Question

The paper referenced in the previous exercise also reported that when each of the 1,178 students who participated in the study was asked if he or she played video games at least once a day, 271 responded yes. The researchers were interested in using this information to decide if there is convincing evidence that more than \(20 \%\) of students ages 8 to 18 play video games at least once a day.

Short answer

There is convincing evidence to suggest that more than 20% of students ages 8 to 18 play video games at least once a day.

Step-by-step solution

Define the Hypotheses

The null hypothesis (\(H_0\)) is that the proportion of students who play video games at least once a day is equal to 20% (or 0.2). The alternative hypothesis (\(H_1\)) is that the proportion of students who play video games at least once a day is greater than 20% (or 0.2). Mathematically, this can be expressed as follows: \(H_0: p = 0.2\), \(H_1: p > 0.2\)

Perform Calculations

Now, calculate the sample proportion (\(p̂\)) by dividing the number of 'yes' responses by the total number of students. \(p̂ = 271/1178 = 0.23\). To conduct the hypothesis test, we need to calculate the test statistic (z) which is given by \(z = (p̂ - p)/(sqrt((p * (1 - p))/n))\) where \(p\) is the proportion under null hypothesis, \(p̂\) is the sample proportion and \(n\) is the sample size. Substituting the given values, we get \(z = (0.23 - 0.2) / (sqrt((0.2 * (1 - 0.2))/1178)) \approx 2.6\).

Interpret the Result

The z-score tells us how many standard deviations away our sample proportion is from the null hypothesis proportion. The z-value obtained is 2.6 which is quite significantly larger than 1.96 (the critical z-value for a 5% level of significance for a one-tailed test). Therefore, we reject the null hypothesis and conclude that there is convincing evidence that more than 20% of students ages 8 to 18 play video games at least once a day.