Question

Suppose the following weights, in pounds, of 28 one-year-old Labradors are collected: 65.4 70.2 60.5 53.2 72.4 62.8 66.9 65.7 58.7 80.3 70.5 72.0 68.4 92.5 60.8 69.8 72. 7 68.5 78.9 82. 7 56.3 67.2 62.6 48.5 80.4 88.0 78.5 76.0

1. To the nearest thousandth, determine the mean of this sample, and to the nearest hundred-thousandth, determine the deviation

2. Determine and properly label the five-number summary of the data, and use it to find the IQR

3. Calculate the interval of values that would be considered to NOT be outliers, and then state the data values, if any, that ARE outliers.

4. Based on the sample mean and deviation, calculate the z-scores of the min, the max, and the two quartiles. Round to 2 decimal places

Answer 1

1)

X	(X - X̄)²
65.4	18.12
70.2	0.29
60.5	83.85
53.2	270.84
72.4	7.52
62.8	47.02
66.9	7.60
65.7	15.66
58.7	120.06
80.3	113.27
70.5	0.71
72	5.49
68.4	1.58
92.5	521.80
60.8	78.45
69.8	0.02
72.7	9.26
68.5	1.34
78.9	85.43
82.7	170.12
56.3	178.4133
67.2	6.0376
62.6	49.8033
48.5	447.6247
80.4	115.4090
88	336.4604
78.5	78.1961
76	40.2318

	X	(X - X̄)²
total sum	1950.4	2810.609
n	28	28

mean =    ΣX/n =    1950.400   /   28   =   69.657
                      
sample variance =    Σ(X - X̄)²/(n-1)=   2810.6086   /   27   =   104.0966
                      
sample std dev =   √ [ Σ(X - X̄)²/(n-1)] =   √   104.0966   =      10.20277

2)

Median=0.5(n+1)th value =    14.5   th value of sorted data
=   69.150  
      
quartile , Q1 = 0.25(n+1)th value=   7.25   th value of sorted data
=   62.65  
      
Quartile , Q3 = 0.75(n+1)th value=   21.75   th value of sorted data
=   77.875  
maximum =    92.5
minimum=   48.5

so, five - number summary is (48.5 , 62.65 , 69.150, 77.875 ,92.5 )

IQR = Q3-Q1 =    15.225

3)

1.5IQR =    22.8375  
      
lower bound=Q1-1.5IQR=   39.8125  
      
upper bound=Q3+1.5IQR=   100.7125  
      
outlier =values outside lower bound and upper bound      
count below lower bound=   0  
count above upper bound=   0  
total outlier =    0  

there is no outliers.

4)

z score of min = (X-µ)/σ = (48.5-69.657)/10.20277 = -2.07

z score of max = (92.5-69.657)/10.20277 = 2.24

z score of Q1 = (62.65-69.657)/10.20277 = -0.69

z score of Q3 = (77.875-69.657)/10.20277 = 0.81