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# exam2_2010 - Statistics 403 Final Exam 1 We are planning a...

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Statistics 403 Final Exam December 17, 2010 1. We are planning a study in which our goal will be to accurately estimate a treatment eﬀect, expressed as the diﬀerence in mean responses between treated and untreated subjects. Our goal is to have the standard error for our estimate be around 20% of the true value. Based on previous studies, we anticipate the true value to be around 2. We also expect, based on previous research, that the treated subjects’ standard deviation is 1, and the untreated subjects’ standard deviation is 2. Our study will enroll twice as many treated subjects as untreated subjects. (a) What total sample size should we obtain? Solution: The standard error of the estimated treatment eﬀect should be 0 . 2 · 2 = 0 . 4, thus the variance of the estimated treatment eﬀect should be around 0 . 16. The estimated treatment eﬀect is ¯ Y T - ¯ Y U , and the variance of the estimated treatment eﬀect is σ 2 T /n T + σ 2 U /n U = 1 / (2 n ) + 4 /n = 9 / (2 n ) , where n is the number of untreated subjects, and thus n + 2 n = 3 n is the total sample size. Thus we need 9 / (2 n ) = 0 . 16 , so n = 9 / (2 · 0 . 16) 28, so the total sample size should be around 84. (b) Consider the power for testing the null hypothesis of zero treatment eﬀect, using the sample size you found in part (a), and assuming that the eﬀect size is 2, as supposed. Will this power be greater than 80%? Explain your reasoning, but you do not need to obtain a numerical value for the power. Solution: When n = 28, the variance of the estimated treatment eﬀect is 9 / 56. If the actual eﬀect size is 1 . 5, then ET = 2 p 9 / 56 4 . 99 Thus the power is P ( T > 2) = P ( T - ET > 2 - ) = P ( Z > 2 - 4 . 99) = P ( Z > - 3) . We know from the 68-95-99 rule that 99% of the probability of a standard normal distribution is between -3 and 3. Thus P ( Z > - 3) is greater than 0.99. So the power is far greater than 80%. 1

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2. Consider the following descriptions of graphs of y (vertical axis) against x (horizontal axis). Match the descriptions of the graphs to the actual graphs below. Each description matches exactly one graph. Two of the graphs match none of the descriptions. Write the letters a-d in the graphs as appropriate.
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exam2_2010 - Statistics 403 Final Exam 1 We are planning a...

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