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