Suppose that the scores of bowlers in a particular league follow a normal distribution such that the standard deviation of the population is 12 . Find the 95% confidence interval of the mean score for all bowlers in this league using the accompanying data set of 40 random scores. Round your answers to two decimal places and use ascending order.
Score
96
103
94
105
91
101
99
89
94
91
82
94
97
99
89
107
97
100
96
92
96
95
99
90
84
97
99
87
102
92
91
90
88
103
94
90
98
91
87
91

To find the 95% confidence interval of the mean score for all bowlers in this league, we need to first calculate the sample mean and sample standard deviation of the given data set.

The sample mean is calculated as follows:

mean = (96 + 103 + 94 + 105 + 91 + 101 + 99 + 89 + 94 + 91 + 82 + 94 + 97 + 99 + 89 + 107 + 97 + 100 + 96 + 92 + 96 + 95 + 99 + 90 + 84 + 97 + 99 + 87 + 102 + 92 + 91 + 90 + 88 + 103 + 94 + 90 + 98 + 91 + 87) / 40

mean = 95.125

The sample standard deviation is calculated as follows:

First, we need to calculate the variance of the data set, which is done by taking the sum of the squared differences between each score and the mean, and dividing by the sample size minus 1:

variance = ((96 - 95.125)^2 + (103 - 95.125)^2 + (94 - 95.125)^2 + (105 - 95.125)^2 + (91 - 95.125)^2 + (101 - 95.125)^2 + (99 - 95.125)^2 + (89 - 95.125)^2 + (94 - 95.125)^2 + (91 - 95.125)^2 + (82 - 95.125)^2 + (94 - 95.125)^2 + (97 - 95.125)^2 + (99 - 95.125)^2 + (89 - 95.125)^2 + (107 - 95.125)^2 + (97 - 95.125)^2 + (100 - 95.125)^2 + (96 - 95.125)^2 + (92 - 95.125)^2 + (96 - 95.125)^2 + (95 - 95.125)^2 + (99 - 95.125)^2 + (90 - 95.125)^2 + (84 - 95.125)^2 + (97 - 95.125)^2 + (99 - 95.125)^2 + (87 - 95.125)^2 + (102 - 95.125)^2 + (92 - 95.125)^2 + (91 - 95.125)^2 + (90 - 95.125)^2 + (88 - 95.125)^2 + (103 - 95.125)^2 + (94 - 95.125)^2 + (90 - 95.125)^2 + (98 - 95.125)^2 + (91 - 95.125)^2 + (87 - 95.125)^2) / (40 - 1)

variance = 198.90625

The sample standard deviation is then the square root of the variance:

standard deviation = sqrt(198.90625)

standard deviation = 14.06

Now that we have the sample mean and sample standard deviation, we can use these values to calculate the 95% confidence interval of the mean score for all bowlers in this league.

We can use the following formula to calculate the confidence interval:

confidence interval = mean +/- (1.96 * (standard deviation / sqrt(sample size)))

Substituting in the values we calculated above, we get:

confidence interval = 95.125 +/- (1.96 * (14.06 / sqrt(40)))

Hence , confidence interval = 95.125 +/- 3.45

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