normal distribution

Question

Assume that x has a normal distribution. Find P(3 ≤ x ≤ 6) given that 𝜇 = 4.7 and 𝜎 = 1.6. (Use 4 decimal places.)

Answer 1

To find the probability P(3 ≤ x ≤ 6) for a normally distributed variable x with mean 𝜇 = 4.7 and standard deviation 𝜎 = 1.6, we will use the standard normal distribution (Z-distribution) and the Z-score formula. The Z-score formula is:

Z = (x - 𝜇) / 𝜎

First, we will calculate the Z-scores for x = 3 and x = 6 using the given mean and standard deviation:

Z₃ = (3 - 4.7) / 1.6 = -1.0625

Z₆ = (6 - 4.7) / 1.6 = 0.8125

Now, we will find the probabilities corresponding to these Z-scores using a standard normal (Z) table or a calculator with a built-in cumulative distribution function (CDF) for the standard normal distribution:

P(Z ≤ -1.0625) = 0.1441

P(Z ≤ 0.8125) = 0.7910

To find the probability P(3 ≤ x ≤ 6), we will subtract the smaller probability from the larger one:

P(3 ≤ x ≤ 6) = P(Z ≤ 0.8125) - P(Z ≤ -1.0625) = 0.7910 - 0.1441 = 0.6469

Thus, the probability P(3 ≤ x ≤ 6) for the given normal distribution is approximately 0.6469, or 64.69%.