Question

Applying Simple Linear Regression (Detecting Changes in Market Share)

The Conference Board produces a Consumer Confidence Index (CCI) that reflects people’s feelings about general business conditions, employment opportunities, and their own income prospects. Some researchers may feel that consumer confidence is a function of the median household income. Shown here are the CCIs for nine years and the median household incomes for the same nine years published by the U.S. Census Bureau. Determine the equation of the regression line to predict the CCI from the median household income. Compute the standard error of the estimate for this model. Compute the value of r². Does median household income appear to be a good predictor of the CCI? Why or why not? Conduct the five step hypothesis test for both the model and regressor, using .05 level of significance.

Please provide the 5 steps for both the model and the regressor test, the Minitab output for each hypothesis test, and state the business implication based upon your analysis. You must use Minitab and the 5 step hypothesis testing process.

CCI             Median Household Income ($1,000)

116.8         37.415

91.5           36.770

68.5           35.501

61.6           35.047

65.9           34.700

90.6           34.942

100.0         35.887

104.6         36.306

125.4         37.005

Please provide the 5 steps and Minitab output, and make a decision about the data. You must use Minitab and the 5 step hypothesis testing process.

Here is an example of reporting results in APA style:

Social support significantly predicted depression scores, b = -.34, t(225) = 6.53, p < .01. 2 Social support also explained a significant proportion of variance in depression scores, R = .12, F(1, 225) = 42.64, p < .01.

Use this prototype to report your results:

Median house income did or did not significantly predict CCI, b = insert the slope coefficient, t(insert the degrees of freedom for t) = insert the observed t, p < insert the p-value. Median house income also did or did not explain a significant proportion of variance in CCI, R² = insert r square value, F(insert degrees of freedom for the model)= insert observed F , p < insert pvalue.

To conduct this test in Minitab, please use the following process:

Minitab has a relatively thorough capability to perform regression analysis. To begin, select Stat from the menu bar. Select Regression from the Stat pulldown menu. Select Regression from the Regression pulldown menu. Place the column name or column location of the y variable in Response. Place the column name or column location of the x variable in Predictors. Click OK. Make sure you copy and paste the equation, regression statistics, the ANOVA, and coefficient output.

Answer 1

Minitab output:

Analysis of Variance

Source DF Adj SS Adj MS F-Value P-Value
Regression 1 2829 2829.0 15.43 0.006
Median 1 2829 2829.0 15.43 0.006
Error 7 1283 183.3
Total 8 4112

Model Summary

S R-sq R-sq(adj) R-sq(pred)
13.5389 68.80% 64.34% 50.08%

Coefficients

Term Coef SE Coef T-Value P-Value VIF
Constant -599 176 -3.41 0.011
Median 19.22 4.89 3.93 0.006 1.00

Regression Equation

CCI = -599 + 19.22 Median

Equation of regression line :

CCI = -599 + 19.22 Median household income

Standard error of the estimate for this model is 13.5389

Coefficient of determination r² = 68.80%

Now we check whether median household income appear to be a good predictor of the CCI:

Hypothesis:

Test statistic, t =  3.93 and p-value = 0.006

Since p-value is less than 005, we reject null hypothesis and conclude that median household income appear to be a good predictor of the CCI

Test for overall significance:

H₀ : Fitted regression model is not statistically significant.

H₁ : Fitted regression model is statistically significant.

Test statistic, F = 15.43 and p-value = 0.006

Since p-value is less than 005, we reject null hypothesis and conclude that Fitted regression model is statistically significant.