Question

For each of the following hypothesis testing scenarios, indicate whether or not the appropriate hypothesis test would be for a difference in population means. If not, explain why not. Scenario 1: The authors of the paper "Adolescents and MP3 Players: Too Many Risks, Too Few Precautions" (Pediatrics [2009]: e953-e958) studied 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. You would like to determine if there is convincing evidence that the proportion of Dutch boys who listen to music at high volume is greater than this proportion for Dutch girls. Scenario 2: The report "Highest Paying Jobs for \(2009-10\) Bachelor's Degree Graduates" (National Association of Colleges and Employers, February 2010 ) states that the mean yearly salary offer for students graduating with accounting degrees in 2010 is \(\$ 48,722\). A random sample of 50 accounting graduates at a large university resulted in a mean offer of \(\$ 49,850\) and a standard deviation of \(\$ 3,300\). You would like to determine if there is strong support for the claim that the mean salary offer for accounting graduates of this university is higher than the 2010 national average of \(\$ 48,722\). Scenario 3: Each person in a random sample of 228 male teenagers and a random sample of 306 female teenagers was asked how many hours he or she spent online in a typical week (Ipsos, January 25,2006 ). The sample mean and standard deviation were 15.1 hours and 11.4 hours for males and 14.1 and 11.8 for females. You would like to determine if there is convincing evidence that the mean number of hours spent online in a typical week is greater for male teenagers than for female teenagers.

Short answer

Scenario 1 does not require a test for difference in population means, it needs difference in population proportions. Scenarios 2 and 3 require a test for difference in population means.

Step-by-step solution

Scenario 1 Analysis

In this scenario, the test is about the proportion of Dutch boys who listen to music at high volume s compared to Dutch girls. Because we are comparing proportions on a categorical variable - boys who listen to music at a high volume vs girls who listen to music at a high volume - we are looking at testing difference in population proportions, not means. So hypothesis test for difference in population means would not be appropriate for this.

Scenario 2 Analysis

In this scenario, the claim being tested is about the difference in the mean yearly salary offer for students graduating with accounting degrees in 2010. Since the problem is about determining whether the mean salary offer is different from the known national average, and salary being a numerical variable allows us to compute mean, this situation calls for a test of difference in population means.

Scenario 3 Analysis

In this situation, the goal is to see if there is evidence that the mean number of hours spent online in a typical week is greater for male teenagers than for female teenagers. Since the variable under study is number of hours spent online - a numerical value which we can compute a mean for - the appropriate hypothesis test would be a test for a difference in population means.

Key concepts

Difference in Population Means

Understanding the difference in population means is a foundational concept in inferential statistics, particularly when comparing two or more groups. When statisticians refer to the 'population,' they mean the entire group of individuals or items they are interested in studying. The 'mean' is the average measure.

For instance, in Scenario 2, the focus is on assessing whether the mean salary offer for graduates from a specific university differs from the national average. In such cases, a t-test for independent samples might be used if the variance between the two groups is assumed to be equal, and a Welch’s t-test if not. The null hypothesis (\( H_0 \textrm{: } \textmu_1 = \textmu_2 \)) in this scenario posits no difference in mean salary offers, while the alternative hypothesis (\( H_1 \textrm{: } \textmu_1 > \textmu_2 \)) suggests that there is a significant difference. If after conducting the appropriate test the results demonstrate statistical significance, we could reject the null hypothesis, supporting the claim that the mean salary offer at the university is different from the national average.

Scenario 3 also presents a case for examining the difference in means—specifically, the amount of time spent online between male and female teenagers. Here, the null hypothesis would imply no difference in average online hours (\( \textmu_{\text{males}} = \textmu_{\text{females}} \)), while the alternate hypothesis would argue that males spend more time online on average (\( \textmu_{\text{males}} > \textmu_{\text{females}} \)).

Difference in Population Proportions

When dealing with categorical data, particularly where we're interested in the proportion of a certain attribute within two or more groups, we analyze the difference in population proportions. A widely used hypothesis test for this purpose is the two-proportion z-test.

In Scenario 1, researchers are examining whether a higher proportion of Dutch boys listen to music at a high volume compared to Dutch girls, which is a categorical variable (high volume vs not). Therefore, the suitable hypothesis test examines the difference in proportions, not means. The null hypothesis would state that the proportions are equal (\( p_1 = p_2 \textrm{, where } p \textrm{ is the population proportion} \)), and the alternative hypothesis suggests a higher proportion in boys (\( p_{\text{boys}} > p_{\text{girls}} \)). To analyze this, the proportion of each group exhibiting the behavior is computed and then compared using the z-test. If the p-value obtained is lower than the chosen alpha level (often 0.05), then it suggests a statistically significant difference in proportions.

This approach is pivotal in many real-world contexts, such as clinical trials, marketing research, or opinion polling, where understanding differences in proportions can guide decisions and strategies.

Statistical Significance

Statistical significance is a determination about the strength of the evidence against the null hypothesis provided by the data. In simple terms, it evaluates whether the observed differences or effects in a study are likely due to genuine effects rather than random chance.

In hypothesis testing, we calculate a p-value, which represents the probability of obtaining the observed results, or more extreme, assuming that the null hypothesis is true. If this p-value is less than the predetermined significance level (commonly set at 0.05), then we may declare the results statistically significant and reject the null hypothesis.

In Scenarios 2 and 3, determining statistical significance is crucial for making inferences about the population means. For Scenario 1, statistical significance would help establish if the difference in proportions between boys and girls listening to music at a high volume is meaningful and not a product of random variation. It’s important to remember, however, that statistical significance does not imply practical significance; even if the test indicates a significant difference, it must still be evaluated in context to determine its real-world relevance and impact.

Numerical Variable Analysis

Numerical variable analysis deals with variables that take on quantifiable values. In statistical testing, particularly for differences in means, these variables are crucial since they allow for calculations of averages and variances among populations or samples.

Both Scenario 2 and Scenario 3 utilize numerical variable analysis to interpret data. In Scenario 2, the numerical variable is the salary offer extended to graduates, and in Scenario 3, it is the hours spent online by teenagers. Descriptive statistics, like mean and standard deviation, assist in summarizing these numerical variables to draw comparisons. The analysis applies tests such as the t-test to determine if the observed differences in means are unlikely to have occurred under the null hypothesis.

Deep knowledge of numerical variables also helps in assessing assumptions relevant to the chosen statistical tests, such as normality, homogeneity of variance, and independence. Ensuring these conditions are met reinforces the reliability of the hypothesis test conducted on numerical data.