Question

An insurance company checks police records of 582 accidents selected at random and finds that teenagers were at the wheel in 91 of them.

1. a) Develop a 95% confidence interval for the percentage of all auto accidents that involve

   teenage drivers. Give an interpretation of this interval

2. b) A state insurance regulator claims that one of every five auto accidents involves a teenage

   driver. Does your confidence interval of part (a) support or contradict this regulator’s claim?

   Explain.

3. c) If we were to increase the confidence level to 99% while keeping the sample size at 582,

   would the confidence interval become narrower or wider?

Answer 1

Solution:

Given:

Sample size = n = 582

x = Number of auto accidents that involve teenage drivers = 91

Part a) Develop a 95% confidence interval for the percentage of all auto accidents that involve teenage drivers. Give an interpretation of this interval.

where

We need to find z_c value for c=95% confidence level.

Find Area = ( 1 + c ) / 2 = ( 1 + 0.95) /2 = 1.95 / 2 = 0.9750

Look in z table for Area = 0.9750 or its closest area and find z value.

Area = 0.9750 corresponds to 1.9 and 0.06 , thus z critical value = 1.96

That is : Z_c = 1.96

Thus

.

Thus

An interpretation of this interval.

We are 95% confident that the true percentage of all auto accidents that involve teenage drivers is between 12.69% and 18.59%.

Part b)  A state insurance regulator claims that one of every five auto accidents involves a teenage driver. Does your confidence interval of part (a) support or contradict this regulator’s claim? Explain

Claim: p = 1/5 = 0.2 = 20%

Since 95% confidence interval is which is less than claimed percentage 20%, thus confidence interval of part (a) contradict this regulator’s claim.

Part c) If we were to increase the confidence level to 99% while keeping the sample size at 582, would the confidence interval become narrower or wider?

The confidence interval would become wider, since as confidence level increases keeping all other values same, length of confidence interval increases.

Thus length of 99% confidence interval would be more than that of 95% confidence interval.

Thus correct answer is: Wider.