Solutions to Practice Problems for Part VI 1. A company sets different ...

Solutions to Practice Problems for Part VI

1. A company sets different prices for a particular stereo system in eight
different regions of the country. The accompanying table shows the numbers
of units sold and the corresponding prices (in hundreds of dollars).
|SALES |420 |
| | |
|Regression Statistics |
|Multiple R |0.9371370|
| |27 |
|R Square |0.8782258|
| |06 |
|Adjusted R |0.8579301|
|Square |08 |
|Standard Error |12.742275|
| |75 |
|Observations |8 |

|ANOVA | | | | | |
| |df|SS |MS |F |Significan|
| | | | | |ce F |
|Regressi|1 |7025.8064|7025.8064|43.271523|0.00059213|
|on | |52 |52 |18 |5 |
|Residual|6 |974.19354|162.36559| | |
| | |84 |14 | | |
|Total |7 |8000 | | | |

| |Coefficien|Standard |t Stat |P-value |
| |ts |Error | | |
|Interc|644.516129|36.68873299|17.5671405|2.18343E-|
|ept | | |5 |06 |
|PRICE |-42.580645|6.473082556|-6.5781093|0.0005921|
| |16 | |92 |35 |

a) Plot these data, and estimate the linear regression of sales
on price.
Here is an Excel-generated scatter plot. You could unscientifically
estimate a regression line with a ruler and a pencil, drawing the line so
that it "fits" the pattern of dots.
[pic]
The estimated regression line, from the Excel output, is:
Sales (Units) = 644.52 - 42.58(Price in $100)

b) What effect would you expect a $100 increase in price to have
on sales?
A $100 increase in the price will be expected to cause a 42.58 unit drop in
sales.
2. On Friday, November 13, 1989, prices on the New York Stock Exchange fell
steeply; the Standard and Poors 500-share index was down 6.1% on that day.
The accompanying table shows the percentage losses (y) of the twenty-five
largest mutual funds on November 13, 1989. Also shown are the percentage
gains (x), assuming reinvested dividends and capital gains, for these same
funds for 1989, through November 12.
|y |x |y |x |y |x |
|4.7 |38.0 |6.4 |39.5 |4.2 |24.7 |
|4.7 |24.5 |3.3 |23.3 |3.3 |18.7 |
|4.0 |21.5 |3.6 |28.0 |4.1 |36.8 |
|4.7 |30.8 |4.7 |30.8 |6.0 |31.2 |
|3.0 |20.3 |4.4 |32.9 |5.8 |50.9 |
|4.4 |24.0 |5.4 |30.3 |4.9 |30.7 |
|5.0 |29.6 |3.0 |19.9 |3.8 |20.3 |
|3.3 |19.4 |4.9 |24.6 | | |
|3.8 |25.6 |5.2 |32.3 | | |

a) Estimate the linear regression of November 13 losses
on pre-November 13, 1989, gains.
Here is the Excel output:
|SUMMARY OUTPUT | | | | | |
| | | | | | |
|Regression Statistics | | | | |
|Multiple R |0.7337257| | | | |
| |13 | | | | |
|R Square |0.5383534| | | | |
| |22 | | | | |
|Adjusted R |0.5182818| | | | |
|Square |32 | | | | |
|Standard Error |0.6424829| | | | |
| |17 | | | | |
|Observations |25 | | | | |
| | | | | | |
|ANOVA | | | | | |
| |df |SS |MS |F |Significanc|
| | | | | |e F |
|Regression |1 |11.07156114|11.071561|26.8216625|2.99579E-05|
| | | |14 |1 | |
|Residual |23 |9.494038861|0.4127842| | |
| | | |98 | | |
|Total |24 |20.5656 | | | |
| | | | | | |
| |Coefficie|Standard |t Stat |P-value | |
| |nts |Error | | | |
|Intercept |1.8853446|0.506748146|3.7204766|0.00112323| |
| |34 | |3 |2 | |
|Gains |0.0895658|0.017294171|5.1789634|2.99579E-0| |
| |82 | |59 |5 | |


The estimated regression line is:
Losses = 1.885 + 0.0896(Gains)
b) Interpret the slope of the sample regression line.
Large mutual funds lost about 1.885% on November 13 (the intercept), plus
an additional loss of about 0.09% for every 1% in value gained in 1989
before November 13 (the slope). In other words, the amount of value a large
mutual fund lost on November 13 depended on how much value had been gained
before November 13.
3. For a period of 11 years, the figures in the accompanying table
were found for annual change in unemployment rate and annual change
in mean employee absence rate due to own illness.
| |Change In |Change In Mean Employee |
| |Unemployment |Absence Rate Due To Own |
|Year |Rate |Illness |
|1 |-.2 |+.2 |
|2 |-.1 |+.2 |
|3 |+1.4 |+.2 |
|4 |+1.0 |-.4 |
|5 |-.3 |-.1 |
|6 |-.7 |+.2 |
|7 |+.7 |-.1 |
|8 |+2.9 |-.8 |
|9 |-.8 |+.2 |
|10 |-.7 |+.2 |
|11 |-1.0 |+.2 |

Excel Regression output:
|SUMMARY OUTPUT | | | | | |
| | | | | | |
|Regression Statistics | | | | |
|Multiple R |0.80517941| | | | |
| |3 | | | | |
|R Square |0.64831388| | | | |
| |6 | | | | |
|Adjusted R |0.60923765| | | | |
|Square |2 | | | | |
|Standard Error |0.20732548| | | | |
| |9 | | | | |
|Observations |11 | | | | |
| | | | | | |
|ANOVA | | | | | |
| |df |SS |MS |F |Significanc|
| | | | | |e F |
|Regression |1 |0.713145275|0.71314527|16.5910019|0.002786228|
| | | |5 | | |
|Residual |9 |0.386854725|0.04298385| | |
| | | |8 | | |
|Total |10 |1.1 | | | |
| | | | | | |
| |Coefficien|Standard |t Stat |P-value | |
| |ts |Error | | | |
|Intercept |0.04485190|0.063473424|0.70662493|0.49768444| |
| |4 | |5 |3 | |
|Unemployment |-0.2242595|0.055057259|-4.0732053|0.00278622| |
| |2 | |59 |8 | |


a) Estimate the linear regression of change in mean employee
absence rate due to own illness on change in unemployment rate.
Change in Absence Rate = 0.0449 - 0.2243(Change in Unemployment
Rate)
b) Interpret the estimated slope of the regression line.
A one-percent increase in the unemployment rate is associated with a
0.2243% decrease in the absence rate. (Note that the intercept is not
statistically significant.)

4. Refer to the data of Exercise 2. Test against a two-sided alternative
the null hypothesis that mutual fund losses on Friday, November 13, 1989,
did not depend linearly on previous gains in 1989.
We perform this test by looking at the p-value associated with the
regression coefficient for the variable "Gains" (see regression output;
this p-value is 2.996E-05, or 0.00002996).
The null hypothesis that the "Gains" coefficient is zero can be rejected
with a very small probability of Type I error (alpha ( 0.00003).

5. An attempt was made to evaluate the forward rate as a predictor of the
spot rate in the Canadian treasury bill market. For a sample of seventy-
nine quarterly observations, the estimated linear regression:
y = .00027 + .7916x
was o