A random variable is normally distributed with a mean of 24 and a standard devia

Question

A random variable is normally distributed with a mean of 24 and a standard deviation of 4. If an observation is randomly selected from the distribution, a. What value will be exceeded 10% of the time? b, what value will be exceeded 75% of the time? C. Determine two values of which the smaller has 5% of the values below it and the larger has 5% of the values above it. d. What value will 25% of the observations be below? a. The value 29.2 will be exceeded 10% of the time. Round to one decimal place as needed.) b. The value 21.3 will be exceeded 75% of the time. Round to one decimal place as needed.) c The value has 5% of the values below it and the value (Round to one decimal place as needed.) has 5% of the values above it.

Explanation / Answer

c) P(X < x) = 0.05

or, P((X - mean)/sd < (x - 24)/4) = 0.05

or, P(Z < (x - 24)/4) = 0.05

or, (x - 24)/4 = -1.645

or, x = -1.645 * 4 + 24

or, x = 17.42

P(X > x) = 0.05

or, P((X - mean)/sd > (x - 24)/4) = 0.05

or, P(Z < (x - 24)/4) = 0.95

or, (x - 24)/4 = 1.645

or, x = 1.645 * 4 + 24
or, x = 30.58

d) P(X < x) = 0.25

or, P((X - mean)/sd < (x - 24)/4) = 0.25

or, P(Z < (x - 24)/4) = 0.25

or, (x - 24)/4 = -0.67

or, x = -0.67 * 4 + 24

or, x = 21.32

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