A study of hospitalized patients who develop pneumonia reported that 1 in 5 (20%) are readmitted to the hospital within 30 days after discharge (“Comparison of Therapist-Directed and Physician- Directed Respiratory Care in COPD Subjects with Acute Pneumonia,” Respiratory Care [2015]: 151–154). The study reported that 15 out of n = 162 hospital patients who had been treated for pneumonia using a respiratory therapist protocol were readmitted to the hospital within 30 days after discharge. These data can be used to decide if the proportion readmitted is less than 0.20.

a. What hypotheses should be tested?

b. Discuss whether the conditions necessary for a large-sample hypothesis test for one proportion are satisfied.

c. The exact binomial test can be used even in cases when the sample size condition for the large-sample test is met. Use the output at the bottom of the page from the Shiny app “Exact Binomial Test for One Proportion” to carry out an exact binomial test for one proportion, using the null hypothesis that the population proportion of subjects who will be readmitted to a hospital within 30 days after following a respiratory therapist protocol for treatment of pneumonia is equal to 1/5 = 0.2.

d. Calculate the P-value for the same hypotheses using the large-sample approach for testing one population proportion. Compare this P-value to the P-value obtained in Part (b). Would the same conclusion be reached in either case? Explain.