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MATH 214 Lab 4: Discrete Random variables In this lab we will be examining different types of distributions and random samples from these distributions. Label the columns in your Minitab worksheet as follows: Old Name C1 C2 C3 C4 C5 C6 C7 New Name Uniform Values 1 Uniform probability 1 Uniform values 2 Uniform probability 2 Uniform sample B Sample P Sample For each part you will first explore the theoretical distribution, then select several random samples from corresponding distribution and see what happens as we change sample size. Part I A. We begin by looking at a discrete uniform distribution; i.e., each integer between a and b is equally likely to be selected. Generate the probability distribution function for the discrete uniform distribution for a = 1 and b = 30. Store the integers 1-30 in the column Uniform Values 1 → Calc → Make Patterned Data → Simple Set of Numbers → → Uniform Values in the Store window Type 1 in the “From First Value” window Type 30 in the “To Last Value” window → OK Store the discrete uniform probabilities in the column Uniform probability 1 → Calc → Probability Distributions → Integer → Probability, so that the radio button next to it is filled → Type 1 in the “Minimum Value” window → Type 30 in the “Maximum Value” window → → Uniform Values in the “Input Column” window Type “Uniform probability 1” in the Optional Storage window → OK Repeat the above but this time use “step of 5” when storing integers in C2 (Uniform values 2) and use “make patterned data” option to type 0.167 to 6 cells in column “Uniform probability 2”. Now graph the two probability distribution functions →Graph → Scatter plot → in Y window →→Uniform probability 1 →→Uniform Values 1 in the X window Repeat for uniform probabilities and values in the next line → Multiple graphs → in separate panels of same graph → same Y Use the graph to describe the distribution, paying careful attention to the shape and center. Can you explain why dots are on different height for the two distributions? B. Select a random sample from the discrete uniform distribution. Store a random sample of size 100 in the column Uniform Sample → Calc → Random Data → Integer Type 100 in Generate Rows of Data window → Store in Columns Box→→Uniform Sample →Minimum Value window, Type 1 →Maximum Value window, Type 30 → OK Construct Histogram for the sample in Uniform Sample → Graph, Histogram, etc. What values are included in each class in the histogram? C. Repeat Part B two more times. Compare the histograms of the 3 samples to the graph of the theoretical distribution. How well do the histograms resemble the theoretical distribution? Part II. Next, we study the binomial distribution with n = 50 and p = .3. A. Generate the binomial probability distribution function and make the corresponding graph. →Graph → Probability Distribution plot → View Single Choose Binomial in drop down window Enter 50 in number of trials and 0.3 in event probability → OK Describe the distribution, paying careful attention to the shape and measures of central tendency (can you estimate mean and/or median). B. Generate a random sample of size 100 from a binomial distribution with n = 50 and p = 0.6 Store a random sample of size 100 in the column B Sample → Sample → Calc → Random Data → Binomial Type 100 in the Generate Rows of Data →→B Sample in the Store in Columns window Type 50 in the Number of Trials window Type .6 in the Probability of Success window → OK Construct histogram for the sample in B Sample. To more easily compare your histogram to the graph of the theoretical distribution, make sure that the locations of the bars of the histogram coincide with the locations of the bars in the chart of the theoretical distribution. C. Repeat part B two more times from a binomial distribution with n = 50 and p = 0.6. Compare the graphs of the three samples to the graph of the theoretical distribution. How well do the histograms resemble the distribution? Part III. Consider the Poisson distribution with λ = 3. Find in a textbook or online description and/or formula of Poisson distribution. A. The Poisson distribution is different from the previous distributions; it assigns probabilities to all non-negative integers. But for large integers those probabilities are virtually 0, so in practice we can ignore them. We will ignore values larger than 25. Generate the Poisson probability distribution function and make the corresponding graph. →Graph → Probability Distribution plot →View Single Choose Poisson in drop down window Enter 3 in mean window → OK Describe the distribution paying careful attention to the shape and measures of central tendency (can you guess values of mean and/or median). B. Generate random samples of size 100 from the Poisson distribution with mean 3 and store the random sample in column P Sample → Calc→ Random Data → Poisson Type 100 in the Generate Rows of Data window →→P Sample in the Store window, Type 3 in the Mean window → OK Construct the histogram for this sample. C. Repeat Part B two more times. Compare the graphs of the three samples to the graph of the theoretical distribution. How well do the histograms resemble the distribution? Part IV. Drawing conclusions about population from a sample. Download the Samples.mtw file from the Labs page to your P drive; here you will find six data sets. These data sets were generated by selecting random samples from the binomial, Poisson, or uniform distribution. Samples 1, 2, and 3 were selected from (not necessarily in this order): a. a Discrete Uniform Distribution, on [0, 20], or b. a Binomial Distribution, with n = 50 and p = .6, or c. a Poisson Distribution, with λ = 10. Samples 4, 5, and 6 were selected from (not necessarily in this order): a. a Discrete Uniform Distribution, on [0, 20], or b. a Binomial Distribution, with n = 20 and p = .6, or c. a Poisson Distribution, with λ = 11. For each of the data sets create a histogram. Using these histograms and the probability distribution functions (histograms) you generated earlier, try to classify each data set as a sample from binomial, uniform, or Poisson. Explain what led to your decision for each data set. You may also find it helpful to look at the descriptive statistics of the samples, especially mean and standard deviation. Part V. The number of data points in your data set probably affects your ability to identify the theoretical distribution. Create some examples that show how increasing and decreasing the number of data points affects your ability to identify a binomial distribution. I would suggest creating a new worksheet, label 4 to 5 columns as Sample 1, Sample 2, … and then generate binomial data samples for each column using a different number of points for each (but keep the number of trials and probability of a success the same in each case). Make sure that you use very small samples (say of size 10) and very large samples (say of size 1000 or 10000). Graph them all on the same graph. Or group samples Part VI Applications. For each random variable family (uniform, binomial, and Poisson) in today’s lab, find (in the textbook or make one up) a realistic example of such a variable. Also use Minitab to calculate probability for your example of the form: P(0.5 < X < 3.7) and interpret the probability. Your values might be different from 0.5 and 3.7; pick some that seem reasonable for your random variable. → Graph → Probability Distribution plot → View Probability Choose the corresponding probability and its characteristics, switch to shaded area Check the radio button for X value and choose middle Enter 0.5 in X value 1 and 3.7 in X value 2 → OK