Answer to Question #287470 in Statistics and Probability for Lisisia

Question

The time (in seconds) that a random sample of employees took to complete a task is as follows: 

63

229

165

77

49

74

67

59

66

 

102

81

72

59

74

61

82

48

70

86

a.      Plot a stem & leaf diagram

b.     Compute the arithmetic mean, mode, median and quartiles

c.      Compute 80^th percentile

d.     Compute the variance and standard deviation

e.      Compute coefficient of variation

f.      Construct a box-plot for the data. Is the data skewed, explain? Identify any outliers in the data

Answer 1

Least to Greatest Values:

48, 49, 59, 59, 61, 63, 66, 67, 70, 72, 74, 74, 77, 81, 82, 86, 102, 165, 229

a.

steam | leaf

4 | 8 9

5 | 9 9

6 | 1 3 6 7

7 | 0 2 4 4 7

8 | 1 2 6

10 | 2

16 | 5

22 | 9

b.

mean:

$\mu=\sum x_i/n=83.37$

In the given data, the observation 59,74 occurs maximum number of times (2)

mode:

$M=59,74$

median:

m= value of ((n+1)/2)th observation

= value of ((19+1)2)th observation

= value of 10th observation

=72

first quartile:

Q₁=((n+1)/4)th value of the observation

=(20/4)th value of the observation

=5th value of the observation

=61

third quartile:

Q₃=(3(n+1)/4)th value of the observation

=(3⋅20/4)th value of the observation

=15th value of the observation

=82

c.

 P₈₀=(80(n+1)/100)th value of the observation

=(80⋅20/100)th value of the observation

=16th value of the observation

=86

d.

Sample Variance:

$s^2=\frac{\sum (x_i-\mu)^2-(\sum (x_i-\mu))^2/n}{n-1}=1881.25$

Sample Standard deviation:

$s=\sqrt{1881.25}=43.37$

e.

Coefficient of Variation:

$s/\mu=43.37/83.37=0.52$

f.

The data skewed to the left side: in Least to Greatest dataset the most part of data in the left side.

for outliers:

lower bound:

$Q_1-1.5IQR=61-1.5\cdot21=29.5$

lower bound:

$Q_3+1.5IQR=82+1.5\cdot21=113.5$

outliers: 165, 229