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Chapter 9: Q. 53. (page 583)

Teen drivers Refer to Exercise 51.

a. Construct and interpret a 95% confidence interval for the true proportion p of all teens in the state who passed their driving test on the first attempt. Assume that the conditions for inference are met.

b. Explain why the interval in part (a) provides more information than the test in Exercise 51.

Short Answer

Expert verified

Part a) 0.6068<p<0.7692

There are 95%confident that the true proportion of all teens in the state that passed their driving test on the first attempt is between 0.6068and 0.7692

Part b) There is sufficient proof to reject the DMV's claim.

Step by step solution

01

Part a) Step 1: given information

n=125p=60%=0.60x=86c=95%

02

Part a) Step 2: Formula used

p^=xnE=za2路p^(1-p^n

03

Part a) Step 3: The objective is to construct and interpret a  confidence interval for the true proportion of all teens in the state that passed their driving test on the first attempt.

The sample proportion is

p^=xn=86125=0.688

For confidence level determined za/2=z0.025using table II (look up 0.025in the table, the score is then they found z-core with opposite sign):

z伪/2=1.96

The margin of error is:

E=za/2路p^(1-p^)n=1.96脳0.688(1-0.688)125=0.0812

The confidence interval is as follows:

p^-E<p<p^+E=0.688-0.0812<p<0.688+0.0812=0.6068<p<0.7692

The true proportion of all teens in the state who passed their driving test on the first try is estimated to be between 0.6068 and 0.7692

04

Part b) Step 1: Given information

The result from exercise part (a):

0.6068<p<0.7692
05

Part b) Step 2: The objective is to explain the interval in part (a) provides more information than the test exercise 51

The confidence interval is not 60% (or 0.60), as claimed by the DMV. As a result, there is sufficient evidence to reject the DMV's claim.

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