ANALYZING THE PRICE OF GASOLINE The assignment this week presents a problem where the American Automobile Association (AA) generates a report on gasoline prices that it distributes to newspapers throughout the state. It further states that on February 18, 1999, the AAA called a random sample of fifty-one stations to determine that day’s price of unleaded gasoline. The following data (in dollars) was given in this report: Table 1 – Prices of Unleaded Gasoline at 51 Stations 1. 07 1. 31 1. 18 1.
01 1. 23 1. 09 1. 29 1.
10 1. 16 1. 080. 96 1.
66 1. 21 1. 09 1. 02 1. 04 1. 01 1.
03 1. 09 1. 111. 11 1.
17 1. 04 1. 09 1. 05 0. 96 1. 32 1.
09 1. 26 1. 111. 03 1. 20 1. 21 1.
05 1. 10 1. 04 0. 97 1. 21 1. 07 1.
170. 98 1. 10 1. 04 1. 03 1. 12 1.
10 1. 03 1. 18 1. 11 1. 091. 06 Create a data array with the gasoline price data data array is defined as “data that have been sorted in ascending or descending order” (Shannon, Groebner, Fry, & Smith, 2002, 72).
The following section presents the data presented in Table 1 as a data array. Data Array 0. 96, 0. 96, 0.
97, 0. 98, 1. 01, 1. 01, 1. 02, 1.
03, 1. 03, 1. 03, 1. 03, 1. 04, 1.
04, 1. 04, 1. 04, 1. 05, 1.
05, 1. 06, 1. 07, 1. 07, 1. 08, 1.
09, 1. 09, 1. 09, 1. 09, 1. 09, 1. 09, 1.
10, 1. 10, 1. 10, 1. 10, 1. 11, 1. 11, 1.
11, 1. 11, 1. 12, 1. 16, 1. 17, 1.
17, 1. 18, 1. 18, 1. 20, 1. 21, 1. 21, 1.
21, 1. 23, 1. 26, 1. 29, 1. 31, 1. 32, 1.
66 Data Analysis Given the data presented in the previous sections, the next few sections use two histograms to estimate the number of prices that are at least $1. 15. The first histogram presents the data using five classes and the second uses fifteen. Histogram #1 Data Used in Histogram #1 (5 classes) Range 0. 70 # of Classes 5 Class Width 0. 1400 Bin # Classes Frequency Relative Frequency Cumulative Frequency Cumulative Relative Frequency 1 0.
9600
37 46 0. 903 1. 2400
984 1. 3800
6601 1 0. 02 51 1. 00 Histogram #1 (using 5 Classes) Estimate of the Number of Prices that are at least $1. 15 Using the histogram presented in the previous section, the estimate of the number of prices that are at least $1.
15 is five. This is because the only values that can be counted fall into bins three, four, and five. Even though bin two may contain values that are above the $1. 15 threshold, they can not be counted as they are not guaranteed to be above the stated value. Therefore the formula for the estimate is: Estimate = B 3 + B 4 + B 5, where B 3 = 4, B 4 = 0 and B 5 = 1. Histogram #2 Data Used in Histogram #2 (15 classes) Range 0.
70 # of Classes 15 Class Width 0. 0467 Bin # Classes Frequency Relative Frequency Cumulative Frequency Cumulative Relative Frequency 1 0. 9600
08 4 0. 082 1. 0067
25 17 0. 333 1. 0534
27 31 0. 614 1. 1001
715 1. 1468
1935
2869 1 0. 02 47 0. 928 1. 2869
989 1. 3336
3803
00 50 0. 9812 1. 4737
9813 1. 5204
6138 0 0. 00 50 0. 9815 1. 6138
00 Histogram #2 (using 15 Classes) Estimate of the Number of Prices that are at least $1. 15 Using the histogram presented in the previous section, the estimate of the number of prices that are at least $1. 15 is ten. This is because the only values that can be counted fall into bins six through fifteen. Even though bin five may contain values that are above the $1.
15 threshold, they can not be counted as they are not guaranteed to be above the stated value. Therefore the formula for the estimate is: Estimate = B 6 + B 7 + B 8 + B 9 + B 10 + B 11 + B 12 + B 13 + B 14 + B 15 where B 6 = 5, B 7 = 1, B 8 = 3, B 9 = 0, B 10 = 0, B 11 = 0, B 12 = 0, B 13 = 0, B 14 = 0 and B 15 = 1. Rejecting a Hypothesis It was reported that a local radio station asserted that thirty percent of the gas stations were charging $1. 15 or more for gasoline. The following sections use the data and histograms presented in the previous sections to either accept or reject this assertion.
By using the data in the previous sections, thirty percent of the fifty-one sampled stations equal 15. 3 stations. Therefore, in order for the radio station to be correct, the number of stations that are charging $1. 15 per gallon of unleaded gasoline must be greater then or equal to 15. 3. Results Using Histogram #1 Using the data presented in histogram #1, the estimated number of stations that have a price of $1.
15 is 24. Unlike the estimation presented in the previous sections, this estimate must use the data presented in bin #2. This is because the only bins that can be eliminated are the bins that are guaranteed not to be $1. 15.
Even though bin #2 may contain prices that are below $1. 15, it cannot be eliminated from the calculations because it not guaranteed. Therefore, the formula for this calculation is: Estimate = B 2 + B 3 + B 4 + B 5, where B 2 = 19, B 3 = 4, B 4 = 0 and B 5 = 1. The formula to detect whether or not the radio stations assertion should be rejected is: Reject = (15. 3 > B 2 + B 3 + B 4 + B 5) or Reject = (15.
3 > 24) Given this formula, the radio stations assertion cannot be rejected. Results Using Histogram 2 Using the data presented in histogram #2, the estimated number of stations that have a price of $1. 15 is 15. Unlike the estimation presented in the previous sections, this estimate must use the data presented in bin #5. This is because the only bins that can be eliminated are the bins that are guaranteed not to be $1. 15.
Even though bin #5 may contain prices that are below $1. 15, it cannot be eliminated from the calculations because it not guaranteed. Therefore, the formula for this calculation is: Estimate = B 5 + B 6 + B 7 + B 8 + B 9 + B 10 + B 11 + B 12 + B 13 + B 14 + B 15 where B 5 = 5, B 6 = 5, B 7 = 1, B 8 = 3, B 9 = 0, B 10 = 0, B 11 = 0, B 12 = 0, B 13 = 0, B 14 = 0 and B 15 = 1. The formula to detect whether or not the radio stations assertion should be rejected is: Reject = (15 > B 5 + B 6 + B 7 + B 8 + B 9 + B 10 + B 11 + B 12 + B 13 + B 14 + B 15) or Reject = (15. 3 > 15) Given this formula, the radio stations assertion can be rejected.
However, when taking into account a few other factors, an argument can be made that the station was not wrong in making this assertion. The first reason is that if the person analyzing the data at the radio station chose to use a resolution of 0 decimal places, then the 15. 3 stations would be rounded down to 15. If that were the case, then the formula changes to: Reject = (15 > = 15) and therefore the radio station is correct and the assertion cannot be rejected. The second reason is that “if only a slight difference in the references is detected” (Shannon et al. , 2002, 7), then the hypothesis cannot be rejected.
If one chooses to define “slight” as at leas. 3 of a station, then again the radio station is correct and the assertion cannot be rejected. Conclusion For the purposes of the question on whether or not the radio station was correct in their statement that thirty percent of the gas stations were charging $1. 15 or higher for unleaded gasoline, the most appropriate histogram to use was the one with fifteen classes. This is because it provides a greater resolution to the bin that contains the $1.
15 decision point. When using the histogram with only five classes, there is too great a range of values and therefore a greater probability that the decision point bin will contain values that are not appropriate for this analysis. In other words, it will cause the margin of error to be greater then the histogram with fifteen classes. References Shannon, P. , Groebner, D. , Fry, P.
, & Smith, K. (2002). A COURSE IN BUSINESS STATISTICS (3 rd ed. ). Upper Saddle River, N.
J: Prentice Hall.