Answer to Question #94427 in Statistics and Probability for christine

Question

Olestra is a fat substitute approved by FDA for use in snack foods. Because there have been anecdotal reports of
gastrointestinal problems associated with olestra consumption, a randomized, double-blind, placebo-controlled
experiment was carried out to compare olestra potato chips to regular potato chips with respect to GI symptoms
(“Gastrointestinal Symptoms Following Consumption of Olestra or Regular Triglyceride Potato Chips,” J. of the
American Medical Association, 1998: 150-152). Among 529 individuals in the TG control group, 17.6% experienced
an adverse GI event, whereas among the 563 individuals in the olestra treatment group, 15.8% experienced such an
event. Carry out a test of hypotheses at the 5% significance level to decide whether the incidence rate of GI problems
for those who consume olestra chips according to the experimental regimen differs from the incidence rate for the TG control treatment.

Answer 1

Let X be the number of individuals in the TG control group.

Given:

$$\hat{p_1}=17.6\%=0.176$$ $$n_1=529$$

$$\hat{p_2}=15.8\%=0.158$$ $$n_2=563$$

$$\alpha=5\%=0.05$$

Claim: $p_1\not=p_2$

The clain is either the null hypothesis or the alternative hypothesis. The null hypothesis states as an equality.

Since the null hypothesis is not the claim the alternative hypothesis is the claim.

$$H_0:p_1=p_2$$ $$H_1:p_1\not=p_2$$

The sample proportion is the number of successes divided by the sample size:

$$x_1=\hat{p_1}n_1=0.176(529)=93.104$$

$$x_2=\hat{p_2}n_2=0.158(563)=88.954$$

$$\hat{p_p}={x_1+x_2 \over n_1+n_2}={93.104+88.954 \over 529+563}\approx0.1667$$

The critical value at the 5% significance level 

$$z^*=\pm1.96$$

Determine the value of the test statistic:

$$z={\hat{p_1}-\hat{p_2} \over \sqrt{\hat{p_p}(1-\hat{p_p})}\sqrt{{1 \over n_1}+{1 \over n_2}} }$$

$$z={0.176-0.158 \over \sqrt{0.1667(1-0.1667)}\sqrt{{1 \over 529}+{1 \over 563}} }\approx0.7976$$

$$-1.96<0.7976<1.96$$

There is not sufficient evidence to support the claim that the incidence rate of GI problems for those who consume olestra chips according to the experimental regimen differs from the incidence rate for the TG control group.