Question

Suppose that populations of men and women have the following summary statistics for their heights (in
centimeters):
Mean
Standard deviation
Men
HM = 172
M = 7.2
Women
Uw = 162
ow = 5.4
Both distributions are approximately normal. Suppose we randomly select a man and a woman from each
population, and calculate the difference between their heights. We can assume that their heights are
independent
Find the probability that the woman is taller than the man
You may round your answer to two decimal places
IN
P woman taller

Answer 1

Answer:

.13

Step-by-step explanation:

172-162 = 10  

 

We can find the standard deviation of D by adding the variances of the heights and taking the square root:

σ  

D

2

​  

 

σ  

D

2

​  

 

σ  

D

​  

 

​  

 

=σ  

M

2

​  

+σ  

W

2

​  

 

=7.2  

2

+5.4  

2

 

square root of 81

=9

​  

Representing probability with area

When D= M-W =0, their heights are equal. When the man is taller, D is positive, and when the woman is taller, D is negative. Since we know the distribution of the difference D is normally distributed, the probability that the woman is taller than the man can be found by calculating the shaded area below D=0, in the corresponding normal distribution:

A standard normal curve is plotted on a horizontal axis representing D, that goes from negative 17 to 37. The mean, or mu sub D, = 10. The standard deviations, or sigma sub D = 9. The value 0 is marked. The area under the curve to the left of 0 is shaded, representing the probability that the woman is taller. The area to the right represents the probability that the man is taller.

normalcdf:

lower bound: −9999

upper bound: 0

μ=10

σ=9

​  

Answer 2

Using subtraction of normal variables, it is found that there is a 0.1335 = 13.35% probability that the woman is taller than the man.

In a normal distribution with mean $\mu$ and standard deviation $\sigma$, the z-score of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

• It measures how many standard deviations the measure is from the mean.  
• After finding the z-score, we look at the z-score table and find the p-value associated with this z-score, which is the percentile of X.
• When two measures are subtracted, the mean is the subtraction of the means while the standard deviation is the square root of the variances.

In this problem:

• Men have mean weight of 172 cm, women of 162 cm, hence, for the distribution of the differences, the mean is $\mu = 162 - 172 = -10$.

• The standard deviation is the square root of the sum of the variances, hence $\sigma = \sqrt{7.2^2 + 5.4^2} = 9$

The probability of the women being taller is P(X > 0), which is 1 subtracted by the p-value of Z when X = 0.

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{0 - (-10)}{9}$

$Z = 1.11$

$Z = 1.11$ has a p-value of 0.8665.

1 - 0.8665 = 0.1335

0.1335 = 13.35% probability that the woman is taller than the man.

A similar problem is given at https://brainly.com/question/22934264