Researchers studying the effect of antibiotic treatment compared to symptomatic treatment for acute sinusitis randomly assigned 166 adults diagnosed with sinusitis into two groups. Participants in the antibiotic group received a 10-day course of an antibiotic, and the rest received symptomatic treatments as a placebo. These pills had the same taste and packaging as the antibiotic. At the end of the 10-day period patients were asked if they experienced improvement in symptoms since the beginning of the study. The distribution of responses is summarized below. Antibiotic Placebo Total Yes 72 60 132 No 13 21 31 Total 85 81 166
Researchers studying the effect of antibiotic treatment compared to symptomatic treatment for acute sinusitis randomly assigned 166 adults diagnosed with sinusitis into two groups. Participants in the antibiotic group received a 10-day course of an antibiotic, and the rest received symptomatic treatments as a placebo. These pills had the same taste and packaging as the antibiotic. At the end of the 10-day period patients were asked if they experienced improvement in symptoms since the beginning of the study. The distribution of responses is summarized below.
| Antibiotic | Placebo | Total | |
| Yes | 72 | 60 | 132 |
| No | 13 | 21 | 31 |
| Total | 85 | 81 | 166 |
A simulation was conducted under the assumpton that the proportion of self-reported improvement with the antibiotic treatment is equal to the proportion of self-reported improvement with the palcebo. 100,000 simulated differences were generated to construct the null distribution shown. The value ˆpa,∼p^a,∼ represents the proportion of self-reported improvement with antibiotic treatment, and ˆpp,∼p^p,∼ is the proportion of self-reported improvement with the placebo treatment.
b) Calculate the observed difference between the proportion of self-reported improvement with antibiotic and proportion of self-reported improvement with placebo.
Round to the nearest ten-thousandth: ˆpa−ˆppp^a-p^p=
c) Estimate the p-value using the figure above.
d) Determine the conclusion of the hypothesis test.
- We should not reject the null hypothesis.
- We should reject the null hypothesis.
The data for proportion of self reported improvement with antibiotics and proportion of self reported improvement with placebo given in the question;
| Antibiotic | Placebo | Total | |
| Yes | 72 | 60 | 132 |
| No | 13 | 21 | 34 |
| Total | 85 | 81 | 166 |
Let is the proportion of self reported improvement with antibiotics and is the proportion of self reported improvement with placebo.
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