Let's solve each part step-by-step using the information provided:

Given Data:

• Mean ($\mu$) = 2762
• Standard Deviation ($\sigma$) = 609

Part (a): Distribution of $X$

Since the amount of calories consumed by customers follows a normal distribution: $X \sim N(2762, 609^2)$ Thus, $X$ is normally distributed with mean 2762 and variance $609^2$.

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Part (b): Probability that the customer consumes less than 2425 calories

To find $P(X < 2425)$:

1. Calculate the z-score: $z = \frac{X - \mu}{\sigma} = \frac{2425 - 2762}{609} = -0.5534$
2. Use the z-score to find the probability from the standard normal distribution table: $P(X < 2425) = P(Z < -0.5534) \approx 0.2894$

So, the probability that a customer consumes less than 2425 calories is approximately 0.2894.

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Part (c): Proportion of customers consuming over 3039 calories

To find $P(X > 3039)$:

1. Calculate the z-score: $z = \frac{3039 - 2762}{609} = 0.4557$
2. Use the z-score to find the probability from the standard normal distribution table: $P(X > 3039) = 1 - P(Z < 0.4557) \approx 1 - 0.6756 = 0.3244$

So, the proportion of customers consuming over 3039 calories is approximately 0.3244.

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Part (d): Fewest calories to receive the Piggy award (top 2% of customers)

To find the calories that correspond to the top 2%, we need to find the z-score associated with the 98th percentile (since top 2% are the highest values):

1. The z-score for the 98th percentile is approximately 2.0537.
2. Convert the z-score back to the original calorie scale: $X = \mu + z \cdot \sigma = 2762 + (2.0537)(609) = 4011.24$

Rounding to the nearest calorie, the fewest calories a person must consume to receive the Piggy award is 4011 calories.

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Summary of Answers:

• (a) $X \sim N(2762, 609^2)$
• (b) Probability that the customer consumes less than 2425 calories: 0.2894
• (c) Proportion of customers consuming over 3039 calories: 0.3244
• (d) Fewest calories for the Piggy award: 4011 calories

Would you like further explanation on any part of this solution?

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Related Questions:

1. What is the probability that a customer consumes between 2500 and 3000 calories?
2. How would the answers change if the mean calorie consumption increased to 2800?
3. What is the z-score of a customer consuming exactly 2762 calories?
4. If the standard deviation decreased to 500, how would that affect the distribution of $X$?
5. What percentage of customers consume between 2200 and 3200 calories?

Tip:

To quickly estimate probabilities for normal distributions, remember that approximately 68% of data falls within one standard deviation of the mean, 95% within two, and 99.7% within three.