Question

The paper "Teens and Distracted Driving" (Pew Internet \& American Life Project, 2009) reported that in a representative sample of 283 American teens ages 16 to \(17,\) there were 74 who indicated that they had sent a text message while driving. For purposes of this exercise, assume that this sample is a random sample of 16 - to 17 -year-old Americans. Do these data provide convincing evidence that more than a quarter of Americans ages 16 to 17 have sent a text message while driving? Test the appropriate hypotheses using a significance level of 0.01 . (Hint: See Example 10.11 )

Short answer

No, the data do not provide convincing evidence to support the claim that more than a quarter of American teens aged 16 to 17 have sent a text message while driving.

Step-by-step solution

Setup the Hypotheses

Firstly, set up the null hypothesis (H0) and the alternative hypothesis (H1). The null hypothesis is H0: p = 0.25 and the alternative hypothesis is H1: p > 0.25, where p represents the proportion of teens aged 16 to 17 who text while driving.

Calculate Test Statistic

Next, calculate the test statistic using the formula: z = (p̂ - p0) / sqrt[(p0(1 - p0) / n], where p̂ is the sample proportion, p0 is the hypothesized population proportion under H0, and n is the sample size. In this case p̂ = 74 / 283 = 0.2615, p0 = 0.25, and n = 283. Substituting these in the formula gives z = (0.2615 - 0.25) / sqrt[(0.25(1 - 0.25) / 283] = 0.5115.

Calculate P-value

Now, calculate the p-value. The p-value is the probability that we would observe a test statistic as extreme as our calculated z-score, assuming the null hypothesis is true. In this case, we use a z-table or calculator to find the area to the right of our test statistic (0.5115) in a standard normal distribution, which gives a p-value of approximately 0.3042.

Make Conclusion

Finally, compare the p-value with the given significance level (α=0.01). In this case, the p-value is much larger than the significance level; therefore, we do not reject the null hypothesis. There isn't enough evidence to support the claim that more than 25% of American teenagers aged 16 to 17 text while driving.

Key concepts

Null Hypothesis

In the realm of hypothesis testing in statistics, a null hypothesis is a statement that there is no effect or no difference, and it serves as a starting point for testing. It is denoted as H0. For example, in the context of the exercise provided, the null hypothesis states that the proportion (p) of 16 to 17 year old Americans who text while driving is 25% (H0: p = 0.25).

Establishing the null hypothesis is a fundamental step because it creates a benchmark against which the alternative hypothesis is compared. It assumes the status quo and puts the onus on the researcher to provide evidence to support any claim of departure from this norm.

Alternative Hypothesis

Conversely, the alternative hypothesis suggests that there is a statistically significant effect or a difference. It is the statement researchers are trying to provide evidence for and it is denoted as H1 or Ha. In the provided exercise, the alternative hypothesis posits that more than a quarter (25%) of the same age group texts while driving (H1: p > 0.25).

The alternative hypothesis is the researcher's assertion which is considered only after it is concluded that the null hypothesis does not hold based on the data. This hypothesis is the main subject of verification in any statistical test and provides the direction for the kind of test to be performed, whether one-tailed or two-tailed.

Test Statistic

A test statistic is a calculated number from the sample data that, under the null hypothesis, follows a known probability distribution. It measures how far off the observed data is from the null hypothesis. The calculation of the test statistic takes into account the sample proportion, the hypothesized proportion under the null hypothesis, and the sample size.

As outlined in the given exercise, the test statistic is calculated using the formula:
\( z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \)
where:\( \hat{p} \) is the sample proportion, \( p_0 \) is the hypothesized proportion under H0, and n is the sample size. This statistic helps to determine how extreme the sample data is, given the null hypothesis is true.

P-value

The p-value is a key component of hypothesis testing and represents the probability of obtaining test results at least as extreme as the observed results, assuming that the null hypothesis is true. It is a decimal between 0 and 1 and is used as a benchmark against the significance level to decide whether to reject the null hypothesis.

In the context of the exercise, the p-value is obtained by finding the area to the right of the test statistic (in this case a z-score) in the standard normal distribution. If this p-value is less than the significance level, the null hypothesis is rejected; otherwise, we fail to reject the null hypothesis.

Significance Level

The significance level (denoted as α) represents the threshold against which the p-value is compared to determine the strength of the evidence against the null hypothesis. Common levels of significance are 0.05, 0.01, and 0.10.

In the exercise, a significance level of 0.01 indicates that there is a 1% risk of concluding that a difference exists when there is no actual difference. It is chosen before examining the data to decide the critical region. Since the p-value in the exercise was greater than 0.01, we do not have sufficient evidence to reject the null hypothesis, indicating that we do not have strong enough evidence to support the claim that more than 25% of teenagers text while driving.