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M&M Sampling

M&M Sampling

Life is like a box of chocolates—
you never know what you're going to get.                                        —Forrest Gump

M&Ms

Mr. Mars claims there are more blue M&Ms than brown M&Ms in a bag of M&Ms. Today, we test that claim with a random sample. In statistics, a sample is a set of things or people chosen to represent the entire population of those things or people. We saw voting samples predict the results of the last presidential election when only a small percentage of votes had been counted. Sampling is a statistical technique used when we cannot count everything or everyone in a population. Every 10 years, the U.S. Constitution calls for a census to count everyone in the United States; but a census is time consuming and costly. Our federal government would save a lot of money if it randomly sampled data from the entire population like the Nielson TV ratings are sampled.

A random sample is a way of choosing data that gives everything or everyone in the population an equal chance of being included in the sample. In today' lab, we will take a random sample of M&Ms and compare our sample to what Mr. Mars claims for all bags of M&Ms. Here are the percentages claimed by Mr. Mars for all 6 colors of M&Ms: 

RED     13%
ORANGE  20%
YELLOW  14%
GREEN   16%
BLUE    24%
BROWN   13%
To test Mr. Mars' claim, we will do a multinomial test called Goodness-Of-Fit. Multinomial tests, in contrast to binomial tests, are used by statisticians when they have more than one category of observations. Because a bag of M&Ms has 6 colors, we have 6 categories of observations to measure. To perform a Goodness-Of-Fit test, we use a Chi-Square table. A statistics table, like a Chi-Square table, is a list of expected probabilities. Two Chi-Square tables are at the bottom of this page. We will compare our sample to what is expected according to these tables. Before we use these tables, however, we must compute the Chi-Square Test Statistic.

The Chi-Square Test Statistic is found like this:
  1. First count your sample M&Ms in each color category. This is your Observed count.
  2. Next multiply the expected percentages of each color times each of your observed counts. This is your Expected count.
  3. Then find the difference between each Observed count and its Expected count.
  4. Square each difference to make sure the number is positive.
  5. Divide by the Expected count.
  6. Add the results from all the categories together;
  7. And you have what is called your Chi-Square Test Statistic.
  8. You are now ready to compare your test statistic to the numbers in the Chi-Square tables.

chi formula



DIRECTIONS—CHI-SQUARE TEST
    For 4 points, do a Chi-Square test with your bag of M&Ms like this:
  1. Open your sample bag of M&Ms, but do NOT sample them—yet.
  2. Count the frequency of each color.
  3. Enter these counts in the mathlet below in the OBSERVED column.
  4. Click in a TOTALS box to see the total number of M&Ms in your bag.
  5. Multiply the number of M&Ms in your bag by each of the percentages above.
  6. Enter the corresponding number of M&Ms in the EXPECTED column.
  7. Enter the decimal part of the number also.
  8. Press the ENTER key to compute the corresponding Chi-Square entry.
  9. After all your cells are filled, click in a TOTALS box and press ENTER.
  10. Make sure the total in your OBSERVED column equals the total in your EXPECTED column!
  11. Copy your table to your lab sheet and answer all questions below.
  12. The TOTAL in the last column is your Chi-Square Test Statistic.
  13. Compare this number to the numbers on row 5 in Chi-Square Table 1.
  14. Look on row 5 because 5 is 1 less than 6. That's the way Chi-Square tests work...
  15. The number 5 is called the degrees of freedom when we have 6 categories.
  16. The degrees of freedom are always 1 less than the number of categories.
  17. The numbers at the top of each column tell us the probability that our frequencies are purely by chance.
  18. Look on row 5. Write down the number for P=.05. This means that 5% of the time, if your Chi-Square test statistic is greater than this number, then your bag could have the frequency distribution that it does.
  19. Look on row 5. Write down the number for P=.01. This means that 1% of the time, if your Chi-Square test statistic is greater than this number, then your bag could have the frequency distribution that it does.
  20. Look on row 5. Write down the number for P=.001. This means that one in a thousand times, if your Chi-Square test statistic is greater than this number, then your bag could have the frequency distribution that it does.
  21. If our Chi-Square test number is greater than any of the numbers on row 5, then it is likely that your color frequencies are unusual.
  22. If not, then your color frequencies do not differ significantly from Mr. Mars' frequency claims.
  23. So was your bag unusual? YES OR NO?
  24. Now go Chi-Square Table 2 and look on row 5.
  25. On row 5, find the closest number to your Chi-Square test number. Write it down.
  26. Write down also the probability number at the top of the column where you found this number.
  27. This number is the probability that your sample bag has the usual color frequencies.
  28. For example, if your Chi-Square test number is 3.14, then 2.67460 is the closest number on row 5.
  29. The corresponding probability is .750.
  30. This means that 75% of the time your color frequencies would be no more unusual than they are.
  31. Write your TOTALS on the board in the group table.
  32. Did anyone have an unusual distribution of M&Ms?
  33. Mr. Mars claims that the number of M&Ms in a bag can be found by multiplying 31 times the number of ounces in a bag.
  34. Our sample bags weigh 1.69 ounces.
  35. How many M&Ms should we expect in our sample bags?
  36. Look at the group totals on the board.
  37. Does it look like the our group average approximates the number of M&Ms that Mr. Mars claims?
  38. Do you think that the difference between our group average and the claimed number is significantly unusual?






Histogram

HISTOGRAM





Above is a frequency analyzer or histogram, commonly known as a bar chart. It will make a histogram out of any data you paste into its text area. The data displayed is from our ESP experiment where we guessed card suits. Try it out with a few prices of gasoline around your area. Pick a width for each bar and a starting point. Click the BARS button to view the bar graph and descriptive statistics in lower left. Compare histograms with the same data but different class widths till you get one you think best describes the data.



DIRECTIONS—BAR CHART
    For 3 points, enter data for the color frequencies of your M&M bag like this:
  1. Erase the previous data in the DATA area on the left.
  2. For each RED M&M, enter a 1.
  3. For each ORANGE M&M, enter a 2.
  4. For each YELLOW M&M, enter a 3.
  5. For each GREEN M&M, enter a 4.
  6. For each BLUE m&M, enter a 5.
  7. For each BROWN M&M, enter a 6.
  8. Enter 1 for the CLASS WIDTH.
  9. Enter 1 for the CLASS START.
  10. Press the BARS button.
  11. Since this data is nominal, not ordinal, the STATISTICS have no meaning.
  12. Use a ruler to copy your bar chart to your lab paper.
  13. Draw your bars with respect to the vertical and horizontal tics on the graph.
  14. Choose a vertical tic increment that makes bars as tall as the graph area allows.
  15. For most of our samples, the vertical increment should be 2 4 6 8 10 12 14.
  16. Label the vertical axis with these increments.
  17. Label between tics on the horizontal axis: RED ORANGE YELLOW GREEN BLUE BROWN
  18. If you do not undersand these directions, ask!




PieChart

Pie Maker





Above is a pie chart maker. Use it for pre-classified data where you already know the frequency of each category. When you enter data in the text area, be sure to name each class, or the results will be unpredictable. And don't use spaces in the class names. The pie maker is confused easily. Enter the radius for the pie chart in the PIE SIZE box. Enter the font size in the FONT SIZE box. Each time you click the PIE button, the pie chart is randomly colored. Or you can select colors by entering the slice number in the SLICE box, and the hexadecimal code for the color in the COLOR box. To return to random colors, enter the slice number in the SLICE box, but erase the digits in the COLOR box.



DIRECTIONS—PIE CHART
    For 3 points, enter data for the color frequencies of your M&M bag like this:
  1. Enter the color label in the DATA area on the left side.
  2. Enter the frequency count of that color next to the color label.
  3. Be sure to put a space between colors and counts.
  4. Enter 1 in the SLICE box.
  5. Enter ff0000, the hexadecimal code for RED in the Color box.
  6. Press Pie button.
  7. Enter 2 in the SLICE box.
  8. Enter ff8800, the hexadecimal code for ORANGE in the Color box.
  9. Press Pie button.
  10. Enter 3 in the SLICE box.
  11. Enter ffff00, the hexadecimal code for YELLOW in the Color box.
  12. Press Pie button.
  13. Enter 4 in the SLICE box.
  14. Enter 00ff00, the hexadecimal code for GREEN in the Color box.
  15. Press Pie button.
  16. Enter 5 in the SLICE box.
  17. Enter 0000ff, the hexadecimal code for BLUE in the Color box.
  18. Press Pie button.
  19. Enter 6 in the SLICE box.
  20. Enter 993333, the hexadecimal code for BROWN in the Color box.
  21. Press Pie button.
  22. Use a ruler to copy the pie chart to your lab paper.
  23. Draw each slice of pie the size of its percentage.
  24. Use a protractor if you have one.


Comments:

From wHolt - 11/14/08 9:31 AM

Table 1
Chi-Square
Statistics


df P = 0.05 P = 0.01 P = 0.001
1 3.84 6.64 10.83
2 5.99 9.21 13.82
3 7.82 11.35 16.27
4 9.49 13.28 18.47
5 11.07 15.09 20.52
6 12.59 16.81 22.46
7 14.07 18.48 24.32
8 15.51 20.09 26.13
9 16.92 21.67 27.88
10 18.31 23.21 29.59
11 19.68 24.73 31.26
12 21.03 26.22 32.91
13 22.36 27.69 34.53
14 23.69 29.14 36.12
15 25.00 30.58 37.70
16 26.30 32.00 39.25
17 27.59 33.41 40.79
18 28.87 34.81 42.31
19 30.14 36.19 43.82
20 31.41 37.57 45.32
21 32.67 38.93 46.80
22 33.92 40.29 48.27
23 35.17 41.64 49.73
24 36.42 42.98 51.18
25 37.65 44.31 52.62
26 38.89 45.64 54.05
27 40.11 46.96 55.48
28 41.34 48.28 56.89
29 42.56 49.59 58.30
30 43.77 50.89 59.70
31 44.99 52.19 61.10
32 46.19 53.49 62.49
33 47.40 54.78 63.87
34 48.60 56.06 65.25
35 49.80 57.34 66.62
36 51.00 58.62 67.99
37 52.19 59.89 69.35
38 53.38 61.16 70.71
39 54.57 62.43 72.06
40 55.76 63.69 73.41
41 56.94 64.95 74.75
42 58.12 66.21 76.09
43 59.30 67.46 77.42
44 60.48 68.71 78.75
45 61.66 69.96 80.08
46 62.83 71.20 81.40
47 64.00 72.44 82.72
48 65.17 73.68 84.03
49 66.34 74.92 85.35
50 67.51 76.15 86.66
51 68.67 77.39 87.97
52 69.83 78.62 89.27
53 70.99 79.84 90.57
54 72.15 81.07 91.88
55 73.31 82.29 93.17
56 74.47 83.52 94.47
57 75.62 84.73 95.75
58 76.78 85.95 97.03
59 77.93 87.17 98.34
60 79.08 88.38 99.62
61 80.23 89.59 100.88
62 81.38 90.80 102.15
63 82.53 92.01 103.46
64 83.68 93.22 104.72
65 84.82 94.42 105.97
66 85.97 95.63 107.26
67 87.11 96.83 108.54
68 88.25 98.03 109.79
69 89.39 99.23 111.06
70 90.53 100.42 112.31
71 91.67 101.62 113.56
72 92.81 102.82 114.84
73 93.95 104.01 116.08
74 95.08 105.20 117.35
75 96.22 106.39 118.60
76 97.35 107.58 119.85
77 98.49 108.77 121.11
78 99.62 109.96 122.36
79 100.75 111.15 123.60
80 101.88 112.33 124.84
81 103.01 113.51 126.09
82 104.14 114.70 127.33
83 105.27 115.88 128.57
84 106.40 117.06 129.80
85 107.52 118.24 131.04
86 108.65 119.41 132.28
87 109.77 120.59 133.51
88 110.90 121.77 134.74
89 112.02 122.94 135.96
90 113.15 124.12 137.19
91 114.27 125.29 138.45
92 115.39 126.46 139.66
93 116.51 127.63 140.90
94 117.63 128.80 142.12
95 118.75 129.97 143.32
96 119.87 131.14 144.55
97 120.99 132.31 145.78
98 122.11 133.47 146.99
99 123.23 134.64 148.21
100 124.34 135.81 149.48

From wHolt - 11/14/08 9:16 AM

Chi-Square Table 2
df\area.995 .990 .975 .950 .900 .750 .500 .250 .100 .050 .025 .010 .005
1 0.00004 0.00016 0.00098 0.00393 0.01579 0.10153 0.45494 1.32330 2.70554 3.84146 5.02389 6.63490 7.87944
2 0.01003 0.02010 0.05064 0.10259 0.21072 0.57536 1.38629 2.77259 4.60517 5.99146 7.37776 9.21034 10.59663
3 0.07172 0.11483 0.21580 0.35185 0.58437 1.21253 2.36597 4.10834 6.25139 7.81473 9.34840 11.34487 12.83816
4 0.20699 0.29711 0.48442 0.71072 1.06362 1.92256 3.35669 5.38527 7.77944 9.48773 11.14329 13.27670 14.86026
5 0.41174 0.55430 0.83121 1.14548 1.61031 2.67460 4.35146 6.62568 9.23636 11.07050 12.83250 15.08627 16.74960
 
6 0.67573 0.87209 1.23734 1.63538 2.20413 3.45460 5.34812 7.84080 10.64464 12.59159 14.44938 16.81189 18.54758
7 0.98926 1.23904 1.68987 2.16735 2.83311 4.25485 6.34581 9.03715 12.01704 14.06714 16.01276 18.47531 20.27774
8 1.34441 1.64650 2.17973 2.73264 3.48954 5.07064 7.34412 10.21885 13.36157 15.50731 17.53455 20.09024 21.95495
9 1.73493 2.08790 2.70039 3.32511 4.16816 5.89883 8.34283 11.38875 14.68366 16.91898 19.02277 21.66599 23.58935
10 2.15586 2.55821 3.24697 3.94030 4.86518 6.73720 9.34182 12.54886 15.98718 18.30704 20.48318 23.20925 25.18818
 
11 2.60322 3.05348 3.81575 4.57481 5.57778 7.58414 10.34100 13.70069 17.27501 19.67514 21.92005 24.72497 26.75685
12 3.07382 3.57057 4.40379 5.22603 6.30380 8.43842 11.34032 14.84540 18.54935 21.02607 23.33666 26.21697 28.29952
13 3.56503 4.10692 5.00875 5.89186 7.04150 9.29907 12.33976 15.98391 19.81193 22.36203 24.73560 27.68825 29.81947
14 4.07467 4.66043 5.62873 6.57063 7.78953 10.16531 13.33927 17.11693 21.06414 23.68479 26.11895 29.14124 31.31935
15 4.60092 5.22935 6.26214 7.26094 8.54676 11.03654 14.33886 18.24509 22.30713 24.99579 27.48839 30.57791 32.80132
 
16 5.14221 5.81221 6.90766 7.96165 9.31224 11.91222 15.33850 19.36886 23.54183 26.29623 28.84535 31.99993 34.26719
17 5.69722 6.40776 7.56419 8.67176 10.08519 12.79193 16.33818 20.48868 24.76904 27.58711 30.19101 33.40866 35.71847
18 6.26480 7.01491 8.23075 9.39046 10.86494 13.67529 17.33790 21.60489 25.98942 28.86930 31.52638 34.80531 37.15645
19 6.84397 7.63273 8.90652 10.11701 11.65091 14.56200 18.33765 22.71781 27.20357 30.14353 32.85233 36.19087 38.58226
20 7.43384 8.26040 9.59078 10.85081 12.44261 15.45177 19.33743 23.82769 28.41198 31.41043 34.16961 37.56623 39.99685
 
21 8.03365 8.89720 10.28290 11.59131 13.23960 16.34438 20.33723 24.93478 29.61509 32.67057 35.47888 38.93217 41.40106
22 8.64272 9.54249 10.98232 12.33801 14.04149 17.23962 21.33704 26.03927 30.81328 33.92444 36.78071 40.28936 42.79565
23 9.26042 10.19572 11.68855 13.09051 14.84796 18.13730 22.33688 27.14134 32.00690 35.17246 38.07563 41.63840 44.18128
24 9.88623 10.85636 12.40115 13.84843 15.65868 19.03725 23.33673 28.24115 33.19624 36.41503 39.36408 42.97982 45.55851
25 10.51965 11.52398 13.11972 14.61141 16.47341 19.93934 24.33659 29.33885 34.38159 37.65248 40.64647 44.31410 46.92789
 
26 11.16024 12.19815 13.84390 15.37916 17.29188 20.84343 25.33646 30.43457 35.56317 38.88514 41.92317 45.64168 48.28988
27 11.80759 12.87850 14.57338 16.15140 18.11390 21.74940 26.33634 31.52841 36.74122 40.11327 43.19451 46.96294 49.64492
28 12.46134 13.56471 15.30786 16.92788 18.93924 22.65716 27.33623 32.62049 37.91592 41.33714 44.46079 48.27824 50.99338
29 13.12115 14.25645 16.04707 17.70837 19.76774 23.56659 28.33613 33.71091 39.08747 42.55697 45.72229 49.58788 52.33562
30 13.78672 14.95346 16.79077 18.49266 20.59923 24.47761 29.33603 34.79974 40.25602 43.77297 46.97924 50.89218 53.67196

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Last Modified 11/18/08 8:31 AM

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