# Statistics Chap12, Cases

**Topics:**Regression analysis, Errors and residuals in statistics, Linear regression

**Pages:**11 (2342 words)

**Published:**June 16, 2013

Simple Linear Regression

Case Problem 1: Measuring Stock Market Risk

a.Selected descriptive statistics follow:

Variable N Mean StDev Minimum Median Maximum Microsoft 36 0.00503 0.04537 -0.08201 0.00400 0.08883 Exxon Mobil 36 0.01664 0.05534 -0.11646 0.01279 0.23217 Caterpillar 36 0.03010 0.06860 -0.10060 0.04080 0.21850 Johnson & Johnson 36 0.00530 0.03487 -0.05917 -0.00148 0.10334 McDonald’s 36 0.02450 0.06810 -0.11440 0.03700 0.18260 Sandisk 36 0.06930 0.19540 -0.28330 0.07410 0.50170 Qualcomm 36 0.02840 0.08620 -0.12170 0.03870 0.21060 Procter & Gamble 36 0.01059 0.03707 -0.05365 0.01333 0.08783 S&P 500 36 0.01010 0.02633 -0.03429 0.01034 0.08104

From the descriptive statistics we see that six of the companies had a higher mean monthly return than the market (as measured by the S&P 500): Exxon Mobil, Caterpillar, McDonald’s, Sandisk, Qualcomm, and Procter & Gamble. Microsoft and Johnson & Johnson had lower mean monthly returns.

Using the standard deviation as a measure of volatility, Sandisk was the most volatile stock with a standard deviation of .1954. The stocks of Johnson & Johnson and P & G exhibit less volatility than the other individual stocks. But, all of the individual stocks are more volatile than the market as a whole. The diversification embodied in the S&P 500 reduces its volatility.

b.The estimated regression equation relating each of the individual stocks to the S&P 500 is shown below. The value of [pic]for each equation is also shown.

Microsoft = 0.00040 + 0.458 S&P 500R-Sq = 7.1%

Exxon Mobil = 0.00926 + 0.731 S&P 500R-Sq = 12.1%

Caterpillar = 0.015000 + 1.49 S&P 500R-Sq = 32.9%

Johnson & Johnson = 0.00521 + 0.009 S&P 500R-Sq = 0.0%

McDonald’s = 0.00930 + 1.500 S&P 500R-Sq = 33.8%

Sandisk = 0.04300 + 2.600 S&P 500R-Sq = 12.3%

Qualcomm = 0.01410 + 1.410 S&P 500R-Sq = 18.7%

Procter & Gamble = 0.00548 + 0.507 S&P 500R-Sq = 12.9%

The betas (slope of estimated regression equation) for the individual stocks can be obtained from the regression output.

CompanyBeta

Microsoft .458

Exxon Mobil .731

Caterpillar1.490

Johnson & Johnson .009

McDonald’s1.500

Sandisk2.600

Qualcomm1.410

The beta for the market as a whole is 1. So, any stock with a beta greater than 1 will move up faster than the market when the market goes up. Any stock with a beta less than 1 will not go down as fast as the market in periods where the market declines.

We would expect Sandisk, with a beta of 2.6, to benefit most from an up market. Johnson & Johnson, with a beta of .009 is least affected by the market. The effect of the market going down cannot be expected to exert much downward pressure on shares of Johnson & Johnson.

c. The[pic]values seem to indicate that from 0% to 33.8% of the variability of the returns in these individual stocks is explained by the return for the market.

Case Problem 2: U.S. Department of Transportation

Descriptive statistics for these data are shown below:

N MEAN MEDIAN TRMEAN STDEV SEMEAN PERCENT 42 12.262 12.000 12.184 3.132 0.483 FATAL 42 1.922 1.881 1.906 1.071 0.165

MIN MAX Q1 Q3

PERCENT 8.000 18.000 9.000 15.000

FATAL 0.039 4.100 0.992 2.824

The following scatter diagram suggests a linear relationship between these two variables:

[pic]

Minitab was used to develop the following regression analysis output:

The regression equation is

FATAL = - 1.60 + 0.287 PERCENT

Predictor Coef SE Coef...

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