BUSINESS Statistics
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group_problems_bus_2212nd_assignment.docxNorthern Virginia Community College
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Assignment for Course: |
Business 221 – Business Statistics |
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Submitted to: |
Dr. Mark DAntonio |
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Submitted by: |
(Name of Group goes here) |
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Date of Submission: MM/DD/YYYY
Title of Assignment: Group problems (updated 4 May 2011)
Instructions: Read the questions very carefully because the requirements for each are specific. Each group should turn in one copy of this WORD document (email to me) with the names of the participating group members typed on the bottom of this cover page. Answer all questions fully in this document only. That is, I want to see the question below followed by your answer so I do not have to guess what you are answering. This project is worth 200 points.
CERTIFICATION OF AUTHORSHIP: I certify that the members of the group named above equally contributed to the assignment that is attached. Any assistance received in its preparation is fully acknowledged and disclosed in the paper. Any sources from which the group used data, ideas or words, either quoted directly or paraphrased is also disclosed. Please type names below.
Student's Signature: ______________________________________
Student's Signature: ______________________________________
Student's Signature: ______________________________________
Student's Signature: ______________________________________
Student's Signature: ______________________________________
Student's Signature: ______________________________________
Student's Signature: ______________________________________
Student's Signature: ______________________________________
Northern Virginia Community College - BUS 221 - for Dr. Mark D Antonio
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Date |
Sweet Crude $ |
Brent Crude $ |
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17-May-90 |
18.89 |
17.05 |
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18-May-90 |
18.78 |
17.08 |
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21-May-90 |
18.26 |
16.65 |
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22-May-90 |
17.51 |
16.48 |
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23-May-90 |
16.25 |
15.7 |
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24-May-90 |
16.02 |
15.8 |
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25-May-90 |
16.12 |
15.95 |
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29-May-90 |
18 |
15.48 |
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30-May-90 |
17.88 |
15.98 |
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31-May-90 |
17.47 |
15.3 |
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1-Jun-90 |
17.51 |
15.43 |
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4-Jun-90 |
17.09 |
15.35 |
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5-Jun-90 |
16.41 |
14.78 |
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6-Jun-90 |
16.91 |
14.8 |
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7-Jun-90 |
16.65 |
15.03 |
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8-Jun-90 |
16.78 |
14.68 |
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11-Jun-90 |
16.82 |
14.73 |
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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.
1) 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.
2) Create descriptive statistics including mean, mode, median, range, high, low, variance and standard deviation for the two different oil prices above. Comment in detail on the similarities and differences.
SWEET CRUDE DESCRIPTIVE STATISTICS GENERATED USING THE EXCEL SHEET.
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3.1. Sweet Crude $ |
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Mean |
17.26333333 |
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Standard Error |
0.203019687 |
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Median |
17.24 |
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Mode |
17.51 |
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Standard Deviation |
0.861339586 |
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Sample Variance |
0.741905882 |
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Kurtosis |
-0.618111121 |
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Skewness |
0.403826692 |
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Range |
2.87 |
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Minimum |
16.02 |
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Maximum |
18.89 |
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Sum |
310.74 |
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Count |
18 |
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Largest(1) |
18.89 |
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Smallest(1) |
16.02 |
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Confidence Level (95.0%) |
0.428334095 |
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3.2. Brent Crude $ |
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Mean |
15.62333333 |
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Standard Error |
0.183785539 |
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Median |
15.455 |
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Mode |
#N/A |
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Standard Deviation |
0.779736004 |
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Sample Variance |
0.607988235 |
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Kurtosis |
-0.64977129 |
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Skewness |
0.628566936 |
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Range |
2.4 |
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Minimum |
14.68 |
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Maximum |
17.08 |
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Sum |
281.22 |
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Count |
18 |
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Largest(1) |
17.08 |
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Smallest(1) |
14.68 |
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Confidence Level (95.0%) |
0.387753589 |
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Mean |
Mode |
Median |
Range |
High |
Low |
Variance |
Standard Deviation |
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Sweet Crude $ |
17.2600 |
17.5100 |
17.2400 |
2.8700 |
18.8900 |
16.0200 |
0.7419 |
0.8600 |
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Brent Crude $ |
15.6200 |
- |
15.4600 |
2.4000 |
17.0800 |
14.6800 |
0.5742 |
0.7700 |
SIMILARITIES
Comparison between the two oil companies (Sweet and Brent) using the descriptive statistics in Excel shows several similarities.
a) Regarding Sweet crude, the range is 2.87 while that of the Brent crude is 2.40.This shows that the dispersion in the prices of the two companies is small over different periods of time.
b) In the case of standard deviation, Sweet crude has 0.86134 while Brent crude has 0.77974, which means, both of them are almost equal. The standard deviation measures how concentrated the data are around the mean; the more concentrated, the smaller the standard deviation. As we see here, the two companies have relatively low standard deviations; therefore, many of them in the data are concentrated around their means.
c) The median is the number at which half your measurements are more than that number and half are less than that number. The median for the Sweet Crude is 17.240 while it is 15.455 for the Brent Crude. Closely looking at the data and the median, we can deduce that the median of the two show resemblance in terms of the data above and the data below the mean.
DIFFERENCES
a) In the case of Sweet crude, the mean is 17.26333 while it is 15.62333 in the Brent Crude. This shows about 1.64 differences in the prices of the two companies. So that the different average prices show an implication of prices that are also different since the periods of times taken for the study remain constant for the two companies.
b) The variance of the Sweet crude is 0.741906 while that of the Brent is 0.607988. This difference shows that the variability of the data in the distribution around the mean differs. The variance is almost equal to the SD squared, and therefore it is expressed in the units of the data squared.
c) On one hand, the Sweet crude has a maximum (high) of 18.89 and a minimum (low) of 16.02. On the other hand, the Brent crude has a maximum of 17.08 and a low of 14.68. The differences between the two are 1.81 and 1.34 maximum and minimum respectively. This shows that the Sweet crude is more expensive than the Brent crude. Therefore, they are different in their prices.
3) What is the kurtosis (research kurtosis on your own) and skew of the two oil prices shown above? What does this mean? Explain.
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Kurtosis |
Skewness |
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Sweet Crude $ |
-0.61811 |
0.403827 |
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Brent Crude $ |
-0.64977 |
0.628567 |
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The kurtosis for the Sweet crude is -0.61811 as per the results of the data analysis from the Excel. The Skewness is 0.403827.
The kurtosis for the Brent crude is -0.64977 as per the results of the data analysis generated using Excel. The skewness is 0.628567.
Skewness is a measure of symmetry, or more precisely, the lack of symmetry. A distribution, or data set, is symmetric if it looks the same to the left and right of the center point while Kurtosis is a measure of whether the data are peaked or flat relative to a normal distribution. That is, data sets with high kurtosis tend to have a distinct peak near the mean, decline rather rapidly, and have heavy tails. Data sets with low kurtosis tend to have a flat top near the mean rather than a sharp peak. A uniform distribution would be the extreme case. From the data above having calculated the kurtosis and the skewness, we can deduce that, because both have positive skewness, it shows that the data are skewed to the right and it is an indication of success. The two distributions above with a negative kurtosis values indicate that the distributions have lighter tails and a flatter peak than the normal distribution.
4) Create a histogram for each of the two oil prices shown.
5) Does the data seem closer to being normal (empirical rule) or not? Use your answers from problems 1-5 above to explain.
The empirical rule states that, if a random variable is normally distributed then:
I. Approximately 68% of the observations will lie within plus and minus one standard deviation of the mean.
II. About 95% of the observations will lie within plus or minus two standard deviations of the mean.
III. Practically all, or 99.7% of the observations, will lie within plus and minus three standard deviations of the mean.
By checking if the each set of data pass the three tests stated above, we use the descriptive statistics in question 3 that gives us the following information:
A. Sweet Crude
Mean = 17.26333333
Standard Deviation = 0.861339586
B. Brent Crude
Mean = 15.62333333
Standard Deviation = 0.779736004
The descriptive data for sets A and B above will be tested to verify the empirical rule as follows; (All numbers are rounded off to two decimal places)
Set A
Test 1
Probability of (17.26 – 0.86) <x<17.26 + 0.86) = Probability of (16.4<x<18.12)
Using a calculator function, normalcdf (16.4, 18.12, 17.26, 0.86) = .68268 approximately 68%
Test 2
P (17.26-2(0.86) <x<17.26+2(0.86)) = P (15.54<x<18.98)
= normalcdf (15.54, 18.98, 17.26, 0.86) = 0.954499876 = 95%
Test 3
P (17.26 – 3(0.86) <x<17.26 + 3(0.86))
P (14.68<x<19.84)
normalcdf (14.68, 19.98, 17.26, 0.86) = 0.9978 = 99.7%
Hence proving the descriptive statistics of (set A) being closer to being normally distributed although positively skewed.
Set B
Test 1
P (15.62- 0.78<x< 15.6 +0.78)
P (14.84 <x< 16.38)
= normalcdf (14.84, 16.38, 15.62, 0.78) = 0.676 = 68%
Test 2
P (15.62 – 2(0.78) <x<15.62 + 2(0.78))
P (13.44 <x< 17.16)
= normalcdf (13.44, 17.16, 15.62, 0.78) = 0.973 = 97% which implies about 95% lie within 2 standard deviations.
Test 3
P (15.62 – 3(0.78<x<15.62 + 3 (0.78))
P (13.28, <x<17.96)
normalcdf = (13.28, 17.26, 15.62, 0.78) = .997 = 99.7%
Hence the descriptive statistics of (set B) show the data being closer to being normally distributed although it shows some positive skewness.
6) 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?
z = (x - µ)/
z = (x – 8000)/400
z = (7216 – 8000)/400
z = 0.024998 or 2.499% or approx. 1 in 40
7) 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?
I chose this figure as the warranty due to the 2.5% constraint. It was the closest in proximity to 2.5% or the highest warranty that can be calculated in adherence to the financial constraint.
In regards to some members of the board wanting to utilize a warranty of 7000 hours, this figure is 2.275% and is not the maximum warranty that can be utilized.
z = (x - µ)/
z = (x – 8000)/400
z = (7000 – 8000)/400
z = 0.02275 or 2.275% or approx. 1 in 44
In regards to the marketing people pushing for 7500 hours, this figure is entirely too high, as 7500 hours would result in replacing 10.565% of the compact fluorescent light bulbs.
z = (x - µ)/
z = (x – 8000)/400
z = (7500 – 8000)/400
z = 0.10565 or 10.565% or approx. 1 in 9
8) 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?
Brent Crude $
14-15 15-16 16-17 17-18 18-19 0 10 5 3 0
Per-Barrel oil Price Intervals ($)
Price Frequency
PER BARREL OIL PRICE
Sweet Crude 33010 33011 33014 33015 33016 33017 33018 33022 33023 33024 33025 33028 33029 33030 33031 33032 33035 33036 18.89 18.779999999999987 18.260000000000002 17.510000000000005 16.25 16.02 16.12 18 17.88 17.47 17.510000000000005 17.09 16.41 16.91 16.649999999999999 16.779999999999987 16.82 17.39 Brent Crude 33010 33011 33014 33015 33016 33017 33018 33022 33023 33024 33025 33028 33029 33030 33031 33032 33035 33036 17.05 17.079999999999988 16.649999999999999 16.479999999999986 15.7 15.8 15.950000000000006 15.48 15.98 15.3 15.43 15.35000000000001 14.78 14.8 15.03 14.68 14.73 14.950000000000006DATE
PRICE
PER BARREL OIL PRICE
Sweet Crude 33010 33011 33014 33015 33016 33017 33018 33022 33023 33024 33025 33028 33029 33030 33031 33032 33035 33036 18.89 18.779999999999987 18.260000000000002 17.510000000000005 16.25 16.02 16.12 18 17.88 17.47 17.510000000000005 17.09 16.41 16.91 16.649999999999999 16.779999999999987 16.82 17.39 Brent Crude 33010 33011 33014 33015 33016 33017 33018 33022 33023 33024 33025 33028 33029 33030 33031 33032 33035 33036 17.05 17.079999999999988 16.649999999999999 16.479999999999986 15.7 15.8 15.950000000000006 15.48 15.98 15.3 15.43 15.35000000000001 14.78 14.8 15.03 14.68 14.73 14.950000000000006DATE
PRICE
Sweet Crude $
16-16.7 16.7-17.4 17.4-18.1 18.1-18.8 18.8-19.5 4 1 4 6 1
Per-Barrel oil Price Intervals ($)
Price Frequency
10
Business 221 - NVCC – Group Problems
