Question

The amount of calories consumed by customers at the Chinese buffet is normally distributed with mean 2984 and standard deviation 610. 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 2863 calories.
c. What proportion of the customers consume over 3107 calories?
d. The Piggy award will given out to the 3% of customers who consume the most calories. What is the fewest number of calories a person must consume to receive the Piggy award? calorie) calories.

Answer 1

Answer:

a. The distribution of X is normal.

b. The probability that the customer consumes less than 2863 calories is 0.4213

c. 42.01% of of the customers consume over 3107 calories

d. The fewest number of calories a person must consume to receive the Piggy award is 4131,41 calories.

Step-by-step explanation:

Let X be the calorie of a randomly chosen customer consumes.

a. The distribution of X is normal.

b. The probability that the customer consumes less than 2863 calories =P(x<2863)=P(z<z*) where z* is the z-statistic of X=2863 calorie.

z* can be calculated using the formula

(1) z*=$\frac{X-M}{s}$ where

• X =2863

• M is the mean calories consumed by customers at the Chinese buffet (2984)

• s is the standard deviation (610)

Then z*=$\frac{2863-2984}{610}$ ≈−0.1984

And P(z<z*)=0.4213

c. The proportion of the customers consume over 3107 calories is

1 - the proportion of the customers consume less than 3107 calories.

= 1- P(z<z*) where z* is the z-statistic of X=3107 calorie.

From above formula (1) we get:

z*=$\frac{3107-2984}{610}$ ≈ 0.5799 and 1-0.5799=0.4201

Thus, 42.01% of of the customers consume over 3107 calories

d. The Piggy award will given out to the 3% of customers, and let z* be the z-score of the the fewest number of calories a person must consume to receive the Piggy award, then

P(z>z*)=0.03 (3%) or P(z<z*)=0.97 gives z*= 1.881

From (1), we have the equation 1.881=$\frac{X-2984}{610}$

We get X= (1.881×610)+2984 =4131,41

The fewest number of calories a person must consume to receive the Piggy award is 4131,41 calories.