If weights of men and women can be assumed as Gaussian random variables with the
ID: 3155414 • Letter: I
Question
If weights of men and women can be assumed as Gaussian random variables with the following mean and standar d deviations (in kg) If a man is picked at random, what is the probability he will weigh more than 100kg If a main amd a women are picked at random, what is the probability they will both weigh Dess than 70kg If a woman is picked among the population of all women who weighs less than 60kg, what is the probability she will weigh less than 60kg but more than 50kg '] If 20 men are picked randomly and their weights are averaged, what is the probability the average will be less than 90kg Use the appropriate table in the Appendix of the textbook as neededExplanation / Answer
i.
We first get the z score for the critical value. As z = (x - u) / s, then as
x = critical value = 100
u = mean = 88
s = standard deviation = 11
Thus,
z = (x - u) / s = 1.09
Thus, using a table/technology, the right tailed area of this is
P(z > 1.09 ) = 0.1379 [ANSWER]
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ii.
MAN:
We first get the z score for the critical value. As z = (x - u) / s, then as
x = critical value = 70
u = mean = 88
s = standard deviation = 11
Thus,
z = (x - u) / s = -1.64
Thus, using a table/technology, the left tailed area of this is
P(z < -1.64 ) = 0.0505
WOMAN:
We first get the z score for the critical value. As z = (x - u) / s, then as
x = critical value = 70
u = mean = 75
s = standard deviation = 9
Thus,
z = (x - u) / s = -0.56
Thus, using a table/technology, the left tailed area of this is
P(z < -0.56 ) = 0.2877
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Hence,
P(both less than 70) = 0.0505*0.2877 = 0.01452885 [ANSWER]
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iii.
For P(x<60):
We first get the z score for the critical value. As z = (x - u) / s, then as
x = critical value = 60
u = mean = 75
s = standard deviation = 9
Thus,
z = (x - u) / s = -1.67
Thus, using a table/technology, the left tailed area of this is
P(z < -1.67 ) = 0.0475
For P(50<x<60):
We first get the z score for the two values. As z = (x - u) / s, then as
x1 = lower bound = 50
x2 = upper bound = 60
u = mean = 75
s = standard deviation = 9
Thus, the two z scores are
z1 = lower z score = (x1 - u)/s = -2.78
z2 = upper z score = (x2 - u) / s = -1.67
Using table/technology, the left tailed areas between these z scores is
P(z < z1) = 0.0027
P(z < z2) = 0.0475
Thus, the area between them, by subtracting these areas, is
P(z1 < z < z2) = 0.0448
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hence,
P(50<x<60|x<60) = P(50<x<60)/P(x<60) = 0.0448/0.0475 = 0.943157895 [ANSWER]
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iv.
Here, the standard error is
SE = s/sqrt(n) = 11/sqrt(20) = 2.459674775.
We first get the z score for the critical value. As z = (x - u) / s, then as
x = critical value = 90
u = mean = 88
SE = standard deviation = 2.459674775
Thus,
z = (x - u) / SE = 0.81
Thus, using a table/technology, the left tailed area of this is
P(z < 0.81 ) = 0.791 [ANSWER]
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