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------------------------------------------------ For the following set of data, find the percentage of data within 1 population standard deviation of the mean, to the nearest tenth of a percent.

Data: 72, 69, 82, 74, 69, 78, 68, 72, 69, 82, 74, 69, 78, 68

Answer :

85.7% of the data falls within 1 population standard deviation of the mean.

  • First, we need to calculate the mean (μ):

Sum of all numbers: 72 + 69 + 82 + 74 + 69 + 78 + 68 + 72 + 69 + 82 + 74 + 69 + 78 + 68 = 1024

Number of data points: 14

Mean (μ) = 1024 / 14 = 73.14285714

  • Subtract the mean from each number, square the result, and sum:

(72 - 72.43)² = 0.43² = 0.1849

(69 - 72.43)² = -3.43² = 11.7649

(82 - 72.43)² = 9.57² = 91.5849

(74 - 72.43)² = 1.57² = 2.4649

(69 - 72.43)² = -3.43² = 11.7649

(78 - 72.43)² = 5.57² = 31.0849

(68 - 72.43)² = -4.43² = 19.6449

(72 - 72.43)² = -0.43² = 0.1849

(69 - 72.43)² = -3.43² = 11.7649

(82 - 72.43)² = 9.57² = 91.5849

(74 - 72.43)² = 1.57² = 2.4649

(69 - 72.43)² = -3.43² = 11.7649

(78 - 72.43)² = 5.57² = 31.0849

(68 - 72.43)² = -4.43² = 19.6449

Summing these squared differences:

0.1849 + 11.7649 + 91.5849 + 2.4649 + 11.7649 + 31.0849 + 19.6449 + 0.1849 + 11.7649 + 91.5849 + 2.4649 + 11.7649 + 31.0849 + 19.6449 = 336.98

  • Divide by n:

336.98 / 14 = 24.07

  • Now we can calculate the population standard deviation (σ):

σ = √24.07 = 4.90

  • The range within one standard deviation of the mean is:

Lower bound: 73.14285714 - 4.90 = 68.24285714

Upper bound: 73.14285714 + 4.90 = 78.04285714

  • Count how many data points fall within this range:

72, 69, 74, 69, 78, 68, 72, 69, 74, 69, 78, 68

12 out of 14 data points are within this range

  • Calculate the percentage:

(12 / 14) × 100 = 85.71428571%

  • Rounding to the nearest 0.1%:

= 85.7%