The Central Limit Theorem is both a central result in mathematical statistics and a theorem about central tendencies. It states that the sample mean, i.e., the average, of a large random sample has approximately a normal distribution, regardless of the distribution of values in the population one samples from. More precisely, it states that the standardized value of a random sample mean has an approximately normal distribution with mean 0 and standard deviation 1. For samples from non-normal populations, the approximation improves as the sample size increases.
Each row of digits below is a random sample of 20 digits.
(a) Calculate the sample mean (average) of the 20 digits in each row. For example, for the first row you should get 4.85.
(b) Make a frequency table showing how many times the sample means fall in each of the intervals [0, 1), [1, 2), [2, 3), [3, 4), [4, 5), [5, 6), [6, 7), [7, 8), [8, 9), [9, 10).
(c) Draw the histogram corresponding to this frequency table.
(d) Calculate the standardized values of the row averages. To do this, subtract the theoretical mean, 4.5, and divide by the theoretical standard deviation, 0.64, in each case.
(e) Make a histogram for the standardized row means. Choose intervals of equal length so that you get about eight intervals that occur with a frequency greater than 0.
(f) Write a paragraph telling what you learned from this exercise.
37751 04998 66038 63480
50915 64152 82981 15796
99142 35021 01032 57907
70720 10033 25191 62358
18460 64947 32958 08752
65763 41133 60950 35372
83769 52570 60133 25211
58900 78420 98579 33665
54746 71115 78218 64314
56819 27340 07500 52663
34990 62122 38223 28526
02269 22795 87593 81830
43042 53600 45738 00261
92565 12211 06868 87786
67424 32620 60841 86848
04016 77148 09535 10743
85225 19763 46105 25289
03360 07457 75131 41209
72460 99682 27970 25632
66960 55780 71778 52629
14824 95631 00697 65462
34001 05618 41900 23303
77718 83830 28781 72917
60930 05091 35726 07414
94180 62151 08112 26646
81073 85553 47650 93830
18467 39689 60801 46828
60643 59399 79740 17295
73372 61697 85728 90779
18395 18482 83245 54942