The average amount of money spent for lunch per person in the college cafeteria is $7.11 and the standard deviation is $2.1. Suppose that 42 randomly selected lunch patrons are observed. Assume the distribution of money spent is normal, and round all answers to 4 decimal places where possible.What is the distribution of XX? XX~ N(,) What is the distribution of ¯xx¯? ¯xx¯~ N(,) For a single randomly selected lunch patron, find the probability that this patron's lunch cost is between $7.294 and $7.636. For the group of 42 patrons, find the probability that the average lunch cost is between $7.294 and $7.636. For part d), is the assumption that the distribution is normal necessary? Yes or No

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Asked Dec 6, 2019
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The average amount of money spent for lunch per person in the college cafeteria is $7.11 and the standard deviation is $2.1. Suppose that 42 randomly selected lunch patrons are observed. Assume the distribution of money spent is normal, and round all answers to 4 decimal places where possible.
What is the distribution of
X
X
?
X
X
~ N(,)
What is the distribution of
¯
x

?
¯
x

~ N(,)
For a single randomly selected lunch patron, find the probability that this patron's lunch cost is between $7.294 and $7.636.
For the group of 42 patrons, find the probability that the average lunch cost is between $7.294 and $7.636.
For part d), is the assumption that the distribution is normal necessary? Yes or No

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Expert Answer

Step 1

Central Limit Theorem for mean:

If a random sample of size n is taken from a population having mean µ and standard deviation σ then, as the sample size increases, the sample mean approaches the normal distribution with mean µ and standard deviation σ/sqrt(n).

Finding the distribution of X:

It is given that the amount of money spent for lunch per person in the college cafeteria follows normal distribution with mean $7.11 and standard deviation $2.1.

That is, µ= 7.11, σ = 2.1.

Thus, X~ N (7.11, 2.1).

Step 2

Finding the distribution of X-bar:

Here, X~ N (7.11, 2.1).

A random sample of 42 (n) individuals is considered.

By central limit theorem for mean, the average amount of money spent for launch per person in the college cafeteria follows a normal distribution with mean μ = 7.11 and standard deviation 2.1/sqrt (42) = 0.3240.

Thus, the distribution of X-bar ~ N(7.11, 0.3240).

Finding the probability:

The probability that patron’s lunch cost is between $7.294 and $7.636 is calculated as follows:

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P(7.294 <X <7.636) = P(X<7.636)– P(X<7.294) [using EXCEL formula, = 0.5989 – 0.5349 NORM.DIST(7.636,7.11,2.1,1)", "ENORM.DIST(7.294,7.11,2.1,1)" = 0.0640

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Step 3

Finding the probability:

The probability that average lunch cost is betw...

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P(7.294 <X <7.636) = P(X<7.636)– P(X<7.294) using EXCEL formula, = 0.9478 – 0.7149 "=NORM.DIST(7.636,7.11,0.3240,1)", "=NORM.DIST(7.294,7.11,0.3240,1)" = 0.2328 %3D

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