Gastric freezing was once a recommended treatment for ulcers in the upper intestine. The use of gastric freezing stopped after experiments showed it had no effect. One randomized comparative experiment found that $28$of the $82$gastric-freezing patients improved, while $30$of the $78$patients in the placebo group improved. We can test the hypothesis of “no difference” in the effectiveness of the treatments in two ways: with a two-sample z test or with a chi-square test.

(a) Minitab output for a chi-square test is shown below. State appropriate hypotheses and interpret the P-value in context. What conclusion would you draw?

Chi-Square Test: Gastric freezing, Placebo Expected counts are printed below observed counts Chi-Square contributions are printed below expected counts

(b) Minitab output for a two-sample z test is shown below. Explain how these results are consistent with the test in part (a).

The given data is

The null hypothesis states that there is no difference between the two groups:

$H_0$: There is no difference in the improvement rates for the two treatments.

The alternative hypothesis states that there is a difference between the two groups.

$H_a$: There is a difference in the improvement rates for the two treatments.

The P-value is given in the output as:

$P=0.570=57.0\%$

The P-value is the probability of obtaining the value of the test statistic, or a value more extreme, if the null hypothesis is true.

If there is no association between the variables, then we have a probability of $57.0\%$of obtaining a similar or more extreme sample.

The data given is

Find the hypothesis

$H_0:p_1-p_2=0$

$H_0:p_1-p_2\ne 0$

The P-value is given in the output as:

$P=0.570=57.0\%$

We can deduce from this P-value that there is no difference in the population proportions, which is the case if the variables are unrelated.

As a result, the two-sample z-test and the chi-square test are equivalent, as seen by the similar P-values.