Module 8 Exam Spring 2021

1 Module 8 Exam Statement of Academic Honesty When I submit my answers to this exam, I am declaring that my answers are due to my work, and my work alone. I did not ask for help on any question or problem on this exam from anyone else, including any source (person or other resource) from the internet. I did not request someone else share their work or their answers with me, and I did not share my answers with anyone else. I do understand that I can ask questions of my instructor during the exam. Name: ____ ______________________________________ Confidence Interval for the Difference Between Two Population Proportions, p 1 -p 2 : A confidence interval for the difference between two population proportions, 𝑝𝑝 1 − 𝑝𝑝 2 , has the form: � ( 𝒑𝒑 � 𝟏𝟏 − 𝒑𝒑 � 𝟐𝟐 ) − 𝑬𝑬 , ( 𝒑𝒑 � 𝟏𝟏 − 𝒑𝒑 � 𝟐𝟐 ) + 𝑬𝑬� where 𝑬𝑬 = 𝒛𝒛 𝒄𝒄 𝒔𝒔 , and where the approximate standard error is 𝒔𝒔 = � 𝒑𝒑 � 𝟏𝟏 ( 𝟏𝟏−𝒑𝒑 � 𝟏𝟏 ) 𝒏𝒏 𝟏𝟏 + 𝒑𝒑 � 𝟐𝟐 ( 𝟏𝟏−𝒑𝒑 � 𝟐𝟐 ) 𝒏𝒏 𝟐𝟐 . We need to assume that the sample size will be large enough so that the approximate normality criteria is valid: the number of successes and the number of failures are at least 10 in each group: 𝑛𝑛 1 𝑝𝑝̂ 1 ≥ 10 𝑎𝑎𝑛𝑛𝑎𝑎 𝑛𝑛 1 (1 − 𝑝𝑝̂ 1 ) ≥ 10, 𝑎𝑎𝑛𝑛𝑎𝑎 𝑛𝑛 2 𝑝𝑝̂ 2 ≥ 10 𝑎𝑎𝑛𝑛𝑎𝑎 𝑛𝑛 2 (1 − 𝑝𝑝̂ 2 ) ≥ 10 . The value for z c is based on a Normal distribution: z c = 1.645 for a 90% Confidence Interval; 1.960 for a 95% Confidence Interval; and 2.576 for a 99% Confidence Interval. You may also use the calculator to check the confidence interval: TI: STAT, TESTS; B:2-PropZInt Enter x 1 , n 1 , x 2 , n 2 and the confidence level; calculator gives the interval (Low, High). For Hypothesis Testing: First calculate the pooled proportion 𝒑𝒑 � = 𝑿𝑿 𝟏𝟏 +𝑿𝑿 𝟐𝟐 𝒏𝒏 𝟏𝟏 +𝒏𝒏 𝟐𝟐 , since the null hypothesis is that 𝑝𝑝 1 = 𝑝𝑝 2 . Then for the standard error use 𝝈𝝈 = � 𝒑𝒑 � ( 𝟏𝟏−𝒑𝒑 � ) 𝒏𝒏 𝟏𝟏 + 𝒑𝒑 � ( 𝟏𝟏−𝒑𝒑 � ) 𝒏𝒏 𝟐𝟐 . The z-score for a particular sample result is given by 𝒛𝒛 = 𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒏𝒏𝒄𝒄𝒅𝒅 𝒅𝒅𝒏𝒏 𝒔𝒔𝒔𝒔𝒔𝒔𝒑𝒑𝒔𝒔𝒅𝒅 𝒑𝒑𝒅𝒅𝒑𝒑𝒑𝒑𝒑𝒑𝒅𝒅𝒑𝒑𝒅𝒅𝒑𝒑𝒏𝒏𝒔𝒔 𝝈𝝈 = 𝒑𝒑 � 𝟏𝟏 −𝒑𝒑 � 𝟐𝟐 𝝈𝝈 . The p-value can be calculated from the z-score by using normalcdf(low, high, 0, 1) . Remember that for a two-tail test one must multiply the value from the normalcdf() calculation by 2. • Remember: o If the p-value > significance level, fail to reject the null hypothesis. o If the p-value < significance level, reject the null hypothesis. You may also use the calculator to check the z-score and p-value: TI: STAT, TESTS; 6:2-PropZTest Enter x 1 , n 1 , x 2 , n 2 and the alternative hypothesis; calculator gives the p-value and z-score.

2 1. PEW research surveyed 2561 parents in October 2020 to learn about their concerns regarding remote, hybrid, or in-classroom learning for their children in K-12 schools. • For the 833 low-income parents, 𝑝𝑝̂ 𝐿𝐿𝐿𝐿𝐿𝐿 = 72% of the parents stated that they are concerned or very concerned about their children falling behind in their education. • For the 1311 middle-income parents, 𝑝𝑝̂ 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 = 63% of the parents stated that they are concerned or very concerned about their children falling behind in their education. The difference in sample proportions is 𝑝𝑝̂ 𝐿𝐿𝐿𝐿𝐿𝐿 − 𝑝𝑝̂ 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 = 72% - 63% = 9% = 0.09 . The margin of error for a 95% confidence level for the difference in the two proportions was reported as E = 0.040 = 4.0%. (15 pts) a. What is the 95% confidence interval for the true, population difference between these two groups of parents, p Low – p Medium ? b. Describe, to a non-statistics person, what your confidence interval means for this problem. c. Based on your confidence interval, could the true population proportions of parents who are concerned or very concerned about their children falling behind in their education be equal in these two groups of parents? (Please explain, don’t just say yes or no.)

4 Use the formulas on the front page of this exam. 3. A study reported by the Women’s Health Initiative gives the following values for the occurrence of breast cancer in women. All women in the study were between 50 and 79 years of age. A total of 19,541 women were randomly assigned to a low-fat diet, and another 29,294 were assigned to a control group which ate a normal diet. After 8 years, 655 of the low-fat diet group had developed breast cancer, and 1072 of the women in the control group had developed breast cancer. Status Sample Size Number who subsequently developed breast cancer Low fat diet 19541 655 Normal diet 29294 1072 (37 pts) a. Create both in symbolic form and in a sentence the null hypothesis that should be used to decide if there is a difference between the two diet groups in the proportions of women who developed breast cancer. H 0 : Symbolic: English: b. Create both in symbolic form and in a sentence the alternative hypothesis that should be used to decide if there is a difference between the two diet groups in the proportions of women who developed breast cancer. H 1 : Symbolic: English: c. Calculate the sample proportion, 𝒑𝒑 � 𝑳𝑳 , of women on the low-fat diet who later developed breast cancer, and calculate the sample proportion, 𝒑𝒑 � 𝑵𝑵 , of women on the normal fat diet who later developed breast cancer.

5 d. Calculate the sample difference in proportions, 𝒑𝒑 � 𝑳𝑳 − 𝒑𝒑 � 𝑵𝑵 . e. Calculate the pooled proportion, 𝒑𝒑 � , of women who developed breast cancer. f. Calculate the standard error, σ , of the sampling distribution for differences in sample proportions. g. Calculate the z-score for your sample difference in part d., and use normalcdf() to calculate the p- value for this sample difference. h. Using a significance level of 1% = 0.01, should we reject or fail to reject the null hypothesis? i. Write your conclusion from part h. in a way that a non-statistics person would understand.

7 d. Calculate a 95% confidence interval for the difference in the population proportions, 𝒑𝒑 𝑴𝑴 − 𝒑𝒑 𝑭𝑭 . e. Interpret, in terms a non-statistics person would understand, your confidence interval, explaining what it tells us about the proportions of males and females whose ring finger is longer than their index finger. f. Based on your confidence interval, could the population proportion of all males whose ring finger is longer be equal to the population proportion for all females? (Please explain, don’t just say yes or no.) g. Based on your confidence interval, could the population proportion of all males whose ring finger is longer be 0.10 or 10% larger than the population proportion for all females? (Please explain, don’t just say yes or no.) h. Based on your confidence interval, could the population proportion of all males whose ring finger is longer be 0.10 or 10% smaller than the population proportion for all females? (Please explain, don’t just say yes or no.)