Step 1: The distribution of heights of young men is approximately Normal with a

Question

Step 1: The distribution of heights of young men is approximately Normal with a mean of 70 inches and a standard deviation of 2.5 inches. Use the 68–95–99.7 rule to answer the following question. ___ percent of young men are taller than 77.5 inches. (Give your answer to two decimal places.)

Step 2: Fill in the blanks: The heights of the middle 68 percent of young men fall between ____ and ____. (Give your answers to one decimal place.)

Step 3:
>Fill in the blank:

____ percent of young men are shorter than 65 inches. (Give your answer to one decimal place.)

Explanation / Answer

the distribution is given to be normal

therefore the formula to be used = z = (x-mean)/standard deviation

mean = 70

standard deviation = 2.5

a) p(x>77.5) =

For x = 77.5, z = (77.5 - 70) / 2.5 = 3

Hence P(x > 77.5) = P(z > 3) = [total area] - [area to the left of 3]

1 - [area to the left of 3]

now from the z table we will take the value of z score = 3

by the emperical formula the 100-99.7 - 0.3% will be more then 3 standRD DEVIATIONS.

of which half will be above it

therefore p(x>77.5) = 0.3/2 = 0.15%

b) by the emperical formula the 68% distribution will be within 1 standard deviation

therefore the upper limit = 70+2.5 = 72.5

and lower limit = 70 -2.5 = 67.5

c) p(x<65) =

For x = 65, the z-value z = (65 - 70) /2.5 = -2

Hence P(x < 65) = P(z < -2), now from the z table we will take the value of z score = -2

And that value will be the probability required.

by the emperical formula

100-95% = 5% of distribution will be more then 2 standard deviation,

half of which will be less then the mean

p(x<65) = 5%/2 = 2.5%

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