A random variable is normally distributed with a mean of 25 and a standard deviation of 5. If an observation is randomly selected from the distribution. Required:
a. What value will be exceeded 75% of the time?
b. What value will be exceeded 10% of the time?
c. What value will be exceeded 85% of the time?
d. Determine two values of which the smallest has 25% of the values below it and the largest has 25% of the values above it.

a. What value will be exceeded 75% of the time?

= 21.63

b. What value will be exceeded 10% of the time?

= 31.41

c. What value will be exceeded 85% of the time?

= 19.82

d. Determine two values of which the smallest has 25% of the values below it and the largest has 25% of the values above it.

The two values are: 21.63 and 28.37

Step-by-step explanation:

We solve using the z score formula:

z = (x-μ)/σ, where

x is the raw score

μ is the population mean

σ is the population standard deviation.

A random variable is normally distributed with a mean of 25 and a standard deviation of 5. If an observation is randomly selected from the distribution.

Required:

a. What value will be exceeded 75% of the time?

We are to find the value x

Exceeded by 75% = 100 - 75 = 25

We find the z score of 25% percentile

= -0.674

Hence:

-0.674 = x - 25/5

Cross Multiply

-0.674 × 5 = x - 25

-3.37 = x - 25

x = 25 - 3.37

x = 21.63

b. What value will be exceeded 10% of the time?

We are to find the value x

Exceeded by 90% = 100 - 90 = 10

We find the z score of 90% percentile

= 1.282

Hence:

1.282 = x - 25/5

Cross Multiply

1.282 × 5 = x - 25

6.41 = x - 25

x = 6.41 + 25

x = 31.41

c. What value will be exceeded 85% of the time?

We are to find the value x

Exceeded by 85% = 100 - 85 = 15

We find the z score of 15% percentile

= -1.036

Hence:

-1.036 = x - 25/5

Cross Multiply

-1.036 × 5 = x - 25

-5.18 = x - 25

x = 25 - 5.18

x = 19.82

d. Determine two values of which the smallest has 25% of the values below it and the largest has 25% of the values above it.

This means their z score is 25% below the mean and 25% above the mean

We are to find the value x

We find the z score of 25 % and 75% percentile

= - 0.674 and 0.674

Hence:

For -0.674

-0.674 = x - 25/5

Cross Multiply

-0.674 × 5 = x - 25

-3.37 = x - 25

x = 25 - 3.37

x = 21.63

For 0.674

0.674 = x - 25/5

Cross Multiply

0.674 × 5 = x - 25

3.37 = x - 25

x = 25 + 3.37

x = 28.37

Therefore, the two values are: 21.63 and 28.37

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step-by-step explanation: