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Date
Sweet Crude $
Brent Crude $
1
7-May-90
18
.89
17.05
18-May-90
18.78
17.08
21-May-90
18.26
16.65
22-May-90
17.51
16.48
23-May-90
16.25
15.7
24-May-90
16.02
15.8
25-May-90
16.12
15.95
29-May-90
18
15.48
30-May-90
17.88
15.98
31-May-90
17.47
15.3
1-Jun-90
17.51
15.43
4-Jun-90
17.09
15.35
5-Jun-90
16.41
14.78
6-Jun-90
16.91
14.8
7-Jun-90
16.65
15.03
8-Jun-90
16.78
14.68
11-Jun-90
16.82
14.73
12-Jun-90
17.39
14.95
1) Use the per-barrel oil price data provided above to create a line chart using Excel that plots both prices. Be sure that the date is on the X-axis and the price in dollars is on the Y-Axis. (Chapter 2)
2) Use the per-barrel oil price data provided above to create a Bar chart using Excel that plots both prices. Be sure that the date is on the X-axis and the price in dollars is on the Y-Axis. (Chapter 2)
3) Create descriptive statistics including mean, mode, median, range, high, low, variance and standard deviation for the two different oil prices above. Comment on the similarities and differences. (Chapter 3)
Date
Sweet Crude
Descriptive Statistics for Sweet Crude
Mean:
17.26
Median:
17.24
Mode:
Range:
2.87
Standard Deviation:
.8617
Sample Variance:
.7425
Date
Brent Crude
Descriptive Statistics for Brent Crude
Mean:
15.62
Median:
14.96
Mode:
0
Range:
2.40
Standard Deviation:
1.036
Sample Variance:
1.074
24-May-90
16.02
8-Jun-90
14.68
25-May-90
16.12
11-Jun-90
14.73
23-May-90
16.25
5-Jun-90
14.78
16.41
6-Jun-90
14.80
7-Jun-90
12-Jun-90
14.95
16.78
7-Jun-90
15.03
16.82
31-May-90
15.30
16.91
4-Jun-90
15.35
17.09
1-Jun-90
15.43
17.39
29-May-90
15.48
17.47
23-May-90
15.70
17.51
24-May-90
15.80
22-May-90
17.51
25-May-90
15.95
30-May-90
17.88
30-May-90
15.98
22-May-90
16.48
21-May-90
18.26
21-May-90
16.65
18-May-90
18.78
17-May-90
17.05
17-May-90
18.89
18-May-90
17.08
4) What is the kurtosis (research kurtosis on your own) and skew of the two oil prices shown above? What does this mean? Explain. (Chapter 3)
Sweet Crude skew is 0.35895706 & kurtosis is 2.1013
Brent Crude skew is 0.55872617 & kurtosis is 2.0791
“Two measures called skew and kurtosis are used to view investment returns. Each of these measures tells us how different the distribution we’re looking at (the investment returns) is from a normal distribution. The more it differs from a normal distribution, the less the importance of the standard deviation statistic.
The skew tells us whether the investment returns are shifted one way or another from the mean. A normal distribution has a skew of 0. A positive skew is a good thing in investing, as it tells us there are a greater number of returns greater than the average than there are returns less than the average. Of course, if there’s a negative average where a few very big losing months pull down the average of a bunch of small positive months, a positive skew might not help you.
In contrast, systems have negative skews. This doesn’t mean they are bad systems, rather it tells us they have more months below the monthly average than they do above the monthly average. If the monthly average is still positive, however, this is usually just a case of a system having a few very big winning months and a lot of very small winning months that are below its average monthly gain.
Kurtosis tells us whether or not there is evidence of extreme values happening with greater frequency than we would expect from a normal distribution. A normal distribution has a kurtosis of 0. A large positive reading of kurtosis tells us there are more occurrences of outlier events than we would expect from a normal distribution. Going back to our example, if we got on a plane with a basketball team on board, our graph of heights would show an abnormally large number of readings several standard deviations away from the average. In the investing world, we often label returns with a positive kurtosis as having “fat tails” – referring to the “tails” or end of the curves on the bell curve graph. “ (1)
5) Create a histogram for each of the two oil prices shown. (Chapter 2)
Sweet Crude
|
10 |
||||||||
|
8 |
||||||||
|
6 |
||||||||
|
4 |
||||||||
|
2 |
||||||||
|
$15.00 |
$16.00 |
$17.00 |
$18.00 |
$19.00 |
Brent Crude
|
9 |
|
|
7 |
|
|
5 |
|
|
3 |
|
|
1 |
|
|
$13.00 |
$14.00 |
6) Does the data seem closer to being normal (empirical rule) or not? Use you answers from problems 1-5 above to explain. (Chapter 3)
The data set along with Bell Shape curve will be implied to Empirical Rule. The range of the data fall from the mean can vary from 1 to 3 standard deviations or fall within ±3 standard deviations. The Empirical Rule for normal distribution or bell shaped curve defines that 68% of the observations roughly expected to fall within 1 standard deviation of the mean, 95% of the observations expected to fall within 2 standard deviations of the mean and 99.73% of the data values can be expected fall within three standard deviation interval of the mean.
7) Your employer, Woodbridge Electric Inc., wants to offer a warranty on the new compact fluorescent light bulb that they have produced and tested. You are called into a meeting and operational experts provide the following data: mean bulb life = 8000 hours, standard deviation = 400 hours (assume a normal distribution). The financial people tell you that the firm cannot afford to replace more than 2.5% of the bulbs under warranty. Some members of the board of directors are pressuring you to come up with a warranty of 7000 hours. The marketing people are pressuring you to create a warranty of 7500 hours. Use the data and adhere to the 2.5% financial constraint above to make your calculations and recommend the highest warranty that you can. What do you recommend as a warranty? (Chapter 7)
Warranty of 7216 hours won’t exceed 2.5% replacements. The Table for Standard Normal Distribution is organized as a cumulative ‘area’ from the left corresponding to the standardized variable z. The Standard Normal Distribution is also symmetric (called a Bell Curve) which means it’s an interpretive procedure to Look-Up the area from the Table. For standardized variable z = -1.96 the Table left column shows two (2) significant digits and one (1) additional significant digit in the top row corresponding to a left area = 0.025. Standardized Variable: z = (x – µ)/(σ) = (x – 8000)/(400)) = -1.96
Algebraically solve for x:
x = 7216
8) Why did you choose this figure as the warranty? What percentage of bulbs would need to be replaced if you chose a warranty of 7000 or 7500? Justify your answer statistically. How do you explain this to the board of directors, marketing people and financial people? (Chapter 7)
Warranty of 7000 hours won’t exceed 0.6% replacements.
NORMAL DISTRIBUTION, STANDARDIZED VARIABLE z, PROBABILITY “LOOK-UP”
STANDARDIZED VARIABLE: z = (x – µ)/(σ)
= (7000 – 8000)/(400)) = -2.5
SAMPLE MEAN: x = 7000
POPULATION MEAN: µ = 8000
POPULATION STANDARD DEVIATION: σ = 400
The Table for Standard Normal Distribution is organized as a cumulative area from the left corresponding to the standardized variable z. The Standard Normal Distribution is also symmetric (called a Bell Curve) which means it’s an interpretive procedure to Look-Up the area from the Table. For standardized variable z = -2.5 the Table left column shows two (2) significant digits and one (1) additional significant digit in the top row corresponding to a left ‘area’ = 0.006. And due to Table’s cumulative nature, the corresponding right area = 1 – 0.006
9) The Operational experts of Woodbridge Electric have announced a breakthrough in the production process and the mean of the same bulb has increased to 9000 hours and the standard deviation has decreasing to 200 hours (again assume a normal distribution). Using the same 2.5% constraint how does this affect the warranty? Is this good news or bad for the customers and the firm? (Chapter 7)
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Business 221 – NVCC – Group Problems
