Statistical Analysis
Question One
The file “SALES” contains weekly sales figures for 2012 for a newly introduced 1-litre carton of orange juice at a large suburban supermarket in Perth.
Weekly Sales Histogram for Orange Juice Cartons
| Bin | Frequency | Cumulative % |
| 80 | 0 | 0.00% |
| 100 | 4 | 8.00% |
| 120 | 7 | 22.00% |
| 140 | 10 | 42.00% |
| 160 | 12 | 66.00% |
| 180 | 9 | 84.00% |
| 200 | 5 | 94.00% |
| 220 | 3 | 100.00% |
| More | 0 | 100.00% |
From the above histogram, it is evidenced that following the introduction of this commodity (1 litre Orange Juice) in Perth suburban market, it took the time of the customers to familiarize themselves with the newly launched orange juice. Therefore, the sales slugged in between 83rd and 100th week as evidenced the sales of only four cartons of 1-litre orange juice. However, after the first twenty weeks after the launching of the product in the market, the demand for the orange juice slightly increased as more than 7 units of the commodity was sold in the supermarket. The number of cartons sold increased, showing an upward trend in the market for the orange juice products until. It was 140-160 week that the supermarket sold the highest number of cartons at 12. This was an indication that the company’s sales reached its peak in the 160th week. However, the number of juice cartons sold in the subsequent weeks started declining. Sales volume steadily dropped after the 160th week. The drop in the sale volume for the orange juice was likely attributed to increased competition probably following the entry of other competitors dealing in the similar products that are only slightly differentiated. On the other hand, the declining sales of the orange juice would have been contributed to a fall in effective demand following changes in the consumer income and purchasing powers.
Question Two
The following table contains data for the number of withdrawals from a bank in an industrial area.
| Year | Withdrawals ($ ten thousands) |
| 2000
2001 2002 2003 2004 2005 2006 2007 2008 2009 |
23.6
18.3 26.0 24.9 28.1 25.3 29.1 24.7 30.1 27.8 |
Graphical Representation
Trend Analysis
The bank withdrawals from the industrial area from 2000 to 2009 were unsteady and unpredictable. As graphically demonstrated above, the trend in the withdrawal was zigzag characterized by ups and downs. For instance, in 2000, the withdrawal was $236,000 while in the following year, the amount withdrawn dropped to $183,000. The following year (2002), the figure increased to $260,000. The pattern continued in such a manner that although showing an upward trend in the amount withdrawn, all the odd years reported a slight decline in the industrial withdrawals compared to the even years. However, when the odd years are compared on their own, there was an increase in the amount withdrawn. This was an indication of increased business and operational activities in the industrial area as evidenced by the upward trend in annual withdrawals. The differences in the withdrawals between odd and even years may be attributed to changes in the economic cycles that are characterized by recessions in the odd years and recovery in the subsequent even years, holding other factors constant.
Question Three
The manufacturer of Boston and Vermont asphalt shingles provides its customers with a 20-year warranty on most of its products. To determine whether a shingle will last as long as the warranty period, accelerated-life testing is conducted at the manufacturing plant. Accelerated-life testing exposes a shingle to the stresses it would be subject to in a lifetime of normal use via an experiment in a laboratory setting that takes only a few minutes to conduct. In this test, a shingle is repeatedly scraped with a brush for a short period of time, and the shingle granules removed by the brushing are weighed (in grams). Shingles that experience low amounts of granule loss are expected to last longer in normal use than shingles that experience high amounts of granule loss. In this situation, a shingle should experience no more than 0.8 gram of granule loss if it is expected to last the length of the warranty period. The file “GRANULE” contains a sample of 170 measurements made on the company’s Boston shingles and 140 measurements made on Vermont shingles.
- List the five-number summaries for the Boston shingles and for the Vermont shingles.
| Five Number Similarity | Boston Shingles | Vermont Shingles |
| Minimum | 0.04 | 0.02 |
| First Quartile, Q1 | 0.17 | 0.13 |
| Median | 0.23 | 0.20 |
| Third Quartile, Q3 | 0.32 | 0.28 |
| Maximum | 0.98 | 0.83 |
- Construct Boxplots for the two brands of shingles and describe the shapes of the distributions.
- Comment on the ability of each type of shingle to achieve a granule loss of 0.8 gram or less.
The box-plot analysis indicates that Vermont Shingles have a higher ability to last longer as compared to the Boston Shingles given that it has a lower spread, an indication that it losses less granules that does Boston Shingles.
Question Four
A tyre manufacturer believes that the tread-life of its snow tyres can be described by a Normal Model with a mean of 32,000 miles and standard deviation of 2500 miles.
- If a person buys a set of these tyres, would it be reasonable for him to expect they’ll last 40,000 miles?
Z-score = (X – Xm)/S.D
= (40,000 – 32,000)/2500
= 3.2
- Approximately what fraction of these tyres can be expected to last less than 30,000 miles?
(32,000 – 30,000)/2500 = 0.8
- Approximately what fraction of these tyres can be expected to last between 30,000 and 35,000 miles?
(35,000 – 30,000)/2500 = 2
- Estimate the inter quartile range (IQR) of the tread-lives.
Q1 = 32,000 – 2500 = 29,500
Q3 = 32,000 + 2500 = 34,500
Q3 – Q1 = 34,500 – 29,500 = 5,000
- In planning a marketing strategy, a local tyre dealer wants to offer a refund to any customer whose tyres fail to last a certain number of miles. However, the dealer does not want to take too big a risk. If the dealer is willing to give refunds to no more than 1 of every 25 customers, for what mileage can he guarantee these tyres to last?
(32000/2500) = 12.8
Q5. [8 Marks]
Each year, ratings are complied concerning the performance of new cars during the first 90 days of use. Suppose that the cars have been categorized according to whether a car needs warranty-related repair (Yes, No) and the country in which the company manufacturing a car is (United States or Not United States).
Based on the data collected, the probability that the new car needs a warranty repair is 0.04, the probability that the car was manufactured by a US –based company is 0.60 and the probability that the new car needs a warranty repair and was manufactured by a US –based company is 0.025.
- Construct a contingency table to evaluate the probabilities of warranty-related repair.
| Manufacturers | ||||
| U.S | Not U.S | Total | ||
| Need for Repair in 90 days | Yes | 0.025 | 0.015 | 0.040 |
| No | 0.975 | 0.985 | 1.95 | |
| Total | 1.00 | 1.00 | ||
A new car (within 90 days of sale) is chosen at random. Use your contingency table to answer the following questions:
- Needs a warranty repair?
Probability of warranty repair = 0.40
- Needs a warranty repair and was manufactured by a US –based company?
Warranty repair and US based = 0.025
- Needs a warranty repair or was manufactured by a US –based company?
Repair or US Manufacturer = 0.04 + 0.025 = 0.065
- Needs a warranty repair or was not manufactured by a US –based company?
Repair or Non US manufacturer = 0.04 + 0.015 = 0.055
Question Six
The Australian quarterly total sales of wine (in millions of litres) and a quarterly Wine Price Index (WPI) are listed here for the 10 quarters from December 2002 to March 2005.
| Wine Sales | WPI |
| 97.6
71.4 86.7 92.0 99.2 76.2 87.7 95.3 103.7 76.4 |
156.0
153.7 155.6 157.0 158.3 156.4 159.1 161.0 162.6 159.5 |
- Using Excel, produce a scatterplot of this data, including least-squares line, and full regression statistics.
- Estimate the quarterly total sales of wine when the WPI is 155.0
Regression Equation: WPI = 0.139Q + 145.5
When WPI = 155.0, Q = (155 – 145.5)/0.139 = 68.3 million litres
Total Wines Sales = 68.3 million litres
- If an economist predicts that quarterly total wine sales should reach 105 million litres for a WPI of 165.0, comment on whether the prediction would be a valid use of this analysis.
At Q = 105, WPI = 0.139(105) + 145.5 = 160
Although not accurate, the prediction that at Q = 105, the WPI should be 165.0 is acceptable at S.E = 5. Using a standard error of 5.00 units, then this prediction is statistically acceptable and valid. However, at standard error that is less than 5.00, this prediction is deemed inaccurate and statistically invalid.
- There are some assumptions involved in using a linear regression model. Do you think they are reasonable for this case? Why?
The construction of a linear regression model is founded on multiple assumptions about the relationship, response variables, and the predictor variables. Although these assumptions are statistically important in understanding the relationship between the response and the predictor variables, their application is limited to certain circumstances. These extensions and assumptions make the procedure of estimating the linear regression model more time-consuming and complex (Pindyck and Rubinfeld 78). Besides, the introduction of the underlying assumptions in the regression modeling process requires more detailed and quantitative data in order to accurately predict the regression model. Based on the above limitations and the nature of the data at hand, it is advisable that we relax or entirely eliminate these assumptions, to aid in the understanding and construction of the linear regression model.
Work Cited
Robert, Pindyck and Daniel, Rubinfeld. Econometric Models and Economic Forecasts. John Wiley, 2008. Print.
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