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Statistics: Frequency Distributions

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I have two questions. Please use Microsoft word to write out solution.

1) The scores in Mathematics of 15 students are given below. Find range, sample variance and standard deviation.
93 85 78 79 97 90 82 61 53 63 94 69 84 67 95

2) The following are the prices charged (in cents) for a can of chicken soup at 20 super markets.
80 89 81 78 92 82 79 85 79 85 91 80 81 84 90 79 79 78 92 84
a) Calculate the mean and standard deviation.
b) What percentage of data lies in the interval Î¼ Â± Ïƒ ? In the interval Î¼ Â± 2Ïƒ ?

https://brainmass.com/math/probability/statistics-frequency-distributions-607719

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1) The scores in Mathematics of 15 students are given below. Find range, sample variance and standard deviation.
93 85 78 79 97 90 82 61 53 63 94 69 84 67 95
X X - XÌ… (X - XÌ…)2
93 13.66666667 186.7777778
85 5.666666667 32.11111111
78 -1.333333333 1.777777778
79 -0.333333333 0.111111111
97 17.66666667 312.1111111
90 10.66666667 113.7777778
82 2.666666667 7.111111111
61 -18.33333333 336.1111111
53 -26.33333333 693.4444444
63 -16.33333333 266.7777778
94 14.66666667 215.1111111
69 -10.33333333 106.7777778
84 4.666666667 21.77777778
67 -12.33333333 152.1111111
95 15.66666667 245.4444444
1190 2691.333333
Range = Maximum value - Minimum value
= 97 - 53
= 44
Sample variance = = 192.2380952
Sample standard deviation = = 13.86499532

2) The following are the prices charged (in cents) for a can of chicken soup at 20 super markets.
80 89 81 78 92 82 79 85 79 85 91 80 81 84 90 79 79 78 92 84
a) Calculate the mean and standard deviation.
X X - XÌ… (X - XÌ…)2
80 -3.4 11.56
89 5.6 31.36
81 -2.4 5.76
78 -5.4 29.16
92 8.6 73.96
82 -1.4 1.96
79 -4.4 19.36
85 1.6 2.56
79 -4.4 19.36
85 1.6 2.56
91 7.6 57.76
80 -3.4 11.56
81 -2.4 5.76
84 0.6 0.36
90 6.6 43.56
79 -4.4 19.36
79 -4.4 19.36
78 -5.4 29.16
92 8.6 73.96
84 0.6 0.36
1668 458.8
Mean = = 83.4
Sample standard deviation = = 4.913997194
b) What percentage of data lies in the interval Î¼ Â± Ïƒ ? In the interval Î¼ Â± 2Ïƒ ?
Î¼ Â± Ïƒ = 83.4 Â± 4.91 = (78.49, 88.31)
There are 13 observations lies in the interval Î¼ Â± Ïƒ.
Hence the percentage of data lies in the interval Î¼ Â± Ïƒ = 13/20 = 65%
Empirical rule states that for a normal distribution,
About 68% (precisely 68.27%) of the values lie within 1 standard deviation of the mean.
Î¼ Â± 2Ïƒ = 83.4 Â± (2 * 4.91) = (73.58, 93.22)
There are 20 observations lies in the interval Î¼ Â± 2Ïƒ.
Hence the percentage of data lies in the interval Î¼ Â± Ïƒ = 20/20 = 100%
Empirical rule states that for a normal distribution,
About 95% (precisely 95.45%) of the values lie within 2 standard deviation of the mean.

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