A normal population has mean μ =33 and standard deviation σ =7.(a) What proportion of the population is between 18 and 29?(b) What is the probability that a randomly chosen value will be between 26 and 36?

To answer these questions, we need to convert the given values into z-scores and then use the standard normal distribution (also known as the z-distribution) to find the proportions and probabilities.

(a) To find the proportion of the population between 18 and 29, we first convert these values into z-scores. The z-score is calculated as:

Z = (X - μ) / σ

where X is the value from the population, μ is the mean of the population, and σ is the standard deviation of the population.

For X = 18, the z-score is:

Z1 = (18 - 33) / 7 = -2.14

For X = 29, the z-score is:

Z2 = (29 - 33) / 7 = -0.57

We then look up these z-scores in the standard normal distribution table or use a calculator that can compute probabilities for the standard normal distribution. The proportion of the population between 18 and 29 is the probability that the z-score is between -2.14 and -0.57, which is P(-2.14 < Z < -0.57).

(b) To find the probability that a randomly chosen value will be between 26 and 36, we again convert these values into z-scores.

For X = 26, the z-score is:

Z1 = (26 - 33) / 7 = -1

For X = 36, the z-score is:

Z2 = (36 - 33) / 7 = 0.43

The probability that a randomly chosen value will be between 26 and 36 is the probability that the z-score is between -1 and 0.43, which is P(-1 < Z < 0.43).

Note: The actual proportions and probabilities depend on the specific values in the standard normal distribution table or the calculator used.