AP Statistics · Unit 6 · Inference for Categorical Data: Proportions

A scooter-share company's charging docks are advertised to fully charge at least 90% of the fleet's batteries during the standard 3-hour overnight charging window. The fleet operations manager is concerned that the true proportion of batteries reaching full charge is actually lower than advertised, which would leave scooters unavailable for the morning commute. The manager selects a random sample of 150 scooters from the company's fleet of 2,000 scooters and checks each scooter the next morning; 126 of the 150 sampled scooters had reached full charge ($\hat{p} = 0.84$). The manager will use these data to test $H_0: p = 0.90$ versus $H_a: p < 0.90$, where $p$ is the true proportion of all scooters in the fleet whose batteries reach full charge within the 3-hour overnight window.

(a)(i) Describe what a Type I error would mean in the context of this study, and describe a consequence of making this error for the scooter-share company.
(a)(ii) Describe what a Type II error would mean in the context of this study, and describe a consequence of making this error for the scooter-share company.
(b)(i) What is the value of the test statistic for this test? Show your work.
(b)(ii) The test statistic from part (b)(i) corresponds to a p-value of $0.0072$. Using $\alpha = 0.05$, do these data provide convincing statistical evidence that fewer than 90% of the fleet's scooters reach full charge within the 3-hour window? Justify your answer.
(c) The manager believes that a Type II error would cost the company more than a Type I error. For future monthly tests, should the manager use $\alpha = 0.01$ or $\alpha = 0.10$? Explain how your choice affects the probability of a Type II error.