Question

The article "The Soundtrack of Recklessness: Musical Preferences and Reckless Behavior Among Adolescents" (Journal of Adolescent Research [1992]: \(313-331\) ) described a study whose purpose was to determine whether adolescents who preferred certain types of music reported higher rates of reckless behaviors, such as speeding, drug use, shoplifting, and unprotected sex. Independently chosen random samples were selected from each of four groups of students with different musical preferences at a large high school: (1) acoustic/pop, (2) mainstream rock, $$\begin{array}{ccccccccc} \text { Type of Box } & & & {\text { Compression Strength (Ib) }} & & & & {\text { Sample Mean }} & {\text { Sample SD }} \\ \hline 1 & 655.5 & 788.3 & 734.3 & 721.4 & 679.1 & 699.4 & 713.00 & 46.55 \\ 2 & 789.2 & 772.5 & 786.9 & 686.1 & 732.1 & 774.8 & 756.93 & 40.34 \\ 3 & 737.1 & 639.0 & 696.3 & 671.7 & 717.2 & 727.1 & 698.07 & 37.20 \\ 4 & 535.1 & 628.7 & 542.4 & 559.0 & 586.9 & 520.0 & 562.02 & 39.87 \\ & & & & & & & \overline{\bar{x}} =682.50 & \end{array}$$ (3) hard rock, and (4) heavy metal. Each student in these samples was asked how many times he or she had engaged in various reckless activities during the last year. The following table lists data and summary quantities on driving over \(80 \mathrm{mph}\) that is consistent with summary quantities given in the article (the sample sizes in the article were much larger, but for the purposes of this exercise, we use \(\left.n_{1}=n_{2}=n_{3}=n_{4}=20\right)\) $$\begin{array}{rrrr} \text { Acoustic/Pop } & \text { Mainstream Rock } & \text { Hard Rock } & \text { Heavy Metal } \\ \hline 2 & 3 & 3 & 4 \\ 3 & 2 & 4 & 3 \\ 4 & 1 & 3 & 4 \\ 1 & 2 & 1 & 3 \\ 3 & 3 & 2 & 3 \\ 3 & 4 & 1 & 3 \\ 3 & 3 & 4 & 3 \\ 3 & 2 & 2 & 3 \\ 2 & 4 & 2 & 2 \\ 2 & 4 & 2 & 4 \\ 1 & 4 & 3 & 4 \\ 3 & 4 & 3 & 5 \\ 2 & 2 & 4 & 4 \\ 2 & 3 & 3 & 5 \\ 2 & 2 & 3 & 3 \\ 3 & 2 & 2 & 4 \\ 2 & 2 & 3 & 5 \\ 2 & 3 & 4 & 4 \\ 3 & 1 & 2 & 2 \\ 4 & 3 & 4 & 3 \\ 20 & 20 & 20 & 20 \\ 2.50 & 2.70 & 2.75 & 3.55 \\ .827 & .979 & .967 & .887 \\ 6830 & 0584 & 0351 & 7868 \end{array}$$ Also, \(N=80\), grand total \(=230.0\), and \(\overline{\bar{x}}=230.0 / 80=\) 2.875. Carry out an \(F\) test to determine if these data provide convincing evidence that the true mean number of times driving over 80 mph varies with musical preference.

Short answer

The short answer to the problem depends on the outcome of the F-test. If the P-value is less than 0.05, then we would reject the null hypothesis and conclude that there is evidence supporting that the true mean number of times driving over 80 mph varies with musical preference. If the P-value is greater than 0.05, then we would not reject the null hypothesis and conclude that there is not enough evidence to suggest that the true mean number of times driving over 80 mph varies with musical preference.

Step-by-step solution

Set Up the Hypothesis

Let's set up the hypothesis for this problem. The null hypothesis states that there is no difference among groups. In the context of this problem, the null hypothesis states that all the genre preferences have the same mean number of times driving over 80 mph. This can be written as: \(H_0 : \mu_1 = \mu_2 = \mu_3 = \mu_4\). The alternative hypothesis is that at least one group has a different mean, or \(H_a : \mu\)'s are not all equal.

Compute the Variations

Calculate the total variation, which is the sum of the variance of each individual score from the overall mean, \(\overline{\bar{x}}\). In our problem, the overall mean (grand mean) is already given as 2.875.\nCalculate the within-group variation which is the sum of the variance of each individual score from the mean of its respective group.\nCalculate the between-group variation which is the total variation minus the within group variation.

Compute the Degrees of Freedom

The degrees of freedom for total variation is \(N - 1 = 80 - 1 = 79\) (where N is the total sample size).\nThe degrees of freedom for within-group variation is equal to \(N – k = 80 – 4 = 76\) (where k is the number of groups).\nThe degrees of freedom for between-group variation is equal to \(k - 1 = 4 - 1 = 3\).

Compute the F statistic

Calculate the F statistic which is the ratio of variance between groups and variance within groups. This can be calculated using the formula: \(F = \) (Between-group variation/df\( _{between}\)) / (Within-group variation/df\(_{within}\))

Conclude the Study

Look up the P-value associated with observed value of F in a F-distribution table. If the P-value is less than the chosen significance level (usually 0.05), the null hypothesis that all the genre preferences have the same true mean is rejected, providing evidence that at least one group has a different mean. If the P-value is greater than the significance level, then it is not rejected, showing there's no convincing evidence that at least one group has a different mean.