Homework1_Solution

Homework1_Solution - A and B are not independent. (c) P ( B...

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STA 3024: Homework 1 Solution 1. (a) The explanatory variable, education status, is categorical. (b) The response variable, presence of insomnia, is categorical. 2. Population: all students in the Statistics class. Sample: the 10 students randomly selected from the class. Parameter: average score of the entire class (its value is not given). Statistics: average score of the 10 students in the sample, namely 70. 3. Experimental study, since the patients are randomly assigned to receive one of the two drugs. 4. Let H represent a Head, T a Tail. (a) The sample space, S = { HH,HT,TH,TT } , contains all possible outcomes we can get when tossing a coin twice (e.g. HT represents the outcome : Head in 1st toss, Tail in 2nd toss ). (b) Note that A = { HH,HT,TH } and B = { HT,TH } . By definition, A and B are independent if the equality P ( A B ) = P ( A ) × P ( B ) holds. Now, A B = { HT,TH } , P ( A B ) = 2 4 , P ( A ) = 3 4 , and P ( B ) = 2 4 . Hence P ( A B ) = 2 4 6 = P ( A ) × P ( B ) = 3 4 × 2 4 , and we can conclude that
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Unformatted text preview: A and B are not independent. (c) P ( B | A ) = P ( A B ) P ( A ) = 2 4 3 4 = 2 3 . 5. (a) n = 7 and p = 0 . 4. (b) Let X denote how many of the 7 games the team wins. Assuming X has a binomial distribution, the required probability is P ( X = 4) = 7! 4!3! (0 . 4) 4 (1-. 4) (7-4) = 0 . 19. 6. Hypotheses : H : = 1600 vs. H a : < 1600 Test statistic : t = x- s/ n = 1570-1600 120 / 100 =-2 . 5 . The df value is 100-1 = 99. P-value : This is a left-sided test, so the P-value is the probability to the left of (below) the test statistic,-2 . 5. Based on the t distribution with df = 99, the P-value is 0 . 007. Conclusion : Since P-value < . 05, there is sucient evidence to reject H . We conclude that the mean lifetime of all bulbs produced by the company is less than 1600 hours....
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