Question

The data set below contains the electricity​ costs, in​ dollars, during July 2013 for a random sample of

30 ​one-bedroom apartments in a large city. Complete parts​ (a) and​ (b).

136
91
199
184
114
196
124
132
146
140
138
168
141
155
193
159
86
175
181
148
210
155
98
163
222
149
111
162
170
116

a. Decide whether the data appear to be approximately normally distributed by comparing data characteristics to theoretical properties.

A.​No, the mean and median of the data are too different for the distribution to be normal.

B.No, the range of the data is too great for the distribution to be normal.

C.Yes, the distribution of the data appears to closely resemble a normal distribution.

D.No, the interquartile range of the data is too small for the distribution to be normal

B) Decide whether the data appear to be approximately normally distributed by constructing a normal probability plot.

Answer 1

X	(X - X̄)²
136	258.14
91	3729.14
199	2202.74
184	1019.74
114	1449.07
196	1930.14
124	787.74
132	402.67
146	36.80
140	145.60
138	197.87
168	253.87
141	122.47
155	8.60
193	1675.54
159	48.07
86	4364.80
175	525.94
181	837.14
148	16.54
210	3356.2711
155	8.6044
98	2923.2044
163	119.5378
222	4890.6711
149	9.4044
111	1686.4711
162	98.6711
170	321.6044
116	1300.8044

	X	(X - X̄)²
total sum	4562	34727.867
n	30	30

mean =    ΣX/n =    4562.000   /   30   =   152.0667
                      
sample variance =    Σ(X - X̄)²/(n-1)=   34727.8667   /   29   =   1197.5126
                      
sample std dev =   √ [ Σ(X - X̄)²/(n-1)] =   √   1197.5126   =       34.6051
Median=0.5(n+1)th value =    15.5   th value of sorted data
=   152  
      
range=max-min =    222   -   86   =   136
quartile , Q1 = 0.25(n+1)th value=   7.75   th value of sorted data
=   130  
      
Quartile , Q3 = 0.75(n+1)th value=   23.25   th value of sorted data
=   176.5  
      
IQR = Q3-Q1 =    46.5  
Skewness=   -0.0102
  

C.Yes, the distribution of the data appears to closely resemble a normal distribution.

b)

rank,i	(i-0.5)/n	z score	data
1	0.0167	-2.13	136
2	0.0500	-1.64	86
3	0.0833	-1.38	91
4	0.1167	-1.19	98
5	0.1500	-1.04	111
6	0.1833	-0.90	114
7	0.2167	-0.78	116
8	0.2500	-0.67	124
9	0.2833	-0.57	132
10	0.3167	-0.48	138
11	0.3500	-0.39	140
12	0.3833	-0.30	141
13	0.4167	-0.21	146
14	0.4500	-0.13	148
15	0.4833	-0.04	149
16	0.5167	0.04	155
17	0.5500	0.13	155
18	0.5833	0.21	159
19	0.6167	0.30	162
20	0.6500	0.39	163
21	0.6833	0.48	168
22	0.7167	0.57	170
23	0.7500	0.67	175
24	0.7833	0.78	181
25	0.8167	0.90	184
26	0.8500	1.04	193
27	0.8833	1.19	196
28	0.9167	1.38	199
29	0.9500	1.64	210
30	0.9833	2.13	222

data appear to be approximately normally distributed