Question

There is disagreement among health care professionals about whether health care workers should wear finger rings while performing patient-related work. In particular, plain rings are presumed to have little impact on bacterial transmission by hand. Previous studies have shown that bacteria are transmitted by patient contact in about 25% of patient contacts where no ring is worn. Let p denote the proportion of bacterial transmissions when a plain ring is worn. Investigators wish to determine whether the proportion of bacterial transmissions when wearing plain rings is greater than 0.25.

(a) What is the appropriate null hypothesis in this study?

(b) What is the appropriate alternative hypothesis in this study?

(c) In the context of this study, describe a Type I error and a Type II error

Suppose that the study above was performed with a random sample of n = 121 patient contacts where a plain ring was worn, and 40 of these patient contacts resulted in bacterial transmission.

(a) Describe the shape, center, and spread of the sampling distribution of ˆp if the null hypothesis Ho : p = 0.25 were true.

(b) Is there convincing evidence that the null hypothesis is not true, or is ˆp consistent with what you would expect to see if the null hypothesis were true? Carry out a hypothesis test to answer this question

Answer 1

(a) .
(b) .
(c) Type I error -> We conclude that the proportion of bacterial transmissions when wearing plain rings is greater than 0.25 when actually the proportion is equal to 0.25.
Type II error -> We conclude that the proportion of bacterial transmissions when wearing plain rings is not greater than 0.25 and that it is equal to 0.25 when actually the proportion is indeed greater than 0.25.

(a) The shape of the sampling distribution of will be symmetric and bell shaped, since the distribution will be approximately normal. The center of the sampling distribution will be = = = 0.25.
The spread of the sampling distribution will be = = = 0.0394, where, = 0.25 and = 121.