Part-Time Jobs and GPA Survey
The student newspaper at North Ridge High School surveys 300 students about part-time jobs and grades. Of the 180 students who report having a part-time job, 90 say their grade point average is 3.0 or higher. Of the 120 students who do not have a job, 80 say their GPA is 3.0 or higher. One reporter writes, 'Having a job lowers your GPA,' and another claims the survey 'proves' that working causes lower grades. The editor notices that the survey was conducted by posting a link in the school's online learning system, and that students who work after school may have been less likely to see or answer it. The editor asks the staff to organize the data in a two-way table, calculate conditional probabilities, and evaluate the causal claim before publishing.
1. Build the two-way table and calculate the proportion of students with GPA 3.0 or higher among those with a job and among those without a job. Interpret the difference.
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The table has two rows, job and no job, and two columns, GPA 3.0 or higher and GPA below 3.0. Job holders: 90 high GPA and 90 below, totaling 180. Students without jobs: 80 high GPA and 40 below, totaling 120. Column totals are 170 high GPA and 130 below, totaling 300. Among job holders, the proportion with high GPA is 90/180 = 0.50, or 50%. Among students without jobs, it is 80/120 = 0.667, or about 66.7%. The difference is about 16.7 percentage points. In this sample, students without jobs are more likely to report a GPA of 3.0 or higher. That is an association, but it does not by itself show that a job causes lower grades.
2. A student is selected at random from the survey. Find the probability of having a job or having a GPA of 3.0 or higher. Show the calculation and explain whether the events are mutually exclusive.
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Let J be having a job and G be having a GPA of 3.0 or higher. P(J) = 180/300 = 0.60. P(G) = 170/300 = 0.5667. The overlap, P(J and G), is 90/300 = 0.30. By the addition rule, P(J or G) = P(J) + P(G) - P(J and G) = 0.60 + 0.5667 - 0.30 = 0.8667, or about 86.7%. The events are not mutually exclusive, because 90 students are in both categories. If the events were mutually exclusive, the overlap would be zero and we would not subtract it. The calculation shows that most students in the survey fall into at least one of the two categories.
3. Evaluate the claim that having a job causes lower grades. Identify at least two plausible confounding variables and one sampling concern, and suggest one change to the survey design.
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The survey shows an association between having a job and lower reported GPA, but it cannot establish causation. Students were not randomly assigned to jobs, so other differences between the groups could explain the pattern. Plausible confounders include the number of hours worked per week, study time, motivation or prior academic preparation, grade level, family obligations, and participation in extracurricular activities. A sampling concern is that the survey was posted in the online learning system, so students who work after school, lack reliable internet, or rarely check the system may be underrepresented. GPA is also self-reported and may be inaccurate. A better design would randomly select students from the full enrollment, collect anonymous responses during school time, ask about hours worked, and, with consent, verify GPA from school records. Even then, the study would show association rather than proof of causation.