreasing loss of lives.

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