Question

The article referenced in the previous exercise also reported that \(24 \%\) of the males and \(16 \%\) of the females in the 2006 sample reported owning an MP3 player. Suppose that there were the same number of males and females in the sample of 1112 . Do these data provide convincing evidence that the proportion of females that owned an MP3 player in 2006 is smaller than the corresponding proportion of males? Carry out a test using a significance level of 01 .

Short answer

To answer this exercise, calculate the test statistic and p-value from the given data. If the p-value is less than the significance level (0.01), then there is convincing evidence supporting the hypothesis that the proportion of females that owned an MP3 player in 2006 is smaller than that of males. If the p-value is greater than or equal to the significance level, then there is not sufficient evidence to support this claim. The exact answer will depend on the calculated values.

Step-by-step solution

Identify the hypothesis

In this problem, we are testing whether the proportion of females owning an MP3 players is lesser than males. The null hypothesis would be that the proportions are equal, while the alternative hypothesis is that the proportion of females is less than that of males. Hence, \(H_0: p_f = p_m\) (the proportion of females is equal to the proportion of males) and \(H_1: p_f < p_m\) (the proportion of females is less than the proportion of males).

Calculate the test statistic

The test statistic is calculated using the formula \( Z = \frac{ \hat{p_f} - \hat{p_m}}{ \sqrt{ \hat{p} (1- \hat{p}) ( \frac{1}{n_f} + \frac{1}{n_m}) } }\) where \(\hat{p}\) is the pooled sample proportion, calculated as \(\hat{p} = \frac{x_f + x_m}{n_f + n_m}\). \(x_f\), \(x_m\), \(n_f\) and \(n_m\) represent the number of successful outcomes (owning an MP3 player) and total number of trials for females and males respectively. Here, \(n_f = n_m = 1112 / 2 = 556\) and \(x_f = 0.16 \times 556 = 89\), \(x_m = 0.24 \times 556 = 133.44\). Plugging these into the formula will yield the test statistic.

Calculate the p-value

The p-value is the probability of getting the observed statistic, or a more extreme value, if the null hypothesis is true. In this case, as it is a one-sided test, the p-value is the probability of getting this Z score or more extreme (lesser, as per our alternative hypothesis). This can be calculated from standard normal distribution tables or with a calculator that has this functionality.

Interpret the results

If the p-value is less than the significance level (0.01 in this problem), then we reject the null hypothesis and conclude that there is convincing evidence that the proportion of females that owned an MP3 player in 2006 is smaller than the corresponding proportion of males. However, if the p-value is greater than or equal to the significance level, we fail to reject the null hypothesis and conclude that there is not enough evidence to support the claim that the proportion of females that owned an MP3 player in 2006 is smaller than the corresponding proportion of males.