AP Statistics 3.7: Proportion Test Conclusions

Model Solution and scoring sections

Section 1 — Procedure and hypotheses
Use a one-sample z-test for a population proportion. Let p be the true proportion of students at Karen's high school who use the app for homework at least once per week.
$$H_0:p=0.22,\qquad H_a:p>0.22.$$
An essentially correct section identifies the one-proportion z-test, states both hypotheses with the correct value and direction, and defines the parameter in context.

Section 2 — Conditions and calculations
The sample is a simple random sample of 130 students. Independence is reasonable because 130<0.10(2000). The large-counts condition holds because
$$130(0.22)=28.6\ge10,\qquad130(0.78)=101.4\ge10.$$
Also,
$$\hat p=\frac{38}{130}\approx0.2923,$$$$z=\frac{0.2923-0.22}{\sqrt{0.22(1-0.22)/130}}\approx1.99,\qquad P(Z>1.99)\approx0.023.$$
An essentially correct section checks independence and large counts and reports a correct test statistic and consistent one-sided p-value.

Section 3 — Conclusion
Because 0.023<0.05, reject $H_0$. There is convincing statistical evidence that the population proportion of students at Karen's school who use the app at least weekly for homework is greater than Country W's proportion of 0.22. An essentially correct section compares the p-value with $\alpha$, makes a consistent decision, and gives a nondefinitive conclusion in context.

Overall scoring: three essentially correct sections earn 4; two E and one P earn 3; the official holistic table determines scores 2 and 1 from the remaining E/P combinations.