Question
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?
Answer:
1. a) Frequency table
Normal view of 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 |
Formula view of frequency table
![]()
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
Normal view of 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
|
Formula view of regression model
| 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).
3. Normal view of ANOVA table
| 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 |
|
|
|
Formula view of ANOVA table
| 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).
4. Normal view of Regression model
| 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 |
|
Formula view of Regression model
![]()
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.
![]()
For mobile price:
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.
For advertisingspots:
![]()
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
Inputs
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.
