Assignment 1 is worth 10 marks. The purpose and intent of the assignment is to help you better understand the relations between three different distributions: 1. the distribution of values in the population, 2. the distribution of values in a sample, and 3. the sampling distribution of the mean. Think of the 6/49 lottery as sampling from a population (which it really is!). 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 in that draw. 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. 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. Each draw consists of seven numbers, six regular numbers and one bonus number. For the purposes of this assignment, we will only consider the regular six numbers. The regular numbers are not reported in the order they were drawn but from smallest to largest. This does not make any difference for our purposes. Taking sample after sample tells us something about the variability that arises from chance. Here, we are interested in the mean. The mean of the values in a sample is taken as an estimate of the mean of the values in a population. With all calculations, include the formula that you used. 1In all calculations give 4 decimal places. The population of 49 balls in the hopper 1. Let's start with a few questions about the population. The population is the complete set of elements from which the samples are drawn. a.What is the size of the population? b.Is the population finite or infinite? c.Is the population discrete or continuous? d.What is the level of measurement? Is it nominal, ordinal, interval, or ratio? e.What is the mean of the population? f.What is the median of the population? g.What is the standard deviation of the population? h.Give an appropriate graphical representation of the population. i.What can be said of the shape of the population? Sampling from the 49 balls in the hopper 2. Each draw can be thought of as taking a sample from this population. a.What sampling strategy was used? b.Were the samples taken with or without replacement? c.How many different samples are possible considering that order does not matter? d.What is the size of the samples? e.How big are the samples compared to the size of the population? Is this considered a big sample or a small sample? What are the implications of this? Pick one sample (draw) at random. 3. Select one sample (draw) from among the samples (draws) so far. a.On what date was your sample drawn? b.List the observations that made up this sample? c.What is the mean for your sample? d.What is the standard deviation for your sample? Considering the outcome of taking sample after sample 4. Now, calculate the means for all the samples in our dataset. In Excel, for each sample (draw), calculate the mean for the six regular values. Each sample (draw) will have one mean. a.How many samples were drawn from this population (as of January 15, 2022)? b.How many samples could be drawn this way? c.What is this distribution of sample means called? d.Is it a theoretical or empirical sampling distribution? e.Is this distribution of sample means continuous or discrete? f.The 'event' is that the mean of a sample is equal to a given value. There are many samples that will have the same mean. How many different 'events' are possible? g.What, theoretically, would be the highest sample mean that could ever be observed? 2h.What was the highest sample mean actually observed? i.On which draw was this sample drawn? j.What, theoretically, would be the lowest sample mean that could ever be observed? k.What was the lowest sample mean actually observed? l.On which draw was this sample drawn? m.What is the mean of the empirical (observed) distribution of sample means? n.What is the standard deviation of the distribution of sample means? o.How does the mean of the distribution of sample means compare to the population mean? p.How does the standard deviation of the distribution of sample means compare to the population standard deviation? 5. These next questions require drawing a histogram or bar chart for the sample means. Please include this histogram or bar chart as part of your assignment. One way of generating this histogram is to use the histogram function in Excel. a.What can be said of the shape of the distribution of sample means? Is it normal? b.How does the shape of the distribution of sample means compare to the shape of the population? c.Construct a box plot for the observed distribution of sample means. d.What is the probability that the average of the six numbers drawn next week will be greater than or equal to 30? 6. Let's now use formulas to calculate the mean and standard deviation of the sampling distribution for the mean using the theoretical formulas. a.Using the formula, E(x)=what would be the expected or theoretical value for the mean of the sampling distribution of the mean? b.Using the formula, x=(Nn) (N1)(/n)what would be the expected or theoretical value for the standard deviation of the sampling distribution of the mean? c.How do the empirical values for the mean and standard deviation of the sampling distribution of the mean compare to the theoretical values for the mean and standard deviation of the sampling distribution of the mean? More samples (draws) 7. 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. a.What difference would taking more samples (viz. more draws) have on the mean, standard deviation, and shape of the distribution of sample means? Sampling with replacement 8. Let's assume now that each ball would have been placed back in the hopper before selecting the next ball. a.What impact would this have had on the mean of the distribution of sample means? 3b.What impact would this have had on the standard deviation of the distribution of sample means? Adding a seventh ball 9. Let's assume now that instead of selecting six balls, we had included the seventh extra ball. a.What impact would this have had on the mean of the distribution of sample means? b.What impact would this have on the standard deviation of the distribution of sample means? c.What impact would this have on the shape of the distribution of sample means?