The purpose and intent of the assignment is to help you understand the difference between three different distributions:

1. the distribution for a sample,
2. the distribution for the population, and
3. the sampling distribution of the mean.

Think of the 6/49 lottery as sampling from a population. The population would be the balls in the tumbler with the numbers 1 to 49.

Balls are drawn one at a time. Once a ball is drawn it cannot be drawn again. Six balls are draws plus one extra ball. For the purposes of this assignment only the six regular numbers are used.

In the lottery, the idea is to match the numbers drawn. In this assignment, we are not interested in matching the numbers but in the mean of the six regular numbers.

Let's start with a few questions about the population. The population is the set of elements from which the samples are drawn.

1. What is the size of the population?
2. Is the population finite or infinite?
3. Is the population discrete or continuous?
4. What is the mean of the population?
5. What is the median of the population?
6. What is the standard deviation of the population?
7. What can be said of the shape of the population?

Each draw can be thought of as taking a sample from this population.

The data set consists of the numbers drawn for all the 6/49 draws since the inception of the lottery. The data is found in an Excel spreadsheet on the course website. Originally, each draw consisted of six numbers. In time, a seventh bonus number was addedbut for the purposes of this assignment, however, we will only consider the regular six numbers. The numbers are not reported in the order they were drawn but from smallest to largest. This does not make any difference for our purposes.

1. How many samples were drawn from this population (as of January 28, 2021)?
2. What is the size of the samples?
3. Were the samples taken with or without replacement?
4. How big are the samples compared to the size of the population?

Take one individual sample, let's say the sample for November 23, 2013.

1. What is the mean for that specific sample?
2. What is the standard deviation for that specific sample?

Now, calculate the means for all the samples (you should use Excel for this).

1. What, theoretically, would be the highest sample mean that could be observed?
2. What was the highest sample mean actually observed?
3. On what week was this sample drawn?
4. What, theoretically, would be the lowest sample mean that could be observed?
5. What was the lowest sample mean actually observed?
6. On what week was this sample drawn?

Let's look at the distribution of sample means. These next questions require drawing a histogram for the sample means. Please include this histogram as part of your assignment. One way of generating this histogram is to use the histogram function in Excel.

1. What is this distribution called?
2. What is the mean of the distribution of sample means?
3. What is the standard deviation of the distribution of sample means?
4. How does the mean of the distribution of sample means compare to the population mean?
5. How does the standard deviation of the distribution of sample means compare to the population standard deviation?
6. What can be said of the shape of the distribution of sample means?
7. How does the shape of the distribution of sample means compare to the shape of the population?
8. Construct a box plot for the distribution of sample means.
9. What is the probability that the average of the six numbers drawn next week will be greater than or equal to 30?

Let's now use formulas to calculate the mean and standard deviation of the sampling distribution for the mean.

1. Using the formula,, what would be the expected or theoretical value for the mean of the sampling distribution of the mean? How does it compare to the mean actually observed in #20 above? Comment on the difference.
2. Using the formula,what would be the expected or theoretical value for the standard deviation of the sampling distribution of the mean? How does it compare to the actual standard deviation calculated in #21 above?
3. What can you say about the shape of distribution of sample means? Is it normal?

There are two Lotto 6/49 draws per week. By next year, there will have been 104 more draws. In five years', there will have been 520 more draws.

1. What difference would that make to the mean, standard deviation, and shape of the distribution of sample means?

Let's assume now that each ball would have been placed back in the hopper before selecting the next ball.

1. What impact would this have had on the mean of the distribution of sample means?
2. What impact would this have had on the standard deviation of the distribution of sample means?

Let's assume now that instead of selecting six balls, you had included the seventh extra ball.

1. What impact would this have had on the mean of the distribution of sample means?
2. What impact would this have had on the standard deviation of the distribution of sample means?