Question

Assume a member is selected at random from the population represented by the graph. Find the probability that the member selected at random is from the shaded area of the graph. Assume the variable x is normally distributed. Women Ages 20-34 Total Cholesterol 200 75 241 300 Total cholesterol level (mg/dL) mu equals 180sigma equals 33.1200 less than x less than 241 x A graph titled "Women Ages 20 to 34 Total Cholesterol" has a horizontal x-axis labeled "Total cholesterol level (Milligrams per deciliter)" from 75 to 300. A normal curve labeled mu = 180 and sigma = 33.1 is over the horizontal x-axis. Vertical line segments extend from the horizontal axis to the curve at 200 and 241. The area under the curve between 200 and 241 is shaded and labeled 200 < x < 241. The probability that the member selected at random is from the shaded area of the graph is nothing. ​(Round to four decimal places as​ needed.)

Answer 1

Let the random variable X is defined as

X : Women ages 20-34 Total cholesterol.

X ~ N ( mu = 180 , sigma² = 33.10²)

we have to find the probability that

P ( 200 < X < 241)

by central limit theorem

$Z = \frac{x-\mu }{\sigma }\sim N(0,1)$

$P ( 200 < X < 241) = P \left ( \frac{200-180}{33.1} <\frac{X-\mu }{\sigma }<\frac{241-180}{33.1}\right )$

= P ( 0.6042 < Z< 1.8429)

= P ( Z < 1.8429) - P (Z < 0.6042)

From normal probability table

P ( Z < 1.8429) = 0.9673

P ( Z < 0.6042) = 0.7271

P ( 200 < X < 241) = 0.9673 -0.7271 = 0.2402

Graph :

Since the normal distribution is symmetric about mu.

50% area occured below 180.

Required area occured in right-tailed side.