Does the tutoring program lift scores above 500?

Question

The national mean on a standardized test is 500 with a known standard deviation of $\sigma = 100$ . A tutoring company claims its program raises scores. A sample of 64 tutored students averages 520.

Run a one-sided z-test at the 5% level (do scores increase?): report the z-statistic and the decision.

Accepted Answer

Directional claim (scores go up), \sigma known, so use an upper-tail z-test. Hypotheses: H0: \mu = 500 versus H1: \mu 500. Standard error: 	ext{SE} = \frac{100}{\sqrt{64}} = \frac{100}{8} = 12.5. Test statistic: z = \frac{520 - 500}{12.5} = \frac{20}{12.5} = 1.6. The one-sided 5% critical value is z{0.05} = 1.645. Since 1.6 < 1.645, we fail to reject H0. The one-sided p-value is 1 - \Phi(1.6) = 0.0548, just above 0.05. Conclusion: the 20-point gain falls just short of significance; there is not quite enough evidence that the program helps. Takeaway. A result can be encouraging yet not signif…

Does the tutoring program lift scores above 500?

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