Question

a. About % of the area under the curve of the standard normal distribution is outside the interval z=[−0.3,0.3]z=[-0.3,0.3] (or beyond 0.3 standard deviations of the mean).

b. Assume that z-scores are normally distributed with a mean of 0 and a standard deviation of 1.

If P(−b<z<b)=0.6404P(-b<z<b)=0.6404, find b.

b=

c. Suppose your manager indicates that for a normally distributed data set you are analyzing, your company wants data points between z=−1.6z=-1.6 and z=1.6z=1.6 standard deviations of the mean (or within 1.6 standard deviations of the mean). What percent of the data points will fall in that range?

Answer:  percent (Enter a number between 0 and 100, not 0 and 1 and round to 2 decimal places)

d. A manufacturer knows that their items have a normally distributed lifespan, with a mean of 12.6 years, and standard deviation of 3.1 years.

If you randomly purchase one item, what is the probability it will last longer than 20 years?

e. A company produces steel rods. The lengths of the steel rods are normally distributed with a mean of 224-cm and a standard deviation of 1.8-cm. For shipment, 27 steel rods are bundled together.

Find the probability that the average length of a randomly selected bundle of steel rods is between 223.6-cm and 224.2-cm.
P(223.6-cm < M < 224.2-cm) =

Enter your answer as a number accurate to 4 decimal places. Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.

Answer 1

a.

Now ,

; From standard normal distribution table

b. Now ,

..............(I)

From standard normal distribution table , ...............(II)

From (I) and (II) , we get , b=0.92

c. Now ,

; From standard normal distribution table

d. Let ,

Now ,

; From standard normal distribution table

e. Let , , n=27

Now ,

; From standard normal distribution table