EXAMPLE 6.14

Parent income contribution by school type: The conclusion. In Example 6.9, we found that the difference in the average parent current income contribution between undergraduates going to a private college versus public college was $1639. Because the cost of tuition at a private college is typically higher than the cost at a public college,¹⁴ we had a prior expectation that the parental current income contribution would be higher for undergraduates going to a private college. It is appropriate to use a one-sided alternative in this situation. So, our hypotheses are

H₀: the true mean difference is 0

versus

H_a: the difference between the average parent income contribution of undergraduates at a private college and public college is positive

The standard deviation is $428 (again, we defer details regarding this calculation), and the test statistic is

= 3.81

Because only positive differences in parental contributions count against the null hypothesis, the one-sided alternative leads to the calculation of the P-value using the upper tail of the Normal distribution. In Table A, the largest z is 3.49. This means that for z = 3.81, P < 0.0002. Using software, we can be more precise. The P-value is

P = P(Z ≥ 3.81)

= 0.0001

The calculation is illustrated in Figure 6.10. There is about a 1-in-10,000 chance of observing a difference as large or larger than the $1639 in our sample if the true population difference is zero. This P-value tells us that our outcome is extremely rare. We conclude that the null hypothesis must be false. Because the observed difference is positive, here is one way to report the result: “The data clearly show that the average parent income contribution for undergraduates at a private college is larger than the average parent income contribution for undergraduates at a public college (z = 3.81, P = 0.0001).”