correlation and regression 1, 5(a,c,d) only 4 problems Cumulative Review Exercis

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correlation and regression
1, 5(a,c,d) only 4 problems

Cumulative Review Exercises Please be aware that some of the following problems may require knowledge of concepts pre- sented in previous chapeen. Effectiveness of Diet. Listed below are Meighte of mbient bdbre aard afier tbe Zone diet. (Data are bursed on reswlts from "Cowwpariaow of the Ankinti, ormish, Weighr Watchers, and Zone Diets for Weight Lorn and Hedrr Disesse Rink Reductiera, "by Danringerer al, ournal of the American Medical Association, Vol. 293 Nov 1. Use the data for Exercises 1-5. 183 212 177 200 155 167 1. Diet clinical Trial: statisties Find the mean and standard deviation of the before after" differences. Diet clinical Trial: z Score Using only the weighs before the diet, identify the highest and to In the context of these sample data, is that highest value an "unusual" weight? Why or why not? Diet Clinical Trial: Hypothesis Test Use a 0.05 significance level to test the daim that det clinical Trial: confidence Interval Construct a 95% confidence interval estimate the mean weight of subjects before the dict, Write a brief statement interpreting the confi dence in 5. Diet clinical Trial: correlation Use the beforelafter weights listed above. a. Test for correlation between the before and after weights. a lf each subject were to weigh exactly the same afer the diet as before, what would be the value of the linear correlation coefficient? c. If all subjects were to lose 5% of their weight from the diet, whal would be the value of the linear correlation coefficient found from the beforelafter weights? d. What do the preceding results suggest about the suitability of correlation as a for test- tool

Explanation / Answer

1. Differences of "before - after" is d = 4 14 -3 1 -4 7 3 4
Mean of the differences = (4 + 14 - 3 + 1 - 4 + 7 + 3 + 4)/8 = 3.25
d - mean = 0.75 10.75 -6.25 -2.25 -7.25 3.75 -0.25 0.75
(d - mean)^2 = 0.5625 115.5625 39.0625 5.0625 52.5625 14.0625 0.0625 0.5625
sum of (d - mean)^2 = 227.5
variance of the differences = 227.5/(8-1) = 32.5
Standard deviation of the differences = sqrt(32.5) = 5.7

5. a)

First we need to find the correlation coefficient between before and after data
before data, x = 183 212 177 209 155 162 167 170
after data, y = 179 198 180 208 159 155 164 166
mean of x = 179.375
mean of y = 176.125
x - mean(x) = 3.625 32.625 -2.375 29.625 -24.375 -17.375 -12.375 -9.375
y - mean(y) = 2.875 21.875 3.875 31.875 -17.125 -21.125 -12.125 -10.125
(x - mean(x))^2 = 13.140625 1064.390625 5.640625 877.640625 594.140625 301.890625 153.140625 87.890625
(y - mean(y))^2 = 8.265625 478.515625 15.015625 1016.015625 293.265625 446.265625 147.015625 102.515625
(x - mean(x))((y - mean(y)) = 10.421875 713.671875 -9.203125 944.296875 417.421875 367.046875 150.046875 94.921875
Sum of (x - mean(x))^2 = 3097.875
Sum of (y - mean(y))^2 = 2506.875
Sum of (x - mean(x))((y - mean(y)) = 2688.625
correlation, r = 2688.625/sqrt(3097.875 * 2506.875) = 0.9648

We use t-test for Correlation test
t = r* sqrt(n-2)/ sqrt(1-r^2)
r = 0.9648
n = 8
degree of freedom = n-2 = 8-2 = 6
t = 0.9648* sqrt(8-2)/ sqrt(1-0.9648^2) = 8.986
p-value for t = 8.986 and df=6 is 0.0001063
As p-value is less than 0.05, we conclude that there is a significant correlation between before and after data.

c) If all subjects loose 5% of their weight from the diet, the after data = (before data) * 0.95
After data = 173.85 201.40 168.15 198.55 147.25 153.90 158.65 161.50
As done in part(a), we calculate the correlation coefficient r between before data and new "after data" as
r = 1

d) The correlation is near 1 in both the cases, if "before" and "after" data is same (as in part(b), where the diet is not effective) or the "after" data is 5% less than "before" data (as in part(c) where the diet is effective), we cannot use the correlation measure as a tool for test the effectiveness of the diet.

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