Question

The authors of the paper "Adolescents and MP3 Players: Too Many Risks, Too Few Precautions" (Pediatrics [2009]:e953-e958) concluded that more boys than girls listen to music at high volumes. This conclusion was based on data from independent random samples of 764 Dutch boys and 748 Dutch girls ages 12 to \(19 .\) Of the boys, 397 reported that they almost always listen to music at a high volume setting. Of the girls, 331 reported listening to music at a high volume setting. Do the sample data support the authors' conclusion that the proportion of Dutch boys who listen to music at high volume is greater than this proportion for Dutch girls? Test the relevant hypotheses using a 0.01 significance level.

Short answer

The sample data supports the authors' conclusion if the calculated p-value is less than or equal to the significance level of 0.01.

Step-by-step solution

State the Null Hypothesis and Alternative Hypothesis

The Null Hypothesis (\(H_0\)) is that the proportion of boys who listen to music at high volume is equal to the proportion of girls. The Alternative Hypothesis (\(H_1\)) is that the proportion of boys who listen to music at high volume is greater than the proportion of girls.

Calculate Sample Proportions

The sample proportion of boys (\(p_1\)) who listen to music at high volume is 397 out of 764 and that of girls (\(p_2\)) is 331 out of 748. So, \(p_1 = \frac{397}{764}\) and \(p_2 = \frac{331}{748}\). Calculate these values.

Calculate Combined Sample Proportion

The combined sample proportion (\(p\)) is the total number of successes (listening to high volume music) divided by the total number of trials (total number of boys and girls). So, \( p = \frac{(397 + 331)}{(764 + 748)}\). Calculate this value.

Test Statistic Calculation

The test statistic (\(Z\)) for testing proportion is calculated using the formula \( Z = \frac{(p_1 - p_2)}{\sqrt{p(1 - p)(\frac{1}{n_1} + \frac{1}{n_2})}}\), where \(n_1\) and \(n_2\) are the sample sizes for boys and girls respectively. Plug in the values and calculate the \(Z\) score.

Determine the P-Value

To determine if the sample data supports the authors' conclusion, compare the p-value to the significance level. If the p-value ≤ 0.01 significance level, then reject the null hypothesis.

Key concepts

Statistical Significance

When conducting hypothesis testing in statistics, researchers seek to determine whether their findings reflect a true effect or whether they are due to chance. This is where the concept of statistical significance comes into play. A result is statistically significant if it is unlikely to have occurred by chance alone, according to a certain threshold, known as the significance level. Common significance levels are 0.05, 0.01, or 0.001, which correspond to a 5%, 1%, or 0.1% chance of rejecting the true null hypothesis (a type I error).

In the context of the exercise, using a 0.01 significance level means that there is only a 1% risk that the conclusion reached is incorrect if the null hypothesis is actually true. This stringent level of significance is chosen to ensure that if there is a claim that more boys than girls listen to music at high volumes, it is backed by strong evidence from the data.

Sample Proportions

Sample proportion represents a fraction of the sample with a particular characteristic. It's especially relevant in studies comparing attributes between two different groups. In the exercise, two sample proportions are calculated, one for Dutch boys (\( p_1 \) for boys) and one for Dutch girls (\( p_2 \) for girls), to compare the rates at which they listen to music at high volumes.

To compute each sample proportion, divide the number of individuals with the characteristic by the total number of individuals in that sample. Here, \( p_1 = \frac{397}{764} \) and \( p_2 = \frac{331}{748} \) represent the proportions of boys and girls, respectively, who listen to music at high volumes. These proportions are crucial in hypothesis testing as they serve as estimators for the true population proportions we are interested in comparing.

P-Value

The p-value is a fundamental concept in the field of hypothesis testing designed to measure the strength of evidence against the null hypothesis. After calculating the test statistic, the p-value tells us the probability of observing a result as extreme as the one obtained (or more extreme), assuming that the null hypothesis is true.

The smaller the p-value, the stronger the evidence against the null hypothesis since it indicates that such an extreme result is less likely to occur simply by chance. If the p-value is less or equal to the significance level (in this case, 0.01), it suggests that the observed data are inconsistent with the null hypothesis, and thus, the null hypothesis should be rejected. In our exercise example, comparing the calculated p-value to the 0.01 threshold will help us draw conclusions about music listening habits among boys and girls.

Null and Alternative Hypotheses

In hypothesis testing, we set up two opposing statements—the null hypothesis (\( H_0 \)) and the alternative hypothesis (\( H_1 \) or \( H_a \)). The null hypothesis represents a position of no effect or no difference, serving as the default assumption. It is what we seek to test against the evidence. The alternative hypothesis, on the other hand, embodies the new claim we're testing for—an effect or a difference in a specific direction (greater than, less than, or not equal).

In our study example, the null hypothesis states that there's no difference in the music-listening habits at high volume between boys and girls (\( p_1 = p_2 \)). The alternative hypothesis suggests that there is indeed a difference, specifically that more boys listen to high volumes than girls (\( p_1 > p_2 \)). The testing process involves using data to determine which hypothesis is more likely to be true.