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BST 611 (Beasley) Homework 2 (100 points) Use the following Sample Data for Questions 1 – 5. Male Female Honda 60 70 Toyota 58 46 Nissan 84 56 Mazda 46 26 Mitsubishi 32 24 1. What is the Marginal Probability of randomly selecting a female from this sample of car owners? P(F) = (2 points) 2. What is the Marginal Probability of randomly selecting a Honda owner from this sample of car owners? P(Honda) = (2 points) 3. What is the Conditional Probability of randomly selecting a Nissan owner given that she is female from this sample of car buyers? P(Nissan | F) = (2 points) 4. What is the Joint (Intersection) Probability of randomly selecting a Male Mazda owner from this sample of car buyers? P(M ∩ Mazda) = (2 points) 5. What is the Joint (Union) Probability of randomly selecting a Male or a Mitsubishi owner from this sample of car buyers? P(M U Mitsubishi) = (2 points) 6. Suppose flipping an unbiased coin four times. List all possible outcomes (show the sample space). (5 points) 7. What’s the Joint Probability of flipping 4 consecutive Heads? P(H = 4) = (2 points) 8. What is the frequency distribution for the Sum of Number of Heads? (4 points) Number of Heads 0 Heads 1 Head 2 Heads 3 Heads 4 Heads Frequency 9. What is the Probability of flipping 3 or more Heads in 4 flips? P(H ≥ 3) = (4 points) 1 BST 611 (Beasley) Homework 2 (100 points) 10. Suppose rolling a single (N=1) unbiased six-sided die. Calculate the Mean, Variance, and Standard Deviation from the following frequency distribution. 4. You can do this by hand or use either SPSS (Analyze-Descriptive Statistics-Explore); JMP (Analyze-Distribution); or SAS (PROC UNIVARIATE; PROC BOXPLOT). (4 points) Roll 1 2 3 4 5 6 Freqeuncy 1 1 1 1 1 1 Y = sY2 = sY = 11. Take the data in BST611Assn2-2rolls.xls and Import it into SPSS, JMP, or SAS. 4. Use either SPSS (Analyze-Descriptive Statistics-Explore); JMP (Analyze-Distribution); or SAS (PROC UNIVARIATE; PROC BOXPLOT) to report the following statistics. (12 points) Roll 1 Roll 2 Sum Mean Y1 = Y2 = YSUM = YY = s12 = s22 = 2 sSUM = sY2 = s1 = s2 = sSUM = sY = 12. Using the concept of the Central Limit Theorem, describe how the Means, Variances, and Standard Deviations in Question 11 relate the Mean, Variance, and Standard Deviation in Question 10. (5 points) 13. Using the concept of the Central Limit Theorem, discuss the shape of the distribution of the Mean of Rolls as compared to the shape of the distribution of a Single Roll. (5 points) 14. Use either SPSS (Analyze-Correlate-Bivariate); JMP (Analyze-Multivariate MethodsMultivariate); or SAS (PROC CORR) to compute the Pearson correlation between Rolls 1 and 2. (2 points) Pearson r12 = ___________ 15. What does this correlation (r12) imply? (2 points) 16. Using the concept of the Central Limit Theorem, discuss the Mean, Variance, Standard Deviation, and the shape of the distribution of the Mean of Rolls for rolling N = 16 dice simultaneously. (5 points) 17. If you were to roll N = 16 dice simultaneously, what is the Probability that the Mean Roll would be equal to or exceed 5.375 (sum of 86 or greater); P( Y ≥ 5.375) = (4 points) 2 BST 611 (Beasley) Homework 2 (100 points) 18. Suppose the Distribution of GRE-Quantitative (Q) scores is Normal with a = 500 with = 100. 18.1. What is probability of randomly selecting a person with a score greater than or equal to 625? P(Q ≥ 625) = (4 points) 18.2. What is probability of randomly selecting N=10 people with a Mean score greater than or equal to 525? P( Q ≥ 525) = (4 points) 19. Frattola et al. (2000, Hypertension, 36, 622-628) found the Standard Deviation of 24-hour Diastolic Blood Pressure (DSP) among Diabetics to be 12. The average 24-hour DSP was 76. Assume these values are representative of the population parameters (Y = 76; Y = 12) and the shape the DSP distribution is Normal. Now suppose N = 6 patients were given Lacidipine. What is the probability that these patients will have a mean DSP of 70 or less? P( Y ≤ 70 | [Y = 76; Y = 12]) = (7 points) 20. Suppose a similar situation with the DSP among Diabetics being Normally Distributed with a Standard Deviation of 24-hour DSP among Diabetics to be 12 (Y = 12). Suppose researchers randomly assigned nC = 4 patients to a control condition given a placebo and nT = 6 patients to a treatment condition given Lacidipine. The results showed that the Control group had a Mean DSP of YC = 75 and the Treatment group had a Mean DSP of YT = 71. The assumption is that population means for these groups are expected to be equal if there is No Treatment effect for Lacidipine. If the population means are equal this implies that the difference in population means will be zero [(T - C) = D = 0]. What is the probability that a Mean Difference this extreme or larger occurred by chance assuming No Treatment effects? P( YD ≥ 4 | [D = 0; Y = 12]) = (7 points) 21. Suppose a coin was flipped 40 times, assuming an unbiased coin, what is the probability of get 30 or more Heads? P(H ≥ 30 | [p = .50]) = (7 points) 22. Suppose a researcher was interested in a smoking cessation treatment. Suppose the researcher randomly assigned nC = 8 patients to a control condition given a placebo and nT = 10 patients to a treatment condition given nicotine patches. The assumption is that population proportions of quitting for these groups are expected to be equal if there is No Treatment effect for the nicotine patches. After 4 weeks, the results showed that the Control group had 2 people quit and the Treatment group had 6 people quit. What is the probability that a Proportional Difference this extreme or larger occurred by chance assuming No Treatment effects? P( p̂D ≥ x? | [pD = 0]) = (7 points) 3