Suppose we are interested in estimating the difference in mean treatment respons

Question

Suppose we are interested in estimating the difference in mean treatment responses, denoted d, between two medical treatments. We plan to collect data X1, . . . , Xn and Y1, . . . , Yn on the responses of treated subjects (Xi) and untreated subjects (Yi). As- suming that the Xi and Yi are unbiased estimates of the treatment responses, we initially assume EX ¯ - EY ¯ = d, and propose a study based on n = 20 since we are satisfied with the power for the setting when the raw effect size is d = EX -EY = 0.7. However, it turns out that the measurements are biased downward by 10% so that E(X ¯ - Y ¯ ) = 0.9d. How many additional data points do we need to collect in order to
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get the same sample size power we would have had with unbiased data?
You can assume that the Xi and the Yi are both independent and identically distributed samples. You can also assume that both variances are known to be 1 (i.e. var(X) = var(Y ) = 1).

Explanation / Answer

(2) A 16 pound weight is attached to the lower end ofa coiled spring which hangs vertically from a fixed support. The weight comes to rest in its equilibrium position, thereby stretching the spring 6 inches. The weight is then placed I foot below equilibrium and given an upward velocity of 1 foot per second. The medium it is in offers a resistance in pounds numerically equal to ptimes the instantaneous velocity in feet per second. (a) Determine the value ofP so that the motion would be critically damped. (b) Determine the equation ofmotion for the value of P found in (a) above. (c) Find the time at which the weight attains its extreme displacement from the equilibrium position. (d) Find the position ofthe weight at this instant.

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