Question

Scenario 1: Seat Belt Use ~   The U.S. Department of Transportation collected seat belt use data by stationing observers at randomly selected roadway sites and recording the number of vehicle occupants who were wearing seat belts. A random sample of 1000 vehicle occupants in the Northeast shows that 909 were wearing seat belts, while a random sample of 1000 vehicle occupants in the Midwest showed that 855 were wearing seat belts.

The null hypothesis H_0H 0 ​ believes that the rate at which drivers in the Midwest and Northeast wear seat belts is the same, and that any observed difference across the two samples is attributable to random chance alone. If this was, in fact, the case, what is the best estimate of the rate at which drivers wear seat belts, regardless of whether they live in the Northeast or Midwest? Submit your answer out to four decimal places.

Answer 1

Scenario 1: Seat Belt Use ~   The U.S. Department of Transportation collected seat belt use data by stationing observers at randomly selected roadway sites and recording the number of vehicle occupants who were wearing seat belts. A random sample of 1000 vehicle occupants in the Northeast shows that 909 were wearing seat belts, while a random sample of 1000 vehicle occupants in the Midwest showed that 855 were wearing seat belts.

The null hypothesis H_0H 0 ​ believes that the rate at which drivers in the Midwest and Northeast wear seat belts is the same, and that any observed difference across the two samples is attributable to random chance alone. If this was, in fact, the case, what is the best estimate of the rate at which drivers wear seat belts, regardless of whether they live in the Northeast or Midwest? Submit your answer out to four decimal places.

Two sample proportion test

p1=909/1000=0.909

p2=855/1000=0.855

P = (x1+x2)/(n1+n2)

=(909+855)/2000 =0.882

=3.7429

Table value of z at 0.05 level = 1.96

Rejection Region: Reject Ho if z < -1.96 or z > 1.96

Calculated z =3.7429   falls in the rejection region

The null hypothesis is rejected.

We conclude that there is a difference in the rate at which drivers in the Midwest and Northeast wear seat belts.

Since there is a difference there is no best estimate of the rate at which drivers wear seat belts.

Z Test for Differences in Two Proportions
Data
Hypothesized Difference	0
Level of Significance	0.05
Group 1
Number of Items of Interest	909
Sample Size	1000
Group 2
Number of Items of Interest	855
Sample Size	1000
Intermediate Calculations
Group 1 Proportion	0.909
Group 2 Proportion	0.855
Difference in Two Proportions	0.054
Average Proportion	0.8820
Z Test Statistic	3.7429
Two-Tail Test
Lower Critical Value	-1.9600
Upper Critical Value	1.9600
p-Value	0.0002
Reject the null hypothesis