Question

1) Let   x be a continuous random variable that follows a normal distribution with a mean of 321 and a standard deviation of 41.

(a) Find the value of   x > 321 so that the area under the normal curve from 321 to x is 0.2224.

Round your answer to the nearest integer.
The value of   x is_______

(b) Find the value of x so that the area under the normal curve to the right of x is 0.3745.

Round your answer to the nearest integer.
The value of   x is ______

2) A study has shown that 24% of all college textbooks have a price of $80 or higher. It is known that the standard deviation of the prices of all college textbooks is $10.00. Suppose the prices of all college textbooks have a normal distribution. What is the mean price of all college textbooks?

Round your answer to the nearest integer.

μ=

3) Use a table, calculator, or computer to find the specified area under a standard normal curve.

Round your answers to 4 decimal places.

a) More than a z-score of 2.48; area = _____________

b) More than a z-score of 1.7; area =_____________

c) More than a z-score of -0.41; area = _____________

d) More than a z-score of 00; area = _____________

4)

The highway police in a certain state are using aerial surveillance to control speeding on a highway with a posted speed limit of 55 miles per hour. Police officers watch cars from helicopters above a straight segment of this highway that has large marks painted on the pavement at  1-mile intervals. After the police officers observe how long a car takes to cover the mile, a computer estimates that cars speed. Assume that the errors of these estimates are normally distributed with a mean of  0 and a standard deviation of  3.58 miles per hour.

a. The state police chief has directed his officers not to issue a speeding citation unless the aerial units estimate of speed is at least 66 miles per hour. What is the probability that a car travelling at 61 miles per hour or slower will be cited for speeding?

Round your answer to four decimal places.

The probability that a car travelling at 61 miles per hour or slower will be cited for speeding is ______

b. Suppose the chief does not want his officers to cite a car for speeding unless they are 99% sure that it is travelling at 61 miles per hour or faster. What is the minimum estimate of speed at which a car should be cited for speeding?

Round your answer to the nearest integer.

The minimum estimate of speed is __

Answer 1

Solution:

Let   x be a continuous random variable that follows a normal distribution with a mean of 321 and a standard deviation of 41.

a ]

We have to find the value of   x > 321 so that the area under the normal curve from 321 to x is 0.2224.

P(321<Z<x)= 0.2224

P(Z<x)-P(Z<321)=0.2224

P(Z<x)= 0.2224+P(Z<321)

            =0.2224+0.5            …….

            =0.5224

X=323

In EXCEL,

P(Z<321)= NORMDIST(321,321,41,TRUE)= 0.5

P(Z<x)      = F(x)

x= NORMINV(0.5224,321,41)=323.3033

b] we have to find the value of x so that the area under the normal curve to the right of x is 0.3745.

P(Z>x) =0.3745

1-P(Z< x)=0.3745

P(z< x)=1-0.3745

P(Z<x)= 0.6255

X=NORMINV(0.6255,321,41)

X=334.1183=334

2]

A study has shown that 24% of all college textbooks have a price of $80 or higher. σ =10

P(z>80)=0.24

P(z<80)=1-0.24=0.76

Z= (x-mean)/σ

0.76=(80-mean)/10

0.76*10=80-mean

Mean=80-7.6

Mean=72.4

μ=$72.4