Question

The average amount of money spent for lunch per person in the college cafeteria is $7.42 and the standard deviation is $2.71. Suppose that 16 randomly selected lunch patrons are observed. Assume the distribution of money spent is normal, and round all answers to 4 decimal places where possible.

1. What is the distribution of XX? XX ~ N( , )
2. What is the distribution of ¯xx¯? ¯xx¯ ~ N( , )
3. For a single randomly selected lunch patron, find the probability that this patron's lunch cost is between $8.0238 and $9.0126.
4. For the group of 16 patrons, find the probability that the average lunch cost is between $8.0238 and $9.0126.
5. For part d), is the assumption that the distribution is normal necessary? Yes OR No

Answer 1

a) It is given that distribution of money spent is normal with mean $7.42 and the standard deviation $2.71.

So

b) By central limiting theorem, the distribution of means is also normal with mean equal to population mean and standard deviation equal to

That is

c) Define standard random variable Z as

Using normal table we get

d) Define standard random variable Z as

Using normal table we get

e) Yes the the assumption that the distribution is normal is necessary