MNITAB 6 mike

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Feb 20, 2024

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Assignment 6 Name: Michael Cuautla Sec0on number: 028D Date: 11/7/23 Please answer all ques0ons using the appropriate version of the dataset as assigned by your TA. Each graph must be labeled with a 0tle, axis labels, and a footnote with your name and sec0on number. In this assignment, in order to study the behavior of , you will be genera0ng random samples. Because the samples are random, each individual’s samples will be different. 1. Begin with a popula0on that is NORMAL with a mean of 100 and a standard devia0on of 10. Take 300 random samples of size n=2. To do this, click on: CALC → RANDOM DATA → NORMAL On the input screen that appears, Generate 300 rows of data , store in C1 and C2. Be sure to enter the mean = 100 and standard devia0on = 10. AWer you click on OK , you should see data in the Data Window in C1 and C2. Each row represents a random sample of n=2 data points from the popula0on. Now use ROW STATISTICS to calculate the mean for each sample of n=2. To do this, click on: CALC → ROW STATISTICS. On the input screen that appears, select MEAN . Use C1-C2 as the input variables and store result in C3. Now look at the different columns. Name C3 so that when you graph the histograms, you will be able to iden0fy the sampling distribu0on easily. To do this, click on the Data Window to move the cursor down to it. In the gray cell at the top of C3, type in “Norn=2)” to iden0fy the distribu0on and sample size. Each entry in C3 should be the average of the values in C1 and C2. Use your calculator to verify this by choosing any row and averaging the values in C1 and C2 and confirm that this is the corresponding value in C3. Show your calcula0on below. Column is 104.53 + 97.832 / 2 equals 101.188 What you have created in C3 approximately represents a Sampling Distribu0on of . Find the mean and standard devia0on of the Sampling Distribu0on. To do this, click on: STAT → BASIC STATISTICS → DISPLAY DESCRIPTIVE STATISTICS ¯ X ¯ X

On the input screen that appears, select C3 for the Variable. The results will be in the Session Window . Wait un(l a*er problem 5 to print the Session Window.) Sta(s(cs a. How does the mean of C3 ( ) compared to the mean of the original popula0on, μ ? Recall that the mean of the original popula0on is 100. Note: should be approximately equal to μ ). = 100.32 Are the two values close? YES If so, the results of your simula0on of a sampling distribu0on confirm that: should be approximately equal to μ . b. How does , the standard devia0on of C3, compare to the standard devia0on of the original popula0on, σ ? Note: should be approx. equal to σ / ). For this example, n = 2 and σ = 10. Using your calculator, calculate σ / . 7.11 Compare your calcula0on to the standard devia0on of C3. 6.93 Are the two numbers close? YES If so, the results of your simula0on of a sampling distribu0on confirm that σ should be approx. equal to σ / . c. Now compare the shape of a set of data from the original popula0on, C1, with the shape of the sampling distribu0on, C3. First, in the Worksheet, give C1 the name: “Popula0on Data”. Give C3 the name: “Sample Means n=2)”. Next, click on: GRAPH → HISTOGRAM. Click on the SIMPLE histogram icon. Select both C1 and C3 for the GRAPH VARIABLES. If you then click on the bukon labeled MULTIPLE GRAPHS, you can select SAME Y and SAME X, including bins by clicking on the check boxes. Now both histograms will have the same scale. Put a 0tle on the graphs. The 0tle should iden0fy the popula0on distribu0on: “Underlying Popula0on: Normal”. Do Not use Axis Labels here. Let MINITAB use the column name so that you can tell which column of data the histogram is displaying. Print the two histograms or copy them to a Word document and print the document at the end of this Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3 Maximum "Norm=2)" 300 0 100.32 0.400 6.93 79.69 95.41 100.43 104.64 117.86 μ ¯ x μ ¯ x μ ¯ x μ ¯ x σ ¯ x σ ¯ x n n x n

assignment. In the space below write a summary of the similari0es and differences between the shapes of the two columns. Be specific. Comment on which histograms look normal. Compare the centers and spreads of the two graphs. From the graphs displayed, they both show a normal distribu0on. The sample mean tends to be more clustered in the center of the graph. In terms of the popula0on graph, the data tends to be sort of spread out but again there seems to be more of a cluster around the middle of the graph. 2. Next begin with a popula0on that is not normal. A UNIFORM distribu0on has a ﬂat shape with data that is evenly distributed. It looks like a rectangle. We will use a UNIFORM distribu0on with

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endpoints 70 and 130. This distribu0on has a mean, μ , equal to 100 and a standard devia0on, σ , equal to 17.3. See Fig. 1 below. Fig. 1 ___________________________________ A B The values of A and B represent the endpoints of the distribu0on. In our example, A will be 70 and B will be 130. To start with a clear Data Window, highlight and delete all data cells. This should give you a blank Data Window, but the Session Window should s0ll have the Descrip0ve Sta0s0cs from the last example. Take 300 random samples of size n=2 from a UNIFORM distribu0on on the interval 70 to 130. Use the commands described above in problem 1, except this 0me select UNIFORM distribu0on with lower endpoint 70 and upper endpoint 130. Use ROW STATISTICS to calculate the mean for each sample of n=2 and store results in C3. Name C3 appropriately, i.e. “Unin=2)”. Each entry in C3 should be the average of the values in C1 and C2. Verify this by choosing any row and averaging the values in C1 and C2 and confirm that this is the corresponding value in C3. Show this work below. Column 2 is 102.866+92.264/2 = 110.405 What you have created in C3 is a representa0on of the Sampling Distribu0on of . a. Use DISPLAY DESCRIPTIVE STATISTICS for C3 to find the mean and standard devia0on of the values in C3. Sta0s0cs How does the mean of C3 compare to the mean of the original popula0on? The mean of the original popula0on is 100). = 100.13 Are the two values close? YES If so, the results of your simula0on of a sampling distribu0on confirm that μ should be approximately equal to μ . ¯ X Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3 Maximum "Unin=2" 300 0 100.13 0.972 16.84 70.00 86.62 101.59 113.42 129.82 μ ¯ x x

b. How does , the standard devia0on of C3, compare to the standard devia0on of the original popula0on, σ ? Note: should be approx. equal to σ / ). For this example, n = 2 and σ = 17.3. Using your calculator, calculate σ / . 17.23 Compare your calcula0on to the standard devia0on of C3. 16.84 Are the two numbers close? YES If so, the results of your simula0on of a sampling distribu0on confirm that σ should be approx. equal to σ / . c. Now compare the shape of data from the original popula0on in C1 with the shape of the sampling distribu0on in C3. C3. First, in the Worksheet, give C1 the name: “Popula0on Data”. Give C3 the name: “Sample Means n=2)”. Use the same procedure for making histograms as was described above in problem 1. Use C1 and C3 for the histograms. Put a 0tle on the graphs. The 0tle should iden0fy the popula0on distribu0on: “Underlying Popula0on: Uniform”. Print the two histograms or copy them to a Word document and print the document at the end of this assignment. In the space below write a summary of the similari0es and differences between the shapes of the two columns. Be specific. Be sure to comment on shape, center, and varia0on for each histogram. σ ¯ x σ ¯ x n n x n

Both graphs showcase very similar distribu0ons. The popula0on data set tends to more spread out given the endpoints of 70-130. In terms of the mean I see that the sample is much closer to 105. 3. Con0nue this example with the Uniform distribu0on, but increase the sample size from n=2 to n=30. Again, start with a clear worksheet . This 0me you need to generate 300 rows of data and store in C1-C30. Now each row represents a sample of n=30 data points from the popula0on. Use ROW STATISTICS to calculate the mean for each sample of n=30. The input variables should be C1-C30, and you can store result in C31. Name C31 “Unin=30)”. Each entry in C31 should be the average of the values in C1 through C30. What you have created in C31 is a representa0on of the Sampling Distribu0on of . a. Use the DISPLAY DESCRIPTIVE STATISTICS on C31 to find the mean and standard devia0on of the values in C31. How does the mean of C31 compare to the mean of the original popula0on? The mean of original popula0on is 100). = 100.25 Are the two values close? YES If so, the results of your simula0on of a sampling distribu0on confirm that μ should be approximately equal to μ . b. How does , the standard devia0on of C31, compare to the standard devia0on of the original popula0on, σ ? Note: should be approx. equal to σ / ). For this example, n = 30 and σ = 17.3. Using your calculator, calculate σ / . 3.23 Compare your calcula0on to the standard devia0on of C31. 3.10 Are the two numbers close? YES If so, the results of your simula0on of a sampling distribu0on confirm that should be approx. equal to σ / . c. Now compare the shape of data from the original popula0on in C1 with the shape of the sampling distribu0on in C31. First, in the Worksheet, give C1 the name: “Popula0on Data”. Give C31 the name: “Sample Means n=30)”. Use the same procedure for making histograms as was described above in problem 1. Use C1 and C31 for the histograms. Put a 0tle on the graphs. The 0tle should iden0fy the popula0on distribu0on: “Underlying Popula0on: Uniform”. Print the two histograms or copy them to a Word document and print the document at the end of this assignment. In the space below write a summary of the similari0es and differences between the shapes of the two columns. Be specific. Be sure to comment on shape, center, and varia0on for each histogram. Be specific. x μ ¯ x x σ ¯ x σ ¯ x n n σ ¯ x n

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The sample mean are around 90-120. In contrast to the population date, it seems to be very spread out around the values. Population data varies more unlike the sample mean data as it has its values surround the mean. The data also seems to not have a center.

MNITAB 6 mike

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