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StatisticsQ&A LibraryThe 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 NoQuestion

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¯

?

¯

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

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:

Step 3

**Finding the probability:**

The probability that average lunch cost is betw...

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