Question

QUESTION 13 Using the information below, answer the next 3 questions: The percent of fat calories that a person in America consumes each day is normally distributed with a mean of about 36 and a standard deviation of about ten. Suppose that 16 individuals are randomly chosen. (a) For the group of 16, find the probability that the average percent of fat calories consumed is more than thirty-five (Round to 3 decimal places) QUESTION 14 (b) Find the first quartile for the percent of fat calories. QUESTION 15 (c) Find the first quartile for the average percent of fat calories.

Answer 1

Solution :

mean = $\mu$ = 36

standard deviation = $\sigma$ = 10

n = 16

$\mu$_{$\bar x$} =  $\mu$ = 36

$\sigma$_{$\bar x$} = $\sigma$ / $\sqrt$ n = 10 / $\sqrt$ 16 = 2.5

a) P($\bar x$ > 35) = 1 - P($\bar x$ < 35)

= 1 - P[($\bar x$ - $\mu$ _{$\bar x$} ) / $\sigma$ _{$\bar x$} < (35 - 36) / 2.5]

= 1 - P(z < -0.40)

Using z table,    

= 1 - 0.345

= 0.655

b) The z dist'n First quartile is,

P(Z < z) = 25%

= P(Z < z) = 0.25  

= P(Z < -0.6745 ) = 0.25

z = -0.6745

Using z-score formula,

x = z * $\sigma$ + $\mu$

x = -0.6745 * 10 + 36

x = 29.26

First quartile =Q1 = 29.26

c) Using z-score formula  

$\bar x$ = z * $\sigma$ _{$\bar x$}+ $\mu$ _{$\bar x$}

$\bar x$ = -0.6745 * 2.5 + 36

$\bar x$ = 34.31

First quartile =Q1 = 34.31