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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*= where
- X =2863
- M is the mean calories consumed by customers at the Chinese buffet (2984)
- s is the standard deviation (610)
Then z*= ≈−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*= ≈ 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=
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.