Question

for the data shown, answer the questions. round to 2 decimal places.

x
5.8
6
24.9
20
14.8
16.9
12.3
15.9
2
3.8

compute the sample mean:

compute the median:

compute the sample standard deviation:

Explanation:

Step1: Compute Sample Mean

Sum all data points: \(5.8 + 6 + 24.9 + 20 + 14.8 + 16.9 + 12.3 + 15.9 + 2 + 3.8\)
\(= 5.8+6=11.8\); \(11.8+24.9=36.7\); \(36.7+20=56.7\); \(56.7+14.8=71.5\); \(71.5+16.9=88.4\); \(88.4+12.3=100.7\); \(100.7+15.9=116.6\); \(116.6+2=118.6\); \(118.6+3.8=122.4\)
Number of data points (\(n\)) = 10.
Mean (\(\bar{x}\)) = \(\frac{122.4}{10} = 12.24\).

Step2: Compute Median

Sort data: \(2, 3.8, 5.8, 6, 12.3, 14.8, 15.9, 16.9, 20, 24.9\)
\(n = 10\) (even), so median is average of 5th and 6th terms.
5th term: 12.3; 6th term: 14.8.
Median = \(\frac{12.3 + 14.8}{2} = \frac{27.1}{2} = 13.55\).

Step3: Compute Sample Standard Deviation

First, find deviations from mean (\(x_i - \bar{x}\)) and square them:

• \(2 - 12.24 = -10.24\); \((-10.24)^2 = 104.8576\)
• \(3.8 - 12.24 = -8.44\); \((-8.44)^2 = 71.2336\)
• \(5.8 - 12.24 = -6.44\); \((-6.44)^2 = 41.4736\)
• \(6 - 12.24 = -6.24\); \((-6.24)^2 = 38.9376\)
• \(12.3 - 12.24 = 0.06\); \((0.06)^2 = 0.0036\)
• \(14.8 - 12.24 = 2.56\); \((2.56)^2 = 6.5536\)
• \(15.9 - 12.24 = 3.66\); \((3.66)^2 = 13.3956\)
• \(16.9 - 12.24 = 4.66\); \((4.66)^2 = 21.7156\)
• \(20 - 12.24 = 7.76\); \((7.76)^2 = 60.2176\)
• \(24.9 - 12.24 = 12.66\); \((12.66)^2 = 160.2756\)

Sum of squared deviations: \(104.8576 + 71.2336 + 41.4736 + 38.9376 + 0.0036 + 6.5536 + 13.3956 + 21.7156 + 60.2176 + 160.2756\)
Calculate step-by-step:
104.8576+71.2336=176.0912; +41.4736=217.5648; +38.9376=256.5024; +0.0036=256.506; +6.5536=263.0596; +13.3956=276.4552; +21.7156=298.1708; +60.2176=358.3884; +160.2756=518.664.

Sample variance (\(s^2\)) = \(\frac{\sum (x_i - \bar{x})^2}{n - 1} = \frac{518.664}{9} \approx 57.6293\).
Sample standard deviation (\(s\)) = \(\sqrt{57.6293} \approx 7.59\).

Answer:

Sample Mean: \(12.24\)
Median: \(13.55\)
Sample Standard Deviation: \(7.59\)