1. by hand. Suppose we measured the height of 5000 women and found that the data
ID: 3128839 • Letter: 1
Question
1. by hand. Suppose we measured the height of 5000 women and found that the data were normally distributed with a mean of 66 inches and a standard deviation of 4 inches (note: these are not necessarily accurate numbers for women in the real world). Answer the following questions and show your work: A. What proportion of women can be expected to have heights less than: 54, 61, 68.2, 72 inches? B. What proportion of women can be expected to have heights greater than: 56, 65, 66, 69.4 inches? C. What proportion of women can be expected to have heights between: 55 and 66 inches, 60 and 70 inches? D. How many women (of the 5000) can be expected to have heights between: 63 and 67 inches, 68 and 72 inches? E. What height corresponds to the 43rd percentile? What height corresponds to the 55th percentile? What height corresponds to the 99th percentile? (If necessary, round the numbers, but keep at least two decimal places.)
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True-False questions. Assume that all of the assumptions for correlation and linear regression have been met (including the assumption that the X and Y variables are each normally distributed). In your write-up, just list the sub-question letter (E-J) and whether the statement is True or False – no need to restate the question or to justify your answer.
E. The best-fit regression line to predict Y when you know X will always go through the point ZX = 0 and ZY = 0. F. If a positive correlation exists between X and Y, and the range of X is then greatly restricted, |r| must increase. G. If a positive correlation exists between X and Y, and a new data point is added whose ZX = 3 and ZY = 0, the correlation will decrease. (Note: for G, H, I, and J, assume there are many, many data points in the dataset, so that the introduction of new data points doesn’t change the values of the averages in any meaningful way.) H. If no correlation exists between X and Y, and a new data point is added whose ZX = 2.5 and ZY = 2.5, r will decrease. I. If a positive but imperfect correlation exists between X and Y, and a new data point is added whose ZX = 2.5 and ZY = 2.5, |r| will increase. J. If a negative correlation exists between X and Y, and a new data point is added whose ZX = 2.5 and ZY = 2.5, |r| will decrease.
Explanation / Answer
1. by hand. Suppose we measured the height of 5000 women and found that the data were normally distributed with a mean of 66 inches and a standard deviation of 4 inches (note: these are not necessarily accurate numbers for women in the real world). Answer the following questions and show your work:
A. What proportion of women can be expected to have heights less than: 54, 61, 68.2, 72 inches?
LESS THAN 54:
We first get the z score for the critical value. As z = (x - u) / s, then as
x = critical value = 54
u = mean = 66
s = standard deviation = 4
Thus,
z = (x - u) / s = -3
Thus, using a table/technology, the left tailed area of this is
P(z < -3 ) = 0.001349898 [ANSWER]
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LESS THAN 61:
We first get the z score for the critical value. As z = (x - u) / s, then as
x = critical value = 61
u = mean = 66
s = standard deviation = 4
Thus,
z = (x - u) / s = -1.25
Thus, using a table/technology, the left tailed area of this is
P(z < -1.25 ) = 0.105649774 [ANSWER]
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LESS THAN 68.2:
We first get the z score for the critical value. As z = (x - u) / s, then as
x = critical value = 68.2
u = mean = 66
s = standard deviation = 4
Thus,
z = (x - u) / s = 0.55
Thus, using a table/technology, the left tailed area of this is
P(z < 0.55 ) = 0.708840313 [ANSWER]
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LESS THAN 72:
We first get the z score for the critical value. As z = (x - u) / s, then as
x = critical value = 72
u = mean = 66
s = standard deviation = 4
Thus,
z = (x - u) / s = 1.5
Thus, using a table/technology, the left tailed area of this is
P(z < 1.5 ) = 0.933192799 [ANSWER]
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