Question

Follow the steps below to find out whether the casino’s slot machine is working as intended. Answer all the questions here for your reference. Note: be very careful about the distinction between theoretical (what we calculated should happen) and actual ("empirical", what actually did happen).

a) Go to Appendix B, find the casino data you’ve been assigned to analyze, and paste it below.

Casino E has lost $200 after 2000 spins.

b) We can think of our actual (empirical) data for the casino as taking a sample from the population with sample mean . With this perspective, what is the sample size for your casino?

c) Similar to the question in "Part 1 – Theoretical Calculations", how much do we expect the casino to have made or lost with this sample size n? Hint: what does µ represent?

d) Next, we’ll find what is for your casino, the average player profit per game.

What is the actual total player profit for your specific casino? Again, remember that if the casino has lost $300, then the player (or players) have gained $300, and vice versa.

The average player profit per game will be this value divided by n. Using this, what is ? Interpret your answer.

e) Next, we’re going to use a hypothesis test to determine the likelihood of the machine working as intended. To begin, what are our null and alternative hypotheses? Hint: we’re using means, not proportions, so make sure your answer takes that into consideration.

f) The Central Limit Theorem states that the mean of a random sample is a random variable (let’s call it Y) whose sampling distribution can be approximated by a Normal model.

What three conditions must be satisfied in order for us to use the CLT here? Note that these are not exactly the same as in the hypothesis testing homework, since we’re using a sampling distribution for a mean and not a proportion. You can still find this in the unit 6 notes.

Are these satisfied for our sample? If so, how is each condition satisfied? Note: our parent distribution (from part 1) is very skewed.

g) Remember that every time we take a sample, its mean will be different and can be thought of as a random variable X- and, under certain conditions, we say that this is normally distributed. This is what sampling distributions are all about. What is the mean and standard deviation for our sampling distribution here? In other words, fill in the blanks below. Again, feel free to consult your unit 6 notes.

X ~ N( , )

h) What is the chance that, for your given game and sample size, the player’s average profit is at least x- (your answer from 2d). In other words, find P(X > x) or P(Z > z0 (these are the same thing), where z0 is the test statistic. The result is your p-value.

i) Using a significance level of α = 0.05, should we reject the null hypothesis?

j) Is this enough evidence to support the casino’s concern that the machine is working incorrectly? A simple "yes" or "no" will suffice here.

Answer 1

In statistical analysis for Casino E, the sample size is 2000 spins and the sample mean is $0.10 profit per game for the player. Central Limit Theorem allows us to use normal distribution for our sample mean. A z-test can then be performed to determine whether to reject the null hypothesis, with a commonly used significance level of 0.05.

Step-by-step explanation:

When analyzing the performance of Casino E's slot machine, the sample size for the empirical data is 2000 spins. This means we have taken a sample of 2000 observations from the population of all possible spins. The sample size, denoted n, is essential for further statistical analysis.

The central limit theorem (CLT) is a fundamental principle in statistics that states the distribution of sample means will approximate a normal distribution as the sample size becomes large, regardless of the shape of the population distribution. This theorem allows us to use normal distribution to make inferences about the population mean based on sample data.

In our scenario, Casino E has lost $200 after 2000 spins. Therefore, the average player profit per game would be the total profit divided by the number of spins, which is $200/2000 spins, resulting in an average profit of $0.10 per spin for the player. This is the empirical mean, represented by x-bar.

The null hypothesis (often denoted H0) typically states that there is no effect or no difference, and in the context of casino games, it might state that the casino's slot machine is operating as intended, meaning it's not favoring the house or the players beyond its designed parameters. The alternative hypothesis (H1 or Ha), on the other hand, suggests that the observed data are not consistent with what the model predicts, indicating that the machine may not be functioning correctly.

If the standard deviation of the population is known, a z-test can be conducted to test the null hypothesis. If the population standard deviation is not known and the sample size is large, we can approximate it with the sample standard deviation.

The significance level (α) is the threshold used to determine whether to reject or fail to reject the null hypothesis. A commonly used α-level is 0.05. If the calculated p-value is less than α, we reject the null hypothesis; if it is greater, we fail to reject it.