https://file.aiquickdraw.com/mathbot/file/1728170717857jjkv5wui.jpg

Let's go through the questions based on the provided information:

We are told that the amount of calories consumed by customers at the buffet is normally distributed with:
- Mean \( \mu = 2659 \) calories
- Standard deviation \( \sigma = 512 \) calories

The problem asks us to work with the normal distribution \( X \sim N(\mu, \sigma^2) \), where \( X \) represents the calories consumed by a randomly selected customer.

### Part (a): Distribution of \( X \)
The distribution of \( X \) is given as a normal distribution with the provided mean and standard deviation. 
\[
X \sim N(2659, 512^2)
\]
So, for part (a), the answer is:
\[
X \sim N(2659, 512^2)
\]

### Part (b): Probability that the customer consumes less than 2489 calories
We need to find \( P(X < 2489) \). First, we compute the z-score:
\[
z = \frac{X - \mu}{\sigma} = \frac{2489 - 2659}{512} = \frac{-170}{512} \approx -0.3320
\]
Next, we look up the z-score of \(-0.3320\) in the standard normal distribution table (or use a calculator). The corresponding probability is approximately:
\[
P(Z < -0.3320) \approx 0.3707
\]
So, the probability that the customer consumes less than 2489 calories is approximately \( 0.3707 \).

### Part (c): Proportion of customers consuming more than 2802 calories
Here, we calculate \( P(X > 2802) \). First, compute the z-score:
\[
z = \frac{2802 - 2659}{512} = \frac{143}{512} \approx 0.2793
\]
Using the z-score table or a calculator, \( P(Z < 0.2793) \approx 0.6099 \). Since we want the proportion consuming more than 2802 calories, we compute:
\[
P(X > 2802) = 1 - P(Z < 0.2793) = 1 - 0.6099 = 0.3901
\]
Thus, the proportion of customers consuming more than 2802 calories is approximately \( 0.3901 \).

### Part (d): Piggy award (top 2%)
The Piggy award goes to the top 2% of customers. This means we need to find the calorie amount that corresponds to the 98th percentile of the normal distribution.

Looking up the z-score corresponding to the 98th percentile, we find:
\[
P(Z < z) = 0.98 \quad \Rightarrow \quad z \approx 2.054
\]
Now, we solve for the calorie threshold using the z-score formula:
\[
X = \mu + z \sigma = 2659 + 2.054 \times 512 \approx 2659 + 1051.65 = 3711.65
\]
Rounding to the nearest calorie, the fewest number of calories a person must consume to receive the Piggy award is approximately \( 3712 \) calories.

---

Would you like further details or clarifications? Here are five related questions that expand on the concepts:

1. What is the probability that a customer consumes between 2489 and 2802 calories?
2. If the mean changes to 2700 calories, how would the answers to parts (b) and (c) change?
3. How would the proportion in part (c) change if the standard deviation increased to 600?
4. Can you find the 90th percentile of the calorie distribution?
5. What is the probability that a customer consumes more than 3500 calories?

**Tip**: When working with normal distributions, z-scores allow you to standardize any value and use the standard normal table to find probabilities or percentiles.

The amount of calories consumed by customers at the Chinese buffet is normally distributed with mean 2659 and standard deviation 512. One randomly selected customer is observed to see how many calories X that customer consumes. Round all answers to 4 decimal places where possible. 

a. What is the distribution of X?

b. Find the probability that the customer consumes less than 2489 calories.

c. What proportion of the customers consume over 2802 calories?

d. The Piggy award will be given out to the 2% of customers who consume the most calories. What is the fewest number of calories a person must consume to receive the Piggy award?

Solve a normal distribution problem involving calorie consumption at a Chinese buffet. Calculate probabilities using z-scores, find the proportion of customers consuming more than a certain number of calories, and determine the top 2% calorie threshold for a special award.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation