Question

If X is a normal random variable with mean 26 and S.D. 4. Find the probability that i) 23≤ X ≤ 30 ii) 18 ≤ X ≤ 24

Answer 1

[AI-generated answer — Bot GPT 3.5]

To find the probabilities, we first need to standardize the given intervals using the Z-score formula:

Z = (X - mean) / S.D.

For the first interval:

i) Z1 = (23 - 26) / 4 = -0.75
Z2 = (30 - 26) / 4 = 1.00

Now, we look up the corresponding values in the standard normal distribution table:

P(23 ≤ X ≤ 30) = P(-0.75 ≤ Z ≤ 1.00) = P(Z ≤ 1.00) - P(Z ≤ -0.75)

From the Z-table, we find:
P(Z ≤ 1.00) = 0.8413
P(Z ≤ -0.75) = 0.2266

So, P(23 ≤ X ≤ 30) = 0.8413 - 0.2266 = 0.6147

For the second interval:

ii) Z1 = (18 - 26) / 4 = -2.00
Z2 = (24 - 26) / 4 = -0.50

Now, we calculate the probability:

P(18 ≤ X ≤ 24) = P(-2.00 ≤ Z ≤ -0.50) = P(Z ≤ -0.50) - P(Z ≤ -2.00)

From the Z-table:
P(Z ≤ -0.50) = 0.3085
P(Z ≤ -2.00) = 0.0228

Thus, P(18 ≤ X ≤ 24) = 0.3085 - 0.0228 = 0.2857

Therefore, the probabilities are:
i) P(23 ≤ X ≤ 30) = 0.6147
ii) P(18 ≤ X ≤ 24) = 0.2857