In a study reported in the September 24, 2007, of Archives of Internal Medicine, German researchers described a study conducted on 1162 German adults suffering from chronic low-back pain. These subjects were randomly assigned to one of three groups: real acupuncture, sham acupuncture, or conventional therapy. Researchers found that $47 \%$ of subjects in the real acupuncture group improved compared to $44 \%$ in the sham acupuncture group and $27 \%$ in the conventional therapy group.
a. What kind of test ( $z-$, $t-$, or chi-square) would be appropriate for testing whether the improvement proportions differ significantly among these three groups? (No explanation is needed.)
b. Report the appropriate degrees of freedom for this test.
c. The $p$-value turns out to be very close to zero. Summarize the conclusion you would draw.

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The Answer of In a study reported in the September 24, 2007, of Archives of Internal Medicine, German researchers described a study conducted on 1162 German adults suffering from chronic low-back pain. These subjects were randomly assigned to one of three groups: real acupuncture, sham acupuncture, or conventional therapy. Researchers found that $47 \%$ of subjects in the real acupuncture group improved compared to $44 \%$ in the sham acupuncture group and $27 \%$ in the conventional therapy group.
a. What kind of test ( $z-$, $t-$, or chi-square) would be appropriate for testing whether the improvement proportions differ significantly among these three groups? (No explanation is needed.)
b. Report the appropriate degrees of freedom for this test.
c. The $p$-value turns out to be very close to zero. Summarize the conclusion you would draw.

In a Study Reported in the September 24, 2007, of Archives