12.6 Outliers  (Page 4/11)

1. Draw a scatterplot of the data.
2. Calculate the least squares line. Write the equation in the form ŷ = a + bx .
3. Draw the line on the scatterplot.
4. Find the correlation coefficient. Is it significant?
5. What is the average CPI for the year 1990?

1. See [link] .
2. ŷ = –3204 + 1.662 x is the equation of the line of best fit.
3. r = 0.8694
4. The number of data points is n = 14. Use the 95% Critical Values of the Sample Correlation Coefficient table at the end of Chapter 12. n – 2 = 12. The corresponding critical value is 0.532. Since 0.8694>0.532, r is significant.
   ŷ = –3204 + 1.662(1990) = 103.4 CPI
5. Using the calculator LinRegTTest, we find that s = 25.4 ; graphing the lines Y2 = –3204 + 1.662X – 2(25.4) and Y3 = –3204 + 1.662X + 2(25.4) shows that no data values are outside those lines, identifying no outliers. (Note that the year 1999 was very close to the upper line, but still inside it.)

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If you are interested in seeing more years of data, visit the Bureau of Labor Statistics CPI website ftp://ftp.bls.gov/pub/special.requests/cpi/cpiai.txt; our data is taken from the column entitled "Annual Avg." (third column from the right). For example you could add more current years of data. Try adding the more recent years: 2004: CPI = 188.9; 2008: CPI = 215.3; 2011: CPI = 224.9. See how it affects the model. (Check: ŷ = –4436 + 2.295 x ; r = 0.9018. Is r significant? Is the fit better with the addition of the new points?)

Try it

The following table shows economic development measured in per capita income PCINC.

Year	PCINC	Year	PCINC
1870	340	1920	1050
1880	499	1930	1170
1890	592	1940	1364
1900	757	1950	1836
1910	927	1960	2132

1. What are the independent and dependent variables?
2. Draw a scatter plot.
3. Use regression to find the line of best fit and the correlation coefficient.
4. Interpret the significance of the correlation coefficient.
5. Is there a linear relationship between the variables?
6. Find the coefficient of determination and interpret it.
7. What is the slope of the regression equation? What does it mean?
8. Use the line of best fit to estimate PCINC for 1900, for 2000.
9. Determine if there are any outliers.

a. The independent variable ( x ) is the year and the dependent variable ( y ) is the per capita income.

b.

c. ŷ = 18.61x – 34574; r = 0.9732

d. At df = 8, the critical value is 0.632. The r value is significant because it is greater than the critical value.

e. There does appear to be a linear relationship between the variables.

f. The coefficient of determination is 0.947, which means that 94.7% of the variation in PCINC is explained by the variation in the years.

g. and h. The slope of the regression equation is 18.61, and it means that per capita income increases by $18.61 for each passing year. ŷ = 785 when the year is 1900, and ŷ = 2,646 when the year is 2000.

i. There do not appear to be any outliers.

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95% critical values of the sample correlation coefficient table

References

Data from the House Ways and Means Committee, the Health and Human Services Department.

Data from Microsoft Bookshelf.

Data from the United States Department of Labor, the Bureau of Labor Statistics.

Data from the Physician’s Handbook, 1990.

Data from the United States Department of Labor, the Bureau of Labor Statistics.

Chapter review

To determine if a point is an outlier, do one of the following:

1. Input the following equations into the TI 83, 83+,84, 84+:

   $\begin{array}{l}{y}_1=a+bx\hfill \\ y_2=(2s)a+bx\hfill \\ y_3=-(2s)a+bx\hfill \end{array}$ where s is the standard deviation of the residuals

   If any point is above y ₂ or below y ₃ then the point is considered to be an outlier.

2. Use the residuals and compare their absolute values to 2 s where s is the standard deviation of the residuals. If the absolute value of any residual is greater than or equal to 2 s , then the corresponding point is an outlier.
3. Note: The calculator function LinRegTTest (STATS TESTS LinRegTTest) calculates s .

Use the following information to answer the next four exercises. The scatter plot shows the relationship between hours spent studying and exam scores. The line shown is the calculated line of best fit. The correlation coefficient is 0.69.