Question

Let \(p\) equal the proportion of drivers who use a seat belt in a country that does not have a mandatory seat belt law. It was claimed that \(p=0.14\). An advertising campaign was conducted to increase this proportion. Two months after the campaign, \(y=104\) out of a random sample of \(n=590\) drivers were wearing their seat belts. Was the campaign successful? (a) Define the null and alternative hypotheses. (b) Define a critical region with an \(\alpha=0.01\) significance level. (c) Determine the approximate \(p\) -value and state your conclusion.

Short answer

The hypotheses are H0: \(p = 0.14\) and Ha: \(p > 0.14\). The critical z-value is approximately 2.33. The sample proportion is 0.176. The test statistic is 1.77 corresponding to a p-value of 0.04. Since the p-value is bigger than the significance level (0.01), we fail to reject the null hypothesis. The campaign's effect on the increase of seatbelt use is not statistically significant.

Step-by-step solution

Define the Hypotheses

The null hypothesis (H0) is defined as the situation before the campaign, where the proportion of drivers using seat belts \(p = 0.14\). The alternative hypothesis (Ha) is the supposed situation after the campaign, where the proportion of seat belt usage is larger, \(p > 0.14\). Mathematically, \n- Null Hypothesis (H0): \(p = 0.14\)\n- Alternative Hypothesis (Ha): \(p > 0.14\)

Define the Critical Region

At an alpha level of 0.01, the critical region is the set of values that will lead us to reject the null hypothesis. The critical value on the z-distribution, which corresponds to the given alpha level of 0.01 (one-sided), is found using a standard z-table or a calculator with an inverse normal function. This value (Z(0.01)) is approximately 2.33. Therefore, any z-score greater than 2.33 will lead to the rejection of the null hypothesis.

Find the Sample Proportion and Test Statistic

The sample proportion is given by \(\hat{p}=y/n=104/590=0.176\). The z-score is calculated using the formula: \n\(z = (\hat{p}-p0)/(\sqrt{p0*(1-p0)/n}) = (0.176-0.14)/(\sqrt{0.14*(1-0.14)/590}) = 1.77\)

Determine the p - Value and Draw Conclusions

The p-value is the smallest level of significance at which we would still reject the null hypothesis. It is found by looking up the z-score in the z-table. For 1.77, the p-value is approximately 0.04. Since this p-value is greater than the significance level (0.01), we fail to reject the null hypothesis. Therefore, we cannot conclude that the campaign significantly increased the proportion of drivers using seatbelts.