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The amount of calories consumed by customers at the Chinese buffet is normally distributed with mean 2885 and standard deviation 651. One randomly selected customer is observed to see how many calories X that customer consumes. Round all answers to two decimal places.

A. X ~ N( )
B. Find the probability that the customer consumes less than 2000 calories.
C. What proportion of the customers consume over 5,000 calories?
D. The Piggy award will given out to the 1% of customers who consume the most calories. What is the least amount of calories a person must consume to receive the Piggy award?

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Answer:

a.  X~N(2,885, 651)

b.  0.086291

c.  0.00058

d.  3213.10 calories

Step-by-step explanation:

a. -A normal distribution is expressed in the form X~N(mean, standard deviation).

-Let X a random variable denoting  the number of calories consumed.

-X is a is a normally distributed random variable with mean 2885 and standard deviation 651.

-This distribution is expressed as X~N(2,885, 651)

b. The probability that less than 2000 calories are consumed is calculated using the formula:

[tex]P(X<x)=P(z<\frac{\bar X-\mu}{\sigma})[/tex]

#substitute the given values in the formula to solve for P:

[tex]P(X<x)=P(z<\frac{\bar X-\mu}{\sigma})\\\\=P(Z<\frac{2000-2885}{651})\\\\=P(z<-1.359)\\\\\\=0.08691[/tex]

Hence, the probability of consuming less than 2000 calories is 0.08691

c. The proportion of customers consuming more than 5000 calories is calculated as:

[tex]P(X>x)=P(z>\frac{\bar X-\mu}{\sigma})\\\\=P(Z>\frac{5000-2885}{651})\\\\=P(z>3.2488)\\\\=1-0.99942\\\\=0.00058[/tex]

Hence, the proportion of customers consuming over  5000 calories is 0.00058

d. The least amount of calories to get the award is calculated as:

1% is equivalent to a z value of 0.50399.

-We equate this to the formula to solve for the mean consumption:

[tex]0.01=P(z>\frac{\bar X-\mu}{\sigma})\\\\=P(z>\frac{\bar X-2885}{651})\\\\\1\%=0.50399 \\\\\frac{\bar X-2885}{651}=0.50399 \\\\\bar X=0.50399\times 651+2885\\\\=3213.09[/tex]

Hence, the least amount of calories consumed to qualify for the award is 3213.10 calories.

Cricetus

(A) X [tex]\sim[/tex] N (2885, 651). To understand the calculation, check below.

(B) The probability be "0.09".

(C) The proportion of customer be "0.01".

(D) The least amount of calories be "4401.83".

Probability

According to the question,

Let,

Normally distributed random variable be "X".

Mean = 2885

Standard deviation = 651

(A) X [tex]\sim[/tex] N (2885, 651)

(B) The probability,

→ P (x < 2000) = P[[tex]\frac{(x - \mu)}{\sigma}[/tex] < [tex]\frac{2000-2885}{651}[/tex]]

                        = P (z < -1.36)

By using the z-table,

                        = 0.09

(C) The proportion be:

→ P (x > 5000) = 1 - p (x < 5000)

                       = 1 - p P[[tex]\frac{x- \mu}{\sigma}[/tex] < [tex]\frac{5000 - 2885}{651}[/tex]]

                       = 1 - P(z < 3.25)

By using the z-table, we get

                       = 1 - 0.99

                       = 0.01

(D) The amount of calories,

P (Z > z) = 1%

By using the standard normal table,

z = 2.33

We know the z-score formula,

→ x = z × σ + μ

By substituting the values,

     = 2.33 × 651 + 2885

     = 4401.83

Thus the approach above is correct.        

Find out more information about probability here:

https://brainly.com/question/25870256