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Chapter 9: Problem 27

USA Today (October 14, 2002) reported that \(36 \%\) of adult drivers admit that they often or sometimes talk on a cell phone when driving. This estimate was based on data from a sample of 1004 adult drivers, and a bound on the error of estimation of \(3.1 \%\) was reported. Assuming a \(95 \%\) confidence level, do you agree with the reported bound on the error? Explain.

Short Answer

Expert verified
Whether we agree with the reported bound on the error depends on the comparison in Step 4, where we compare the calculated error of estimation to the reported bound. If they are approximately equal, we can agree with the bound.

Step by step solution

01

Calculate the Sample Proportion

The sample proportion, often denoted as \( p \), is the proportion of the sample that has a certain characteristic. In this case, it is the proportion of adult drivers who admit that they often or sometimes talk on a cell phone when driving. The sample proportion can be calculated as the number of successes over the total sample size. So, in this case we have \( p = 0.36 \)
02

Calculate Standard Deviation

The standard deviation, often denoted as \( \sigma \), of the sample proportion can be calculated using the formula \( \sigma = \sqrt{p(1-p)/n} \), where \( p \) is the sample proportion and \( n \) is the total sample size. So in this case, \( \sigma = \sqrt{0.36 * (1 - 0.36) / 1004} \)
03

Calculate the 95% Confidence Interval

The confidence interval can be calculated as the sample proportion plus or minus the z-value for the 95% confidence level times the standard deviation. The z-value for a 95% confidence interval is usually 1.96. So, the confidence interval is \( 0.36 卤 1.96 * \sigma \)
04

Compare the Error

The error of estimation, often denoted as \( E \), can be calculated as the upper bound of the confidence interval minus the sample proportion. So, \( E = 0.36 + 1.96 * \sigma - 0.36 \). Compare this value to the reported bound on the error of estimation (3.1%)
05

Conclusion

If the calculated error of estimation is approximately equal to the reported bound on the error of estimation (3.1%), meaning a difference less than 0.5% for example, then the reported bound is reasonable. Otherwise, if the difference is greater than this, we would not agree with the reported bound.

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Most popular questions from this chapter

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