Patients with pleural infections (fluid buildup in the chest) were randomly assigned to placebo, the treatment "tPA" (tissue plasminogen activator), the treatment "DNase" (deoxyribonuclease),or a combination of tPA and DNase in a double-blind clinical trial. Some of the patients needed to be given penicillin. The following table shows a cross-classification of the data. Is there evidence of an association between treatment group and use of penicillin?

Here is computer output for a chi-square test to compare the four groups.
X-squared = 0.59, df = 3, p-value = 0.90
(a) State the null and alternative hypotheses in context.
(b) Compute the expected frequency of the Penicillin/ Placebo cell under the null hypothesis.
(c) If α = 0.10, what is your conclusion regarding H0?

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Penicillin No peniin 51 50 48 Total Placebo tPA DNase tPA+DNase Total 12 198 210 49 55 52 51 52