Question

1,

Let's say our population distribution consisting of the number of credits of all current students at a Community College this term is approximately normal. Let's assume that of all the college students, the average number of credits taken this term is 14.4 on average, with a standard deviation of 2.2361 credits. What is the probability that the average amongst a group of 25 students would be less than 13.8 credits?

List the mean and standard deviation of your sampling distribution. Then, write your answer to the probability question in a complete sentence.

2, A bag contains 2 gold marbles, 9 silver marbles, and 21 black marbles. Someone offers to play this game: You randomly select one marble from the bag. If it is gold, you win $4 overall (including the cost to play). If it is silver, you win $2 overall. If it is black, you lose $1 overall.

What is your expected value of actual winnings if you play this game? Round your answer to two decimal places.

Answer 1

We would be looking at the first question here.

Q1) As the underlying distribution is approximately normal here, therefore the mean and standard deviation of the sampling distribution here are computed as:

Ux= = 14.4

$\sigma_{\bar X} = \frac{\sigma}{\sqrt{n}} = \frac{2.2361}{\sqrt{25} } = 0.44722$

therefore 14.4 and 0.44722 are the mean and standard deviation of the sampling distribution of sample mean here.

The probability now is computed here as:

$P(\bar X < 13.8)$

Converting it to a standard normal variable, we have here:

$P(Z < \frac{13.8 - 14.4}{0.44722})$

PZ < -1.3416)

Getting it from the standard normal tables, we have here:

Therefore 0.0899 is the required probability here.