Is the app's reaction time worse than the 250 ms claim?

Question

A software vendor claims its interface responds in a mean of 250 ms. An independent tester runs 16 trials and records a mean of 262 ms with a sample standard deviation of 20 ms. The population standard deviation is unknown.

Test at the 5% level (two-sided) whether the true mean response time differs from 250 ms, and report the t-statistic.

Accepted Answer

Unknown \sigma, so use a one-sample t-test with 16 - 1 = 15 degrees of freedom. Hypotheses: H0: \mu = 250 versus H1: \mu 
eq 250. Standard error: 	ext{SE} = \frac{20}{\sqrt{16}} = \frac{20}{4} = 5 	ext{ ms}. Test statistic: t = \frac{262 - 250}{5} = \frac{12}{5} = 2.4. The two-sided 5% critical value for 15 degrees of freedom is t{0.025,\,15} = 2.131. Since 2.4 2.131, reject H0. The two-sided p-value is about 0.030. Conclusion: there is evidence the interface is slower than the advertised 250 ms. Takeaway. As degrees of freedom grow (15 here), the t-critical value drifts down toward the z …

Is the app's reaction time worse than the 250 ms claim?

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