Question

A survey was conducted to investigate motor fuel octane ratings of several blends of gasoline, and the following data is presented:

88.5	98.8	89.6	92.2	92.7	88.4	87.5	90.9
94.7	88.3	90.4	83.4	87.9	92.6	87.8	89.9
84.3	90.4	91.6	91.0	93.0	93.7	88.3	91.8
90.1	91.2	90.7	88.2	94.4	96.5	89.2	89.7
89.0	90.6	88.6	88.5	90.4	84.3	92.3	92.2
89.8	92.2	88.3	93.3	91.2	93.2	88.9
91.6	87.7	94.2	87.4	86.7	88.6	89.8
90.3	91.1	85.3	91.1	94.2	88.7	92.7
90.0	86.7	90.1	90.5	90.8	92.7	93.3
91.5	93.4	89.3	100.3	90.1	89.3	86.7
89.9	96.1	91.1	87.6	91.8	91.0	91.0

1. Construct stem-and-leaf diagram for these data and compute the sample quartiles. (10 pts)
2. Compute sample median, mode, mean, variance and standard deviation. (5 pts)
3. Construct a frequency distribution and histogram for the ratings by using eight bins. What is the shape of the distribution? (10 pts)

Answer 1

Construct stem-and-leaf diagram for these data and compute the sample quartiles. (10 pts)

We need to sort the data in ascending order first to carry the calculations.

	X	X^2
1	83.4	6955.56
2	84.3	7106.49
3	84.3	7106.49
4	85.3	7276.09
5	86.7	7516.89
6	86.7	7516.89
7	86.7	7516.89
8	87.4	7638.76
9	87.5	7656.25
10	87.6	7673.76
11	87.7	7691.29
12	87.8	7708.84
13	87.9	7726.41
14	88.2	7779.24
15	88.3	7796.89
16	88.3	7796.89
17	88.3	7796.89
18	88.4	7814.56
19	88.5	7832.25
20	88.5	7832.25
21	88.6	7849.96
22	88.6	7849.96
23	88.7	7867.69
24	88.9	7903.21
25	89	7921
26	89.2	7956.64
27	89.3	7974.49
28	89.3	7974.49
29	89.6	8028.16
30	89.7	8046.09
31	89.8	8064.04
32	89.8	8064.04
33	89.9	8082.01
34	89.9	8082.01
35	90	8100
36	90.1	8118.01
37	90.1	8118.01
38	90.1	8118.01
39	90.3	8154.09
40	90.4	8172.16
41	90.4	8172.16
42	90.4	8172.16
43	90.5	8190.25
44	90.6	8208.36
45	90.7	8226.49
46	90.8	8244.64
47	90.9	8262.81
48	91	8281
49	91	8281
50	91	8281
51	91.1	8299.21
52	91.1	8299.21
53	91.1	8299.21
54	91.2	8317.44
55	91.2	8317.44
56	91.5	8372.25
57	91.6	8390.56
58	91.6	8390.56
59	91.8	8427.24
60	91.8	8427.24
61	92.2	8500.84
62	92.2	8500.84
63	92.2	8500.84
64	92.3	8519.29
65	92.6	8574.76
66	92.7	8593.29
67	92.7	8593.29
68	92.7	8593.29
69	93	8649
70	93.2	8686.24
71	93.3	8704.89
72	93.3	8704.89
73	93.4	8723.56
74	93.7	8779.69
75	94.2	8873.64
76	94.2	8873.64
77	94.4	8911.36
78	94.7	8968.09
79	96.1	9235.21
80	96.5	9312.25
81	98.8	9761.44
82	100.3	10060.09
Total	7423.1	672664.3

For stem and leaf plot we will use the integer part as stem and the decimals as the leaf

Stem	Leaf
83	4
84	3 3
85	3
86	7 7 7
87	4 5 6 7 8 9
88	2 3 3 3 4 5 5 6 6 7 9
89	0 2 3 3 6 7 8 8 9 9
90	0 1 1 1 3 4 4 4 5 6 7 8 9
91	0 0 0 1 1 1 2 2 5 6 6 8 8
92	2 2 2 3 6 7 7 7
93	0 2 3 3 4 7
94	2 2 4 7
95
96	1 5
97
98	8
99
100	3

Key : 83 | 4 = 83.4

Sample quartiles are found using

Where N = 82

Therefore  

Quartile	Value	Answer
1st	20.75th = 20th + 0.75 (21st - 20th)	88.575
2nd	41.5	90.4
3rd	62.25	92.2

Compute sample median, mode, mean, variance and standard deviation. (5 pts)

Sample median = 2nd quartile

= 90.4

Mode = value with highest frequency

Since there more than 1 value with having being repeated 3 times there is no mode.

Mean =

= 90.526

SD =

Var = 8.4402

SD = 2.9052

Construct a frequency distribution and histogram for the ratings by using eight bins. What is the shape of the distribution? (10 pts)

For histogram we need continuous classes.So we take a width of 3 for every class starting form 82 to 106. That will give us eight bins

Classes	Frequency
82-85	3
85-88	10
88-91	34
91-94	27
94-97	6
97-100	1
100-103	1
103-106	0
Total	82

As we can that the highest no. of values are concentrated in the centre and the rightand left have relatively low frequencies and look similar. So we can say that the distribution has a symetrical shape.