Question

The researcher studying this situation believes that the variable “number of camera bodies” may exhibit a quadratic relationship rather than a linear relationship. She decides to construct a new model with that variable assuming both a linear value and a quadratic value in the model. Only sales of the camera body should be used as an independent variable in this model. Do not use price of lens or gender. Let x = sales of the camera body. Let y = sales of the lens Thus, this model can be expressed as ŷ = b₀ + b₁x + b₂x². In the spreadsheet, enter a new column of data. This column should be the values of the sales of the camera body squared. Use the proper excel function to do this, or manually square each camera body sales value. Answer the following questions about this model. Use the material in section 15.1 to help you answer the following questions.

1. Find the regression coefficients and the regression equation for this data by using the method of least squares analysis.

2. Provide the meanings of the regression coefficients in this model. Be sure to state these meanings in the context of the problem.

3. Use the appropriate hypothesis tests to ascertain whether the linear model or the quadratic model is the better model in this case. Be sure to justify your work.

Sales of Body	Price of Lens	Gender	Sales of Lens
155	$700	1	122
101	650	1	120
157	725	0	135
180	575	1	95
150	600	0	100
201	750	0	174
99	560	1	118
137	500	0	130
155	675	1	128
165	550	1	166
152	725	0	131
127	750	1	102
217	565	0	165
186	670	0	154
176	600	1	97
123	585	0	129
109	645	0	98
90	575	1	105
176	660	0	120
129	650	1	105

Answer 1

a) Calculate the regression equation for the given data by using Excel. Instructions: • Import the data into Excel work sheet. Go to Addins Megastat → Correlation/Regression Analysis → Regression Choose ales of the camera body as independent variable and sales of lens as dependent variables. Click Ok. The resultant output is as shown below: Regression Analysis r2 0.303 0.550 20.814 n 20 k Dep. Var.les of Lens Std. Error F 7.81 Source Regression Residual Total ANOVA table df MS 1 3,383.9978 18 433.2335 19 p-value .0120 SS 3.383.9978 7,798.2022 11,182.2000 variables Intercept Sales of Body Regression output coefficients std. error 67.5825 20.9602 0.3827 0.1369 confidence interval t(df=18) p-value 95% lower 95% upper 3.224 .0047 23.5468 111.6182 2.795 .0120 0.0950 0.6704 b) From the above output, the regression equation is given by, û = 67.5825 +0.3827x The coefficient of the model is 0.3827. If one unit increases in sales of the camera body then the sales of the lens increases by 0.3827 units. Calculate the quadratic equation for the given data as follows: Instructions: • Import the data into Excel work sheet. Go to Addins → Megastat → Correlation/Regression Analysis → Regression Choose ales of the camera body(X) and X2 as independent variable and sales of lens as dependent variables. Click Ok. . The resultant output is as shown below: Regression Analysis R? Adjusted RP R Std. Error 0.366 0.291 0.605 20.422 n k Dep. Var. 20 2 Sales of Lens F 4.91 ANOVA table df MS 2 2,046.1436 17 417.0537 19 Source Regression Residual Total p-value .0208 SS 4,092.2873 7,089.9127 11.182.2000 variables Intercept Sales of Body X^2 Regression output coefficients std. error t df=17) 166.0259 78.2893 2.121 -1.0043 1.0727 -0.936 0.0046 0.0036 1.303 p-value .04.90 .3623 .2099 confidence interval 95% lower 9596 upper 0.8498 331.2020 -3.2675 1.2590 -0.0029 0.0121 From the above output, the quadratic regression equation is given by, û = 166.0259-1.0043x + 0.0046x? From the above two output, the P-value of the linear regression model is 0.0120 is less than the level of significance 0.05. Thus, the researcher can reject the null hypothesis that the resultant linear regression is not effective. The P-value of the quadratic regression model is 0.0208 is less than the level of significance 0.05. Thus, the researcher can reject the null hypothesis that the resultant quadratic regression is not effective.