Question

Assume X is normally distributed with a mean of 10 and a standard deviation of 2. Determine the following:

(a) ( X< 13)

(b) ( X> 9)

(c) (6 < x< 14)

(d) (2 < x< 4)

(e) (−2

Answer 1

Explanation:

To determine the probabilities associated with these events, we can use the standard normal distribution and apply the z-score formula:

z = (x - μ) / σ

where z is the z-score, x is the value of interest, μ is the mean, and σ is the standard deviation.

(a) P(X < 13):

To find the probability that X is less than 13, we calculate the z-score:

z = (13 - 10) / 2 = 1.5

Using a standard normal distribution table or a statistical calculator, we can find that the probability corresponding to a z-score of 1.5 is approximately 0.9332. Therefore, P(X < 13) ≈ 0.9332.

(b) P(X > 9):

To find the probability that X is greater than 9, we calculate the z-score:

z = (9 - 10) / 2 = -0.5

Using the standard normal distribution table or a statistical calculator, we can find that the probability corresponding to a z-score of -0.5 is approximately 0.3085. However, we are interested in the probability of X being greater than 9, so we subtract this value from 1:

P(X > 9) = 1 - 0.3085 ≈ 0.6915.

(c) P(6 < X < 14):

To find the probability that X falls between 6 and 14, we calculate the z-scores for the lower and upper bounds:

Lower z = (6 - 10) / 2 = -2

Upper z = (14 - 10) / 2 = 2

Using the standard normal distribution table or a statistical calculator, we can find the probabilities associated with these z-scores:

P(Z < -2) ≈ 0.0228

P(Z < 2) ≈ 0.9772

To find the probability between these two z-scores, we subtract the lower probability from the upper probability:

P(6 < X < 14) = P(Z < 2) - P(Z < -2) = 0.9772 - 0.0228 ≈ 0.9544.

(d) P(2 < X < 4):

To find the probability that X falls between 2 and 4, we calculate the z-scores for the lower and upper bounds:

Lower z = (2 - 10) / 2 = -4

Upper z = (4 - 10) / 2 = -3

Using the standard normal distribution table or a statistical calculator, we can find the probabilities associated with these z-scores:

P(Z < -4) ≈ 0.00003167

P(Z < -3) ≈ 0.00134990

To find the probability between these two z-scores, we subtract the lower probability from the upper probability:

P(2 < X < 4) = P(Z < -3) - P(Z < -4) = 0.00134990 - 0.00003167 ≈ 0.00131823.

(e) P(X < -2):

To find the probability that X is less than -2, we calculate the z-score:

z = (-2 - 10) / 2 = -6

Using the standard normal distribution