Let's break down and solve each part of the question, which is based on a normal distribution.

Given:

• Mean $\mu = 2659$
• Standard deviation $\sigma = 512$

The normal distribution for the calories consumed can be expressed as $X \sim N(\mu = 2659, \sigma = 512)$.

a) Distribution of $X$

The distribution is normally distributed with the mean and standard deviation given: $X \sim N(2659, 512^2)$

b) Probability that the customer consumes less than 2489 calories

To find this probability, we need to calculate the z-score for $X = 2489$: $z = \frac{2489 - 2659}{512} = \frac{-170}{512} \approx -0.3320$

Using the standard normal distribution table, a z-score of -0.3320 corresponds to a probability of about 0.3707. Therefore, the probability that the customer consumes less than 2489 calories is: $P(X < 2489) \approx 0.3707$

c) Proportion of customers who consume over 2802 calories

Again, we first calculate the z-score for $X = 2802$: $z = \frac{2802 - 2659}{512} = \frac{143}{512} \approx 0.2793$

Using the standard normal table, a z-score of 0.2793 corresponds to a probability of about 0.6094. Since we are looking for the proportion who consume more than 2802 calories, we subtract this from 1: $P(X > 2802) = 1 - 0.6094 = 0.3906$

Thus, about 39.06% of the customers consume more than 2802 calories.

d) Calories required to be in the top 2% of consumers (Piggy award)

We need to find the z-score that corresponds to the top 2%, which is the 98th percentile of the normal distribution. From the z-score table, the z-score for the 98th percentile is approximately 2.054.

Now, use the z-score formula to solve for the corresponding calorie value: $X = \mu + z \cdot \sigma = 2659 + 2.054 \cdot 512 = 2659 + 1051.648 \approx 3711.6$

Rounding to the nearest calorie, the fewest number of calories a person must consume to receive the Piggy award is: $X \approx 3712$

Do you have any questions or would you like further details? Here are 5 related questions to expand your understanding:

1. How does the z-score relate to the probability in a normal distribution?
2. What is the probability that a customer consumes between 2400 and 2900 calories?
3. How do you interpret the area under the normal curve in relation to real-world scenarios?
4. If the standard deviation was smaller, how would it affect the Piggy award calculation?
5. Can you calculate the calorie threshold for the top 5% of customers?

Tip: When working with normal distributions, always ensure your z-scores are calculated correctly, as small mistakes can lead to incorrect probabilities.