If X is a normal random variable with mean 60, and a standard deviation of 2, find:

A: P (X<58)
B: P(X>63)
C: P(57<X<64)

Calculate Z scores for the various values.

Z = (X - μ )/SD

Look up Z scores in table in the back of your statistics book labeled something like "areas under normal distribution" to find the probabilities.

I hope this helps. Thanks for asking.

To find the probabilities in this case, we can use the standard normal distribution table or a calculator that gives us the cumulative distribution function (CDF) of the normal distribution.

The standard normal distribution has a mean of 0 and a standard deviation of 1. In order to use the standard normal distribution to solve these problems, we need to convert the given values of X to their corresponding z-scores using the formula:

z = (x - mean) / standard deviation

Let's calculate the probabilities step by step:

A: P(X < 58)
First, we convert 58 to a z-score using the formula:
z = (58 - 60) / 2 = -1

Now, we look up the probability corresponding to a z-score of -1 from the standard normal distribution table or use a calculator. The value is approximately 0.1587.

Therefore, P(X < 58) ≈ 0.1587

B: P(X > 63)
Similarly, we convert 63 to a z-score:
z = (63 - 60) / 2 = 1.5

Now, we need to find the probability corresponding to a z-score of 1.5. Since we want P(X > 63) which is the complement of P(X ≤ 63), we subtract the probability corresponding to a z-score of 1.5 from 1.
P(X > 63) = 1 - P(X ≤ 63)

Looking up the probability corresponding to a z-score of 1.5, we find it to be approximately 0.9332.

Therefore, P(X > 63) ≈ 1 - 0.9332 ≈ 0.0668

C: P(57 < X < 64)
To find this probability, we need to calculate the z-scores for both 57 and 64:
z1 = (57 - 60) / 2 = -1.5
z2 = (64 - 60) / 2 = 2

Now, we find the probabilities corresponding to each z-score:
P(X < 57) ≈ P(Z < -1.5) (using the standard normal distribution table or a calculator)
P(X < 57) ≈ 1 - P(Z ≤ -1.5) (since we want the complement)

P(X < 57) ≈ 1 - 0.0668 ≈ 0.9332

P(X < 64) ≈ P(Z < 2)
P(X < 64) ≈ 0.9772 (from the standard normal distribution table or a calculator)

To find P(57 < X < 64), we subtract P(X < 57) from P(X < 64):
P(57 < X < 64) = P(X < 64) - P(X < 57)

P(57 < X < 64) ≈ 0.9772 - 0.9332 ≈ 0.044