Question

Use either the critical-value approach or the P-value approach to perform the required hypothesis test. For several years, evidence had been mounting that folic acid reduces major birth defects. A. Czeizel and I. Dudas of the National Institute of Hygiene in Budapest directed a study that provided the strongest evidence to date. Their results were published in the paper “Prevention of the First Occurrence of Neural-Tube Defects by Periconceptional Vitamin Supplementation” (New England Journal of Medicine, Vol. 327(26), p. 1832). For the study, the doctors enrolled women prior to conception and divided them randomly into two groups. One group, consisting of 2701 women, took daily multivitamins containing 0.8 mg of folic acid, the other group, consisting of 2052 women, received only trace elements. Major birth defects occurred in 35 cases when the women took folic acid and in 47 cases when the women did not. a. At the 1% significance level, do the data provide sufficient evidence to conclude that women who take folic acid are at lesser risk of having children with major birth defects? b. Is this study a designed experiment or an observational study? Explain your answer. c. In view of your answers to parts (a) and (b), could you reasonably conclude that taking folic acid causes a reduction in major birth defects? Explain your answer.

Answer 1

Step 1 a) \(\displaystyle{H}_{{{0}}}:{p}_{{{1}}}={p}_{{{2}}}\)
\(\displaystyle{h}_{{{a}}}:{p}_{{{1}}}{<}{p}_{{{2}}}\) The sample proportion is the number of successes divided by the sample size: \(\displaystyle{w}{i}{d}{e}\ \hat{{{p}}}_{{{1}}}={\frac{{{x}_{{{1}}}}}{{{n}_{{{1}}}}}}={\frac{{{35}}}{{{2701}}}}\approx{0.013}\)
\(\displaystyle{w}{i}{d}{e}\ \hat{{{p}}}_{{{2}}}={\frac{{{x}_{{{2}}}}}{{{n}_{{{2}}}}}}={\frac{{{47}}}{{{2052}}}}\approx{0.023}\)
\(\displaystyle{w}{i}{d}{e}\ \hat{{{p}}}_{{{p}}}={\frac{{{x}_{{{1}}}+{x}_{{{2}}}}}{{{n}_{{{1}}}+{n}_{{{2}}}}}}={\frac{{{35}+{47}}}{{{2701}+{2052}}}}={\frac{{{82}}}{{{4753}}}}\approx{0.017}\) Determine the value of the test statistic: \(z=\frac{\widehat{p}_{1}-\widehat{p}_{2}}{\sqrt{\widehat{p}_{p}(1-\widehat{p}_{p})}\sqrt{\frac{1}{n_{1}}+\frac{1}{n_{2}}}}=\frac{0.013-0.023}{\sqrt{0.017(1-0.017)\sqrt{\frac{1}{2701}+\frac{1}{2052}}}}\approx -2.64\) Determine the P-value using table 2: \(\displaystyle{P}={0.0041}\) If the P-value is smaller than the significance level, reject the null hypothesis: \(\displaystyle{P}{<}{0.01}={1}\%\Rightarrow\) Reject \(\displaystyle{H}_{{{0}}}\)

Step 2 b) An experiment deliberately imposes some trearment on individuals in order to observe their responses. An observational study tries to gather information without disturbing the scene they are observing. Designed Experiment c) Yes, because we rejected the null hypothesis in (a). Answer: a. Yes b. Designed experiment c. Yes