Interpret Statistical Measures

5. The city council of Pine Bluffs is considering increasing the number of police in an effort to reduce crime. Before making a final decision, the council asks the chief of police to survey other cities of similar size to determine the relationship between the number of police and the number of crimes reported. The chief gathered the following sample information.

City Police Number of Crimes City Police Number of Crimes
Oxford 15 17 Holgate 17 7
Starksville 17 13 Carey 12 21
Danville 25 5 Whistler 11 19
Athens 27 7 Woodville 22 6

a. If we want to estimate crimes on the basis of the number of police, which variable is the dependent variable and which is the independent variable?
b. Draw a scatter diagram.
c. Determine the coefficient of correlation.
d. Determine the coefficient of determination.
e. Interpret these statistical measures. Does it surprise you that the relationship is inverse?

6. The owner of Maumee Ford-Mercury-Volvo wants to study the relationship between the age of a car and its selling price. Listed below is a random sample of 12 used cars sold at the dealership during the last year.

Car Age (years) Selling Price ($000) Car Age (years) Selling Price ($000)
1 9 8.1 7 8 7.6
2 7 6.0 8 11 8.0
3 11 3.6 9 10 8.0
4 12 4.0 10 12 6.0
5 8 5.0 11 6 8.6
6 7 10.0 12 6 8.0

a. If we want to estimate selling price on the basis of the age of the car, which variable is the dependent variable and which is the independent variable?
b. Draw a scatter diagram.
c. Determine the coefficient of correlation.
d. Determine the coefficient of determination.
e. Interpret these statistical measures. Does it surprise you that the relationship is inverse?

13. The following sample observations were randomly selected.
X: 4 5 3 6 10
Y: 4 6 5 7 7

a. Determine the regression equation.
b. Determine the value of Y when X is 7

14. The following sample observations were randomly selected.
X: 5 3 6 3 4 4 6 8
Y: 13 15 7 12 13 11 9 5

a. Determine the regression equation.
b. Determine the value of Y when X is 7.

41. The table below shows the number of cars (in millions) sold in the United States for various years and the percent of those cars manufactured by GM.

Year Cars Sold (millions) Percent GM Year Cars Sold (millions) Percent GM
1950 6.0 50.2 1980 11.5 44.0
1955 7.8 50.4 1985 15.4 40.1
1960 7.3 44.0 1990 13.5 36.0
1965 10.3 49.9 1995 15.5 31.7
1970 10.1 39.5 2000 17.4 28.6
1975 10.8 43.1 2003 17.1 27.8

Use a statistical software package to answer the following questions.

a. Is the number of cars sold directly or indirectly related to GM's percent of the market? Draw a scatter diagram to show your conclusion.
b. Determine the coefficient of correlation between the two variables. Interpret the value.
c. Is it reasonable to conclude that there is a negative association between the two variables? Use the .01 significance level.
d. How much of the variation in GM's market share is accounted for by the variation in cars sold?

47. An ANOVA table is:
SOURCE DF SS MS F
Regression 1 50
Error
Total 24 500

a. Complete the ANOVA table.
b. How large was the sample?
c. Determine the standard error of estimate.
d. Determine the coefficient of determination.

10. Johnson Wholesale Company manufactures a variety of products. The prices and quantities produced for April 1994 and April 2005 are:
1994 2005
1994 2005 Quantity Quantity
Product Price Price Produced Produced
Small motor (each) $23.60 $28.80 1,760 4,259
Scrubbing compound (gallon) 2.96 3.08 86,450 62,949
Nails (pound) 0.40 0.48 9,460 22,370

Using April 1994 as the base period, find the value index of goods produced for April 2005.

11. The index of leading economic indicators, compiled and published by the U.S. National Bureau of Economic Research, is composed of 12 time series, such as the average work hours of production in manufacturing, manufacturers' new orders, and money supply. This index and similar indexes are designed to move up or down before the economy begins to move the same way. Thus, an economist has statistical evidence to forecast future trends. You want to construct a leading indicator for Erie County in upstate New York. The index is to be based on 2000 data. Because of the time and work involved, you decide to use only four time series. As an experiment, you select these four series: unemployment in the county, a composite index of county stock prices, the County Price Index, and retail sales. Here are the figures for 2000 and 2005.

2000 2005
Unemployment rate (percent) 5.3 6.8
Composite county stocks 265.88 362.26
County Price Index (1982 _ 100) 109.6 125.0
Retail sales ($ millions) 529,917.0 622,864.0

The weights you assign are: unemployment rate 20 percent, stock prices 40 percent, County Price Index 25 percent, and retail sales 15 percent.

a. Using 2000 as the base period, construct a leading economic indicator for 2005.
b. Interpret your leading index.

Domestic International Total
Sales Sales Sales Employees
Year ($ million) ($ million) ($ million) (thousands)
1997 11,814 10,708 22,522 92.6
1998 12,901 10,910 23,811 96.1
1999 15,532 11,825 27,357 99.8
2000 17,316 11,856 29,172 100.9
2001 19,825 12,492 32,317 101.8
2002 22,455 13,843 36,298 108.3
2003 25,274 16,588 41,862 110.6
2004 27,770 19,578 47,348 109.9

21. Refer to the Johnson & Johnson data. Use 1997 as the base period and compute a simple index of the number of employees for each year from 1998 until 2004. Interpret the trend in the number of employees. (see table above)

Revenue Employees
Year ($ million) (000)
2000 130,385 90.0
2001 126,416 91.0
2002 132,210 96.0
2003 134,187 87.0
2004 152,363 80.0

25. Compute a simple index for the number of employees for GE. Use 2000 as the base period. What can you conclude about the change in the number of employees over the period? (see table above)

26. Compute a simple index for the number of employees for GE using the period 2000-02 as the base. What can you conclude about the change in the number of employees over the period? (see table above)