Question

In: Statistics and Probability

Sales (Y) Calls (X1) Time (X2) Years (X3) Type 48 168 12.3 5 ONLINE 36 131...

Sales (Y)	Calls (X1)	Time (X2)	Years (X3)	Type
48	168	12.3	5	ONLINE
36	131	16.4	4	NONE
46	162	15.7	3	NONE
47	183	13.0	3	ONLINE
44	177	15.3	3	ONLINE
49	181	12.4	2	ONLINE
35	123	19.0	3	NONE
46	169	14.8	3	GROUP
44	158	13.9	1	GROUP
39	146	15.4	3	GROUP
48	178	12.6	4	ONLINE
42	142	17.0	0	ONLINE
45	137	13.0	2	ONLINE
54	195	15.2	2	ONLINE
43	146	16.4	0	ONLINE
44	165	17.4	3	ONLINE
34	121	13.2	2	NONE
44	146	16.5	1	NONE
40	132	18.2	1	NONE
51	182	17.9	2	ONLINE
41	151	18.0	1	NONE
45	146	15.6	3	ONLINE
52	190	13.2	3	ONLINE
39	150	19.4	0	GROUP
41	149	13.2	3	GROUP
45	167	14.5	4	GROUP
46	189	20.0	1	GROUP
47	162	16.4	3	ONLINE
42	147	13.2	3	GROUP
45	171	19.4	2	ONLINE
44	165	15.0	0	ONLINE
50	175	15.1	3	ONLINE
46	161	13.2	3	GROUP
53	188	11.0	2	ONLINE
39	136	17.3	0	NONE
39	135	17.7	1	ONLINE
48	168	15.9	5	ONLINE
46	167	10.1	0	ONLINE
43	150	17.4	3	GROUP
44	151	15.2	2	GROUP
42	141	12.2	3	NONE
39	131	19.4	2	NONE
49	174	18.3	0	ONLINE
41	154	14.5	4	NONE
42	131	20.2	3	GROUP
39	128	15.3	1	GROUP
37	126	13.4	4	NONE
46	180	15.1	4	NONE
45	166	19.5	5	NONE
44	152	16.0	2	ONLINE
50	179	12.8	3	ONLINE
39	140	18.2	1	NONE
43	154	15.3	1	ONLINE
45	164	17.2	3	ONLINE
42	139	18.6	2	NONE
44	165	19.2	2	NONE
45	172	12.6	3	GROUP
41	147	18.5	3	GROUP
43	152	17.2	1	GROUP
48	160	15.8	2	ONLINE
42	159	13.6	4	GROUP
46	186	14.1	3	GROUP
46	150	20.7	2	GROUP
43	155	11.2	3	ONLINE
45	157	16.3	4	ONLINE
48	170	12.1	1	ONLINE
45	175	18.3	2	GROUP
49	186	17.5	1	GROUP
51	181	11.4	4	GROUP
47	171	17.3	2	ONLINE
50	185	16.4	0	ONLINE
39	146	15.8	1	GROUP
42	156	18.6	2	GROUP
46	157	19.3	2	ONLINE
43	163	11.7	1	GROUP
54	175	14.2	1	ONLINE
51	175	12.0	2	ONLINE
50	173	13.3	1	ONLINE
41	140	14.9	3	NONE
43	156	20.5	2	ONLINE
40	146	18.2	2	NONE
42	148	10.5	2	GROUP
50	183	11.7	1	GROUP
49	191	13.1	2	GROUP
40	149	14.2	4	ONLINE
40	143	18.3	2	NONE
47	185	15.2	2	ONLINE
41	136	17.4	3	GROUP
51	198	13.0	1	ONLINE
43	153	13.2	3	GROUP
38	129	15.2	3	NONE
44	158	11.8	3	ONLINE
43	149	12.7	1	GROUP
47	175	13.9	2	GROUP
40	154	16.4	3	GROUP
43	151	14.3	1	GROUP
46	153	22.0	0	ONLINE
46	167	14.8	1	ONLINE
46	167	15.8	0	ONLINE
39	143	17.7	3	NONE

Part C: Regression and Correlation Analysis

Use the dependent variable (labeled Y) and the independent variables (labeled X1, X2, and X3) in the data file. Use Excel to perform the regression and correlation analysis to answer the following.

Generate a scatterplot for the specified dependent variable (Y) and the X1 independent variable, including the graph of the "best fit" line. Interpret.

Determine the equation of the "best fit" line, which describes the relationship between the dependent variable and the selected independent variable.

Determine the coefficient of correlation. Interpret.

Determine the coefficient of determination. Interpret.

Test the utility of this regression model. Interpret results, including the p-value.

Based on the findings in Steps 1-5, analyze the ability of the independent variable to predict the designated dependent variable.

Compute the confidence interval for β1 (the population slope) using a 95% confidence level. Interpret this interval.

Using an interval, estimate the average for the dependent variable for a selected value of the independent variable. Interpret this interval.

Using an interval, predict the particular value of the dependent variable for a selected value of the independent variable. Interpret this interval.

What can be said about the value of the dependent variable for values of the independent variable that are outside the range of the sample values? Explain.

In an attempt to improve the model, use a multiple regression model to predict the dependent variable .Y, based on all of the independent variables. X1, X2, and X3.

Using Excel, run the multiple regression analysis using the designated dependent and three independent variables. State the equation for this multiple regression model.

Perform the Global Test for Utility (F-Test). Explain the conclusion.

Perform the t-test on each independent variable. Explain the conclusions and clearly state how the analysis should proceed. In particular, which independent variables should be kept and which should be discarded. If any independent variables are to be discarded, re-run the multiple regression, including only the significant independent variables, and summarize results with discussion of analysis.

Is this multiple regression model better than the linear model generated in parts 1-10? Explain. Please use the actual data from below in the analysis.

Expert Solution

a)

equation of best fit line => y = 0.2018X1+12.243

Coefficient of correlation = 0.871

Coefficient of Determination = (Coefficient of correlation)² = 0.871² = 0.759

ANOVA
	df	SS	MS	F	Significance F
Regression	1	1307.747	1307.747	309.046	4.58E-32
Residual	98	414.6929	4.231561
Total	99	1722.44

Since p-value is 0 which is less than 0.05, we conclude that there exist a linear relationship between Y and X1.

Confidence interval for β1 = (0.179 , 0.224)

Multiple Regression test:-

SUMMARY OUTPUT

Regression Statistics
Multiple R	0.874471
R Square	0.7647
Adjusted R Square	0.757347
Standard Error	2.054696
Observations	100

ANOVA
	df	SS	MS	F	Significance F
Regression	3	1317.15	439.0499	103.9965	4.76E-30
Residual	96	405.2904	4.221775
Total	99	1722.44

	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	14.75674	2.664714	5.537831	2.67E-07	9.467321	20.04615
Calls (X1)	0.197783	0.011964	16.53132	7.75E-30	0.174034	0.221531
Time (X2)	-0.0938	0.082642	-1.13501	0.259195	-0.25784	0.070243
Years (X3)	-0.19451	0.167938	-1.15821	0.24965	-0.52786	0.138846

Multiple linear regression model is better because it explains 76.4 % variability between the data and the Simple linear regression only explains 75.9% of the variability between the data.

orchestra answered 2 years ago

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