QA 233 (Basic Business Statistics)

Solutions to Practice Problem Set IV

Eddie Fletcher, a financial analyst for the firm of Bink, Brink, Blink, and Dink (BBB&D) has been tracking the price of on share of SMC Oil on a weekly basis for the past six months. Additionally, he has collected information on the average price per gallon of regular unleaded gasoline over the same period. The data he has collected are:

Date	Beginning of Week Stock Price	Average Price Per Gallon of Regular Unleaded Gasoline
6/28/99	42.825	$0.899
7/12/99	43.125	$0.929
7/19/99	42.250	$0.939
7/26/99	43.375	$0.939
8/02/99	43.500	$0.929
8/09/99	43.825	$0.939
8/16/99	44.250	$0.959
8/23/99	44.125	$0.969
8/30/99	45.375	$0.999
9/06/99	45.000	$1.039
9/13/99	44.675	$0.999
9/20/99	46.125	$1.059
9/27/99	45.750	$1.039
10/04/99	46.500	$1.069
10/11/99	46.750	$1.089
10/18/99	46.875	$1.099
10/25/99	47.000	$1.149
11/01/99	46.875	$1.139
11/08/99	48.125	$1.179
11/15/99	47.875	$1.179
11/22/99	47.750	$1.199
11/29/99	47.875	$1.219
12/06/99	48.500	$1.249
12/13/99	48.750	$1.239
12/20/99	49.125	$1.289
12/27/99	49.125	$1.299

 

1. Has Mr. Fletcher collected a census or a sample? Please explain.

Mr. Fletcher selected data only from specific points in time, so he has only collected a sample of prices for gasoline and SMC Oil stock.

2. How many observations has Mr. Fletcher collected?

Mr. Fletcher has collected weekly data on a single entity (stock) over the past six months. Each week over which data have been collected represents one observation, so Mr. Fletcher has collected twenty-six observations.

3. How frequently does BBB&D’s Beginning of Week Stock Price increase from one week to the next?

Note that we only have twenty-five observations for this question (because we are looking at weekly changes in SMC Oil’s stock price). In sixteen instances SMC Oil’s Beginning of Week Stock Price increased from one week to the next. Because this variable (whether SMC Oil’s Beginning of Week Stock Price increased from one week to the next) is nominal, the appropriate measure is the sample proportion. Thus we have that

4. On how many variables has Mr. Fletcher collected data?

Mr. Fletcher has collected data on two variables: SMC Oil’s Beginning of Week Stock Price and Average Price Per Gallon of Regular Unleaded Gasoline.

5. What is the probability that both Average Price Per Gallon of Regular Unleaded Gasoline and BBB&D’s Beginning of Week Stock Price increase from one week to the next? What approach(es) are you using to assess this probability?

Again note that we only have twenty-five observations for this question. In thirteen instances both Average Price Per Gallon of Regular Unleaded Gasoline and SMC Oil’s Beginning of Week Stock Price increased from one week to the next. If we let event A = Average Price Per Gallon of Regular Unleaded Gasoline and event B = SMC Oil’s Beginning of Week Stock Price, we have that

6. At what level is each variable measured? What are the ramifications (i.e., how does this limit what you can do analytically with each variable)?

The two variables (SMC Oil’s Beginning of Week Stock Price and Average Price Per Gallon of Regular Unleaded Gasoline) are both measured on the ratio scale (i.e., they are both quantitative). Thus the basic arithmetic operations of addition and subtraction on their values are valid, so we can compute summary statistical measures such as the mean and standard deviation.

7. Is this data time-series or cross-sectional in nature? Please explain your response.

Mr. Fletcher made a single observation on each of two variables (SMC Oil’s Beginning of Week Stock Price and Average Price Per Gallon of Regular Unleaded Gasoline) at specific points in time, so he has collected time-series data.

8. Have these data been collected observationally or experimentally? Please explain your response.

The data have been collected observationally. There is no indication that any attempt has been made to control the conditions under which the data are collected.

9. What is the probability that either Average Price Per Gallon of Regular Unleaded Gasoline or BBB&D’s Beginning of Week Stock Price increases from one week to the next?

Again note that we only have twenty-five observations for this question. In twenty-one instances either Average Price Per Gallon of Regular Unleaded Gasoline or SMC Oil’s Beginning of Week Stock Price increased from one week to the next. If we let event A = Average Price Per Gallon of Regular Unleaded Gasoline Increases from one Week to the Next and event B = SMC Oil’s Beginning of Week Stock Price Increases from one Week to the Next, we have that

10. From what population has this data been collected?

We have taken a sample from the populations of i) all Prices Per Gallon of Regular Unleaded Gasoline and ii) all SMC Oil’s Stock Prices over the past six months.

11. Does there appear to be a relationship between BBB&D’s Beginning of Week Stock Price and Average Price Per Gallon of Regular Unleaded Gasoline? If so, what is the nature of the relationship? Support your conclusion with the appropriate graphical display and numerical descriptive statistics.

A scatter diagram is the most appropriate graphical display of the relationship between our two ratio-scaled variables (SMC Oil’s Beginning of Week Stock Price and Average Price Per Gallon of Regular Unleaded Gasoline). Such a scatter diagram is provided below:

This scatter diagram suggests there is a a strong positive linear relationship between SMC Oil’s Beginning of Week Stock Price and Average Price Per Gallon of Regular Unleaded Gasoline. The most appropriate numerical measure of the relationship between SMC Oil’s Beginning of Week Stock Price (which we will call ‘y’) and Average Price Per Gallon of Regular Unleaded Gasoline (which we will call ‘x’) is Pearson’s Product Moment Correlation Coefficient (or r_xy). To calculate r_xy we first must calculate the covariance (or s_xy) of SMC Oil’s Beginning of Week Stock Price (which we will call ‘y’) and Average Price Per Gallon of Regular Unleaded Gasoline (which we will call ‘x’). We have that:

thus we have that

which again supports our conclusion that there is a strong positive linear relationship between SMC Oil’s Beginning of Week Stock Price and Average Price Per Gallon of Regular Unleaded Gasoline.

12. Is there an apparent pattern to the values of BBB&D’s Beginning of Week Stock Price and Date of Measurement? Support your conclusion with appropriate graphical display.

A runs plot (a scatter diagram in which the x-axis represents time or order of data collection) is the most appropriate graphical display of the relationship between our ratio-scaled variable (SMC Oil’s Beginning of Week Stock Price) and Date of Measurement. Such a runs plot is provided below:

This runs plot suggests the existence of a fairly strong positive linear relationship between SMC Oil’s Beginning of Week Stock Price and Date of Measurement.

13. Is there an apparent pattern to the values of Average Price Per Gallon of Regular Unleaded Gasoline and Date of Measurement? Support your conclusion with appropriate graphical display.

A runs plot (a scatter diagram in which the x-axis represents time or order of data collection) is the most appropriate graphical display of the relationship between our ratio-scaled variable (Average Price Per Gallon of Regular Unleaded Gasoline) and Date of Measurement. Such a runs plot is provided below:

This runs plot suggests the existence of a strong positive linear relationship between Average Price Per Gallon of Regular Unleaded Gasoline and Date of Measurement.

14. Given that Average Price Per Gallon of Regular Unleaded Gasoline has increased over the past week, what is the probability that BBB&D’s Beginning of Week Stock Price would also increase over the same period of time?

This is a question about conditional probability - specifically “given that Average Price Per Gallon of Regular Unleaded Gasoline has increased over a particular week, what is the probability that SMC Oil’s Beginning of Week Stock Price would also increase over the same period of time?” Lets call event A = Average Price Per Gallon of Regular Unleaded Gasoline increased over a particular week and event B = SMC Oil’s Beginning of Week Stock Price increased over a particular week. Then we need to find

15. What is the probability that Average Price Per Gallon of Regular Unleaded Gasoline increases by at least 0.500 from one week to the next?

Again, we only have twenty-five observations for this question. In no instances did Average Price Per Gallon of Regular Unleaded Gasoline increase by $0.50 from one week to the next, so the probability is zero.

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