Analysis Of Furniture Orders, ANOVA, Regression Models, Hypothesis Testing, Essay.

Determining the distribution of furniture orders

1. Missy Walters owns a mail-order business specializing in clothing, linens, and furniture for children. She is considering offering her customers a discount on shipping charges for furniture based on the dollar-amount of the furniture order. Before Missy decides the discount policy, she needs a better understanding of the dollar-amount distribution of the furniture orders she receives.

Missy had an assistant randomly select 50 recent orders that included furniture. The assistant recorded the value, to the nearest dollar, of the furniture portion of each order. The data collected is listed below (data set also provided in accompanying MS Excel file).

136	281	226	123	178	445	231	389	196	175
211	162	212	241	182	290	434	167	246	338
194	242	368	258	323	196	183	209	198	212
277	348	173	409	264	237	490	222	472	248
231	154	166	214	311	141	159	362	189	260

a. Prepare a frequency distribution, relative frequency distribution, and percent frequency distribution for the data set using a class width of $50.  

b. Construct a histogram showing the percent frequency distribution of the furniture- order values in the sample. Comment on the shape of the distribution.

c. Given the shape of the distribution in part b, what measure of location would be most appropriate for this data set?                                                                     

2. Shown below is a portion of a computer output for a regression analysis relating Y (demand) and X (unit price).

ANOVA
	df	SS
Regression	1	5048.818
Residual	46	3132.661
Total	47	8181.479
	Coefficients	Standard Error
Intercept	80.390	3.102
X	-2.137	0.248

a. Determine whether or not demand and unit price are related. Use α = 0.05.

b. Compute the coefficient of determination and fully interpret its meaning. Be very specific.                                                                                        

c. Compute the coefficient of correlation and explain the relationship between demand and unit price.                                                              

3. The following are the results from a completely randomized design consisting of 3 treatments.

Source of Variation	Sum of Squares	Degrees of Freedom	Mean Square	F
Between Treatments	390.58
Within Treatments (Error)	158.40
Total	548.98	23

Using α = .05, test to see if there is a significant difference among the means of the three populations. The sample sizes for the three treatments are equal.

4. In order to determine whether or not the number of mobile phones sold per day (y) is related to price (x₁ in $1,000), and the number of advertising spots (x₂), data were gathered for 7 days. Part of the Excel output is shown below.

ANOVA	df	SS                        MS              F
Regression		40.700
Residual		1.016
	Coefficients	Standard Error
Intercept	0.8051
x1	0.4977	0.4617
x2	0.4733	0.0387

a. Develop an estimated regression equation relating y to x₁ and x₂.                 

b. At α = 0.05, test to determine if the estimated equation developed in Part a represents a significant relationship between all the independent variables and the dependent variable.

c. At α = 0.05, test to see if β₁ and β₂ is significantly different from zero.           

d. Interpret slope coefficient for X₂.                                                                       

e. If the company charges $20,000 for each phone and uses 10 advertising spots, how many mobile phones would you expect them to sell in a day?      

1. a) Frequency table

Classes	Frequency	Relative Frequency	Percentage  Relative Frequency
100 to 150	3	0.06	6
150 to 200	15	0.3	30
200 to 250	14	0.28	28
250 to 300	6	0.12	12
300 to 350	4	0.08	8
350 to 400	3	0.06	6
400 to 450	3	0.06	6
450 to 500	2	0.04	4
Total	50	1	100

c) Histogram

The asymmetric shape of the histogram indicated above is established from the length of the right tail exceeding that of the left tail. This corresponds to presence of positive skew and potential presence of positive side outliers (Taylor and Cihon, 2014).

c) The suitable central tendency measure needs to be outlined. The choice is between mean and median. Here, median would be the appropriate choice considering that the underlying data is skewed and hence the mean would be vulnerable to extremely high values which is not an issue with median (Lehman and Romano, 2016).

2. Regression Model

ANOVA
	df	SS
Regression	1	5048.818
Residual	46	3132.661
Total	47	8181.479
	Coefficients	Standard Error	t value	p value
Intercept	80.39	3.102	25.916	0.000
X	-2.137	0.248	-8.617	0.000

ANOVA
	df	SS
Regression	1	5048.818
Residual	46	3132.661
Total	=SUM(B3:B4)	=SUM(C3:C4)
	Coefficients	Standard Error	t value	p value
Intercept	80.39	3.102	=B8/C8	=T.DIST(D8,B5,FALSE )
X	-2.137	0.248	=B9/C9	=T.DIST(D9,B5,FALSE)

The relevant hypotheses for performing hypothesis test are listed below.

The slope coefficient corresponding to unit price has a test statistics value of -8.617 which yields the p value as 0.000. Thus, the evidence indicates rejection of H₀ thus paving way for acceptance of H₁ (Koch, 2013).

Conclusion:

The linear relationship between the two variables is significant in statistical terms owing to slope being non-zero.

b) For the regression model, the coefficient of determination is determined as highlighted below:

The given regression model has the capability to account to explain 61.7% changes in the unit demand using price as the suitable predictor variable (Harmon, 2011).

c) For the regression model, the coefficient of correlation is determined as highlighted below:

Considering the above computations, the appropriate value of correlation coefficient is -0.786 and this value has been selected considering that regression line is downward sloping as evident from the slope (Lind, Marchal and Wathen, 2012).

Source of variation	Sum of squares	Degree of Freedom	Mean Square	F	Significance F
Between Treatments	390.58	2	195.29	25.89	0.00
Within Treatment (Error)	158.40	21	7.54
Total	548.98	23

Source of variation	Sum of squares	Degree of Freedom	Mean Square	F	Significance F
Between Treatments	390.58	3-1	=B2/C2	=D2/D3	=F.DIST(E2,C2,C3,FALSE)
Within Treatment (Error)	158.4	24-3	=B3/B3
Total	=SUM(B2:B3)	23

Test statistics (For ANOVA based on the above output) = 25.89

Corresponding p value (For ANOVA based on the above output) = 0.00

Conclusion:

It would not be appropriate that the means across the different populations is same as the statistical evidence suggests that one least one population mean shows a significant deviation (Koch, 2013).

n	7
k	2
ANOVA
	df	SS	MS	F	Significance F
Regression	2	40.7000	20.3500	80.1181	0.0000
Residual	4	1.0160	0.2540
	Coefficients	Standard Error	t value	p value
Intercept	0.8051
X1	0.4977	0.4617	1.0780	0.2060
X2	0.4733	0.0387	12.2300	0.0000

a) Regression equation based on intercept and slope coefficients is given below:

b) Total data has taken for one week i.e. for n = 7 days and hence,

The degree of freedom (Regression) = k = 2

The degree of freedom (Residual) = 7-2-1 =4

The relevant hypotheses for performing hypothesis test are listed below.

Test statistics (For ANOVA based on the above output) = 80.118

Corresponding p value (For ANOVA based on the above output) = 0.00

Conclusion:

The multiple regression model highlighted above is significant owing to existence of atleast one non-zero slope coefficient.

The slope coefficient corresponding to unit price has a test statistics value of 1.078 which yields the p value as 0.206. Thus, the evidence indicates non-rejection of H₀.

Conclusion:

The linear relationship between the two variables is insignificant in statistical terms owing to slope being assumed as zero.

The slope coefficient corresponding to unit price has a test statistics value of 12.23 which yields the p value as 0.000. Thus, the evidence indicates rejection of H₀ thus paving way for acceptance of H₁.

Conclusion:

The linear relationship between the two variables is significant in statistical terms owing to slope being non-zero.

d) Slope coefficient for advertising spots :  

Interpretation: The above coefficient highlights that daily sales of mobile can witness an increase/decrease of 0.4733 units provided the advertising spots undergo an increase/decrease of 1 unit (Harmon, 2011).

e) Regression equation

References

Harmon, M. (2011) Hypothesis Testing in Excel - The Excel Statistical Master 7th ed. Florida: Mark Harmon.

Koch, K.R. (2013) Parameter Estimation and Hypothesis Testing in Linear Models 2nd ed. London: Springer Science & Business Media.

Lehman, L. E. and Romano, P. J. (2016) Testing Statistical Hypotheses 3rd ed. Berlin : Springer Science & Business Media.

Lind, A.D., Marchal, G.W. and Wathen, A.S. (2012) Statistical Techniques in Business and Economics 15th ed. New York: McGraw-Hill/Irwin.

Taylor, K. J. and Cihon, C. (2014) Statistical Techniques for Data Analysis 2nd ed. Melbourne: CRC Press.