OPRE 3360 – SPRING 2014 PRACTICE PROBLEMS EXAM 1 CHAPTER 1-­‐3 Note: These are ADDITIONAL problems in preparation for your first exam. Use them in combination with the problems and material discussed in class, and with the assignments. 1. Consider the following data. 8.9 10.2 11.5 7.8 10.0 12.2 13.5 14.1 10.0 12.2 6.8 9.5 11.5 11.2 14.9 7.5 10.0 6.0 15.8 11.5 a) Construct a frequency distribution. b) Construct a percent cumulative frequency distribution. Class Frequency Cumulative Percent Frequency 6.0 -­‐ 7.9 4 20 8.0 -­‐ 9.9 2 30 10.0 -­‐ 11.9 8 70 12.0 -­‐ 13.9 3 85 14.0 -­‐ 15.9 3 100 20 2. NRF/BIG research provided results of a consumer holiday spending survey (USA Today, December 20, 2005). The following graph reports the dollar amount of holiday spending for a sample of 25 consumers. 7
Frequency
6
5
4
3
2
1
0
0-249
250-499 500-749 750-999 1000-12491250-14991500-17591750-19992000-2249
Holiday Spending
a) Comment on the shape of the histogram This histogram is skewed to the right. That is evident in the graph because of the long right tail. b) The skewness value for this data is 1.15, the mean is $738 and the median is $610. Are these values supporting your conclusion in part a)? Explain briefly. Yes, these values support the conclusion in part a). Skewness greater than 1 indicates than the distribution is highly skewed to the right, and as the values of Mean > Median we know that the mean has been inflated by the large values in the right end of the 1 distribution. In cases like this, the median is a better indicator of where the center of the distribution is. 3. Data on market value and profits for a sample of 50 Fortune 500 companies are shown in the following crosstabulation. Profit ($1000s) Market Value 0-­‐300 300-­‐600 600-­‐900 900-­‐1200 Total ($1000s) 0-­‐8000 23 4 27 8000-­‐16000 4 4 2 2 12 16000-­‐24000 2 1 1 4 24000-­‐32000 1 2 1 4 32000-­‐40000 2 1 3 Total 27 13 6 4 50 a) Compute the row percentages for your crosstabulation. Profit ($1000s)
Market Value
($1000s)
0-300
300-600
600-900
900-1200
Total
0-8000
85.19
14.81
0.00
0.00
100
8000-16000
33.33
33.33
16.67
16.67
100
16000-24000
0.00
50.00
25.00
25.00
100
24000-32000
0.00
25.00
50.00
25.00
100
32000-40000
0.00
66.67
33.33
0.00
100
b) Comment on any relationship between the variables There appears to be a positive relationship between Profit and Market Value. As
profit goes up, Market Value goes up.
4. According to an annual consumer spending survey, the average monthly Bank of America Visa credit card charge was $1838 (U.S. Airways Attaché Magazine, December 2003). A sample of monthly credit card charges provides the following data. 236 1710 1351 825 7450 316 4135 1333 1584 387 991 3396 170 1428 1688 a)
b)
c)
d)
e)
Compute the mean and median. Compute the first and third quartiles. Compute the range and interquartile range. Compute the variance and standard deviation. Do the data contain outliers? 2 a.
x=
Σxi 27000
=
= 1800
n
15
Median 8th position = 1351
b. Q1:
⎛ 25 ⎞
i=⎜
⎟15 = 3.75
⎝ 100 ⎠
4th position: Q1 = 387
Q 3:
⎛ 75 ⎞
i=⎜
⎟15 = 11.25
⎝ 100 ⎠
12th position: Q3 = 1710
c. Range = 7450 - 170 = 7280
IQR = Q3 - Q1 = 1710 - 387 = 1323
d.
s2 =
Σ( xi − x ) 2 51, 454, 242
=
= 3, 675,303
n −1
15 − 1
s = 3,675,303 = 1917
e. We just focus on the two largest values, and calculate the z-scores
z=
x − x 4135 − 1800
=
= 1.22
s
1917
z=
x − x 7450 − 1800
=
= 2.95 do not indicate outliers.
s
1917
3 5. For the following observations, calculate the correlation coefficient and indicate what kind of relationship (if any) exists between women's height (inches) and annual starting salary ($1000). a
b
Height'(x)
Salary'(y)
64
63
68
65
67
66
65
64
66
45
40
39
38
42
45
43
35
33
Variance
Std.Dev(x)
Std.BDev(y)
Corr.BCoeff.
c
xi0mean(x)
+1.33
+2.33
2.67
+0.33
1.67
0.67
+0.33
+1.33
0.67
d
yi0mean(y)
5
0
+1
+2
2
5
3
+5
+7
Sums
e
c'x'd
+6.67
0.00
+2.67
0.67
3.33
3.33
+1.00
6.67
+4.67
+1.00
f
g
(xi0mean(x))^2 (yi0mean(y))^2
1.78
25
5.44
0
7.11
1
0.11
4
2.78
4
0.44
25
0.11
9
1.78
25
0.44
49
20.00
142
+0.125
1.58
4.21
+0.019
As the correlation coefficient is -­‐0.019, a value very close to zero, there is no linear relationship between women’s height and salary. 6. Automobiles traveling on a road with a posted speed limit of 55 miles per hour are checked for speed by a state police radar system. Following is a frequency distribution of speeds. Speed (miles per hour) Frequency 50–54 40 55–59 145 60–64 175 65–69 75 70–74 15 Total 450 a) What is the mean speed of the automobiles traveling on this road? b) Compute the variance and the standard deviation. 4 fi
40
145
175
75
15
450
Mi
52
57
62
67
72
f i$ M i
2080
8265
10850
5025
1080
27,300
-8.67
-3.67
1.33
6.33
11.33
a.
x=
27,300
= 60.67
450
b.
s2 =
s = 22.72 = 4.77 10,200.01
= 22.72 449
5 75.1689
13.4689
1.7689
40.0689
128.3689
3006.756
1952.9905
309.5575
3005.1675
1925.5335
10,200.01