Assume the random variable X is normally distributed with mean mu=50 and standard deviation sigma=7. Compute the probability. P(X>39)

Question

Assume the random variable X is normally distributed with mean

mu equals 50μ=50

and standard deviation

sigma equals 7σ=7.

Compute the probability. Be sure to draw a normal curve with the area corresponding to the probability shaded.

Upper P left parenthesis Upper X greater than 39 right parenthesisP(X>39)

Suppose that a random variable X is normally distributed with mean

muμ

and standard deviation

sigmaσ.

The area below the normal curve represents a proportion or probability.

To help visualize the area of​ interest, start by drawing a standard normal curve. Label the given values for X and mu .and μ.

3939 5050       A normal curve has a horizontal axis with two labeled coordinates, 39 and 50. The curve's peak is near the top of the graph at horizontal coordinate 50. Two vertical line segments run from the horizontal axis to the curve at 39 and 50.

Now shade the area of interest. Which of the following is​ correct?

A.

3939

5050

•  
•  
•  

A normal curve has a horizontal axis with two labeled coordinates, 39 and 50. The curve's peak is near the top of the graph at horizontal coordinate 50. Two vertical line segments run from the horizontal axis to the curve at 39 and 50. The area under the curve to the right of the vertical line segment at 39 is shaded.Your answer is correct.

B.

3939

5050

•  
•  
•  

A normal curve has a horizontal axis with two labeled coordinates, 39 and 50. The curve's peak is near the top of the graph at horizontal coordinate 50. Two vertical line segments run from the horizontal axis to the curve at 39 and 50. The area under the curve to the left of the vertical line segment at 39 is shaded.

C.

3939

5050

•  
•  
•  

A normal curve has a horizontal axis with two labeled coordinates, 39 and 50. The curve's peak is near the top of the graph at horizontal coordinate 50. Two vertical line segments run from the horizontal axis to the curve at 39 and 50. The area under the curve between the vertical line segments is shaded.

Upper P left parenthesis Upper X greater than 39 right parenthesisP(X>39)

is the area of the shaded region in this graph. Use technology to find the area of the shaded region. Use the Tech Help button for further assistance.The area is

1.5711.571.

​(Round to four decimal places as​ needed.)

Answer 1

Z score=(39-50)/7=-11/7=-1.57.

This Z score corresponds to a probability of 0.0582, but this is the area to the left of Z=-1.57, corresponding to P(X<39). The total area under the normal distribution curve is 1. The area to the right of Z=-1.57 is 1-0.0582=0.9418, which is P(X>39). Note that the probability corresponding to a Z score of 1.57 is also 0.9418, the area to the left of Z=1.57, because of symmetry. P(-1.57)=1-P(1.57).