To solve these questions, let's break down each part of the problem:

Given:

• Mean $\mu = 35$
• Standard deviation $\sigma = 6$

We are working with a normal distribution, so we can use the Z-score formula: $Z = \frac{X - \mu}{\sigma}$

Part (a): Proportion of the population between 21 and 31

1. Find the Z-scores for $X = 21$ and $X = 31$.

   • For $X = 21$: $Z = \frac{21 - 35}{6} = \frac{-14}{6} = -2.3333$
   • For $X = 31$: $Z = \frac{31 - 35}{6} = \frac{-4}{6} = -0.6667$

2. Use a Z-table or standard normal distribution calculator to find the probabilities corresponding to these Z-scores:

   • $P(Z < -2.3333) \approx 0.0099$
   • $P(Z < -0.6667) \approx 0.2525$

3. The proportion of the population between 21 and 31 is: $P(-2.3333 < Z < -0.6667) = 0.2525 - 0.0099 = 0.2426$

Part (b): Probability that a randomly chosen value will be between 28 and 38

1. Find the Z-scores for $X = 28$ and $X = 38$.

   • For $X = 28$: $Z = \frac{28 - 35}{6} = \frac{-7}{6} = -1.1667$
   • For $X = 38$: $Z = \frac{38 - 35}{6} = \frac{3}{6} = 0.5$

2. Find the probabilities corresponding to these Z-scores:

   • $P(Z < -1.1667) \approx 0.1210$
   • $P(Z < 0.5) \approx 0.6915$

3. The probability of a randomly chosen value being between 28 and 38 is: $P(-1.1667 < Z < 0.5) = 0.6915 - 0.1210 = 0.5705$

Final Answers

1. The proportion of the population between 21 and 31 is approximately 0.2426.
2. The probability that a randomly chosen value will be between 28 and 38 is approximately 0.5705.

Would you like more detailed explanations on any step?

Here are 5 related questions to explore further:

1. How would the answers change if the standard deviation were different?
2. What is the probability that a value will be less than 28 in this distribution?
3. How do we interpret a Z-score in terms of probability?
4. How do the properties of the normal distribution affect these probabilities?
5. How can these calculations be applied in real-world scenarios?

Tip: When working with normal distributions, calculating Z-scores helps translate values into a common scale, making it easier to find probabilities using standard tables or calculators.