Question

If X is a normal random variable with mean m = 50 and standard

deviation s = 2, find a. Pr( X < 53 ) = b. Pr( 48 < X < 51
)

Answer 1

To find the probabilities, use the standard normal distribution and calculate the corresponding z-scores. For Pr(X < 53), the probability is approximately 0.9332. For Pr(48 < X < 51), the probability is approximately 0.5328.

Step-by-step explanation:

To find the probabilities, we can use the standard normal distribution. We can convert the given values to z-scores using the formula z = (x - mean) / standard deviation.

a. Pr( X < 53 )

To find this probability, we need to calculate the z-score for X = 53. z = (53 - 50) / 2 = 1.5

Then, we can use a standard normal table or a calculator to find the probability corresponding to this z-score.

From the table or calculator, we find that the probability is approximately 0.9332.

Therefore, Pr( X < 53 ) = 0.9332

b. Pr( 48 < X < 51)

To find this probability, we need to calculate the z-scores for X = 48 and X = 51.

For X = 48, z = (48 - 50) / 2 = -1

For X = 51, z = (51 - 50) / 2 = 0.5

Then, we can use a standard normal table or a calculator to find the probabilities corresponding to these z-scores.

From the table or calculator, we find that the probability for z = -1 is approximately 0.1587 and the probability for z = 0.5 is approximately 0.6915.

Therefore, Pr(48 < X < 51) = 0.6915 - 0.1587 = 0.5328.