Question

Cell Phones and Driving In the author’s home town of Madison, CT, there were 2733 police traffic stops in a recent year, and 7% of them were attributable to improper use of cell phones. Use a 0.05 significance level to test the claim that the sample is from a population in which fewer than 10% of police traffic stops are attributable to improper cell phone use.

Short answer

There is enough evidence to support the claim that fewer than 10% of police traffic stops are attributable to improper cell phone use.

Step-by-step solution

Given information

Out of 2733 police traffic stops, 7% were due to improper use of cell phones.

z-test

The z-test is used to test whether the sample proportion is equal to the given value of proportion.

Hypotheses

The null hypothesis is as follows:

The sample proportion of traffic stops due to improper use of cell phones is equal to 10%.

\({H_0}:P = 0.10\)

The alternative hypothesis is as follows:

The sample proportion of traffic stops due to improper use of cell phones is equal to 10%.

\({H_0}:P < 0.10\)

The test is left-tailed.

If the absolute value of the test statistic is greater than the critical value, the null hypothesis is rejected.

Important computations

Sample proportion (p) of the traffic stops due to improper use of cell phones is given to be equal to 7% or 0.07.

Population proportion (P) of the traffic stops due to improper use of cell phones is given to be equal to 10% or 0.10.

Thus, the population proportion (Q) of the traffic stops that were due to reasons other than improper use of cell phones is given as:

\(\begin{aligned} Q &= 1 - P\\ &= 1 - 0.10\\ &= 0.90\end{aligned}\)

The sample size (n) is equal to 2733.

Test statistic

The test statistic value is computed below:

\(\begin{aligned} z &= \frac{{p - P}}{{\sqrt {\frac{{PQ}}{n}} }}\;\; \sim N\left( {0,1} \right)\\ &= \frac{{0.07 - 0.10}}{{\sqrt {\frac{{\left( {0.10} \right)\left( {0.90} \right)}}{{2733}}} }}\\ &= - 5.2278\end{aligned}\)

Thus, the test statistic is – 5.2278.

Critical value and p-value

The critical value of z corresponding\(\alpha = 0.05\)for a left-tailed test is equal to -1.6449.

By using the standard normal table, the p-value is 0.000.

Since the absolute value of the test statistic is greater than the critical value and the p-value is less than 0.05, the null hypothesis is rejected.

Conclusion

There is enough evidence to support the claim thatfewer than 10% of police traffic stops are attributable to improper cell phone use.