1690 HW 9

Part A 1. Life rating in Greece a. The population parameter of interest is the life rating. The value of the point estimate is 25% of 100 randomly sampled Greeks in 2011 (or 25). b. The conditions required for constructing a confidence interval are as follows: i. The sample observations are independent. This condition is met by collecting a random sample of Greeks. ii. At least 10 successes and 10 failures are expected in the sample. This can be assumed based off the population size. c. We are 95% confident that the proportion of Greeks who are “suffering” lies between 20.20% and 29.80%. d. Is the current year result significantly different from 17%? i. The hypotheses can be described as follows:  H 0 : p 1 -p 2 = 0  H A : p 1 – p 2 ≠ 0 ii. The assumptions necessary to conduct the hypothesis test are:  Independent subjects  At least 10 successes and 10 failures are expected in the sample (success-failure condition) iii. The test statistic has been calculated as follows:  z = ^ p − p 0 √ p 0 ( 1 − p 0 ) n = 0.25 − 0.17 √ 0.17 ( 1 − 0.17 ) 100 = 0.08 √ 0.17 ( 0.83 ) 100 = 0.08 √ 0.1411 100 = 0.08 √ 0.001411 = 0.08 0.037563 iv. The STATA Command used to calculate the p-value was: prtesti 100 0.25 0.17, level (95). The resulting p-value was found to be 0.0332. v. There is enough statistical evidence to say that the current year result is significantly different from 17%. e. I would reach the same conclusion because the null value (17) is not contained within the confidence interval leading me to also reject the null hypothesis in both scenarios. 2. Prenatal Vitamins and Autism a. The point estimate for the proportion of mothers of children with autism who did not use prenatal vitamins is 0.44. b. The point estimate for the proportion of mothers of children with typical development who used prenatal vitamins is 0.69. c. The hypotheses to test for a difference of proportions of using prenatal vitamins during the three months before pregnancy between mothers of children with autism and mothers of children with typical development can be described as: i. Ho: p 1 = p 2 ii. HA: p 1 ≠ p 2 d. 90% confidence interval i. The conditions for a confidence interval are:

 Independence within and between groups  Success-Failure Conditions ii. The confidence interval was calculated as: iii. We are 90% confident that the true difference of proportions of using prenatal vitamins during the three months before pregnancy between mothers of children with autism and mothers of children with typical development is between -0.322 and -0.178. e. Difference between two proportions i. The conditions for a difference of two proportions are:  Independent subjects  Success-Failure Conditions ii. ^ p pooled = n 1 ^ p 1 + n 2 ^ p 2 n 1 + n 2 = ( 254 ) ( 0.44 ) + ( 229 ) ( 0.69 ) 254 + 229 = 111.76 + 158.01 483 = 269.77 483 = 0.56 z = ( ^ p 1 √ ^ p pooled ( 1 − ^ p pooled )( 1 n 1 ¿ + 1 n 2 )= ( 0.44 − 0.69 ) − 0 √ ( 0.56 ) ( 1 − 0.56 ) ( 1 254 + 1 229 ) = − 0.25 √ ( 0.56 ) ( 0.44 ) ( 0.00394 + 0 iii. Using STATA, the p-value was computed to be 0.0000. iv. There is enough statistical evidence to suggest that there is a significant difference between the two proportions. f. The title of this article is not appropriate because the results of our statistical test told us to reject our null hypothesis in favor of the alternative. Our null hypothesis was that the difference of proportions was not different – meaning that our alternative hypothesis was that they are different. Additionally, this title can be misleading for the average reader. A better title would be: “Prenatal Vitamin Use Before Pregnancy is not Associated with Autism”. g. Bonus Questions i. The most appropriate statistical test is the Chi-Squared test. The hypotheses can be described as follows:  H0: There is no association between prenatal vitamin use and autism  HA: There is an association between prenatal vitamin use and autism ii. The expected frequencies for each cell have been calculated below. Autism Typical Development Total No Vitamin ( 254 ) ( 181 ) 483 = 95.2 ( 229 )( 181 ) 483 = 85.8 181 Vitamin ( 254 )( 302 ) 483 = 158.8 ( 229 )( 302 ) 483 = 143.2 302

Part B 1. A frequency table for tumor5scale has been generated below. 2. A frequency table for tumorsize_cat showing the frequencies and relative frequencies within each tumor group can be seen below. 3. Is there an association between tumor size group and recurrence? a. The hypotheses can be described as: i. H0: p 1 =p 2 ii. HA: p 1 ≠p 2 b. A hypothesis test for the difference of two proportions will be used to answer the research question. c. STATA Output:

d. Because the p-value is less than the 0.05 significance level, we would reject the null hypothesis. There is enough statistical evidence to suggest that recurrence rates differ by tumor size group. 4. Contingency Table Methods a. The hypotheses can be written as: i. H0: Tumor size group and recurrence are independent (not associated). ii. HA: Tumor size group and recurrence are not independent (are associated). b. The Chi-Squared Test will be used to answer the research question. The assumptions for this test are as follows: i. Independent subjects ii. All expected counts ≥10 c. STATA Output: d. There exists enough statistical evidence to suggest that recurrence rates and tumor size groups are not independent (are dependent). 5. The results from B-3 and B-4 are similar in that they yield the same result based on slightly different hypotheses. The hypotheses differ in appearance and are essentially opposite of each other. Additionally, the normality assumptions for by tests are the same. The Chi- squared test in part B-4 requires an additional step that was not required in part B-3.