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HI6007 Statics For Business Decisions

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HI6007 Statics For Business Decisions

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Course Code: HI6007
University: Holmes Institute

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Country: Australia

Question:
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. (3 marks)
 
b. Construct a histogram showing the percent frequency distribution of the furnitureorder values in the sample. Comment on the shape of the distribution. (3 marks) c. Given the shape of the distribution in part b, what measure of location would be most appropriate for this data set? (2 marks) HI6007 T12018 Group Assignment Page 2 Question 2 6 marks 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. (2 marks)
b. Compute the coefficient of determination and fully interpret its meaning. Be very specific. (2 marks)
c. Compute the coefficient of correlation and explain the relationship between demand and unit price. (2 marks)
 
Question 3 6 marks 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. HI6007 T12018 Group Assignment Page 3 Question 4 10 marks 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. a. Develop an estimated regression equation relating y to x1 and x2. (2 marks) 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. (2 marks) c. At α = 0.05, test to see if β1 and β2 is significantly different from zero. (2 marks) d. Interpret slope coefficient for X2. (2 marks) 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? (2 marks
Answer:

1. Response to Question No. 1.
a. The table below shows the per cent frequency distribution for the data set using a class width of $50-  

The shape of the histogram displays that the distribution of the dataset is skewed to the right that is, the dataset is positively skewed, as the right side of the histogram is longer than the left side. The majority of the values is clustered to the left side of the x-axis (the smaller values) with increasingly fewer values at the right side of the x-axis (the larger values).

There are three widely used measures of location which are Mean, Median, and Mode. However, the median is more preferable as the appropriate measure of location when the data is skewed. Mode is also taken into account sometimes but median is the most appropriate in such cases.

2. Response to Question No. 2.
            The required calculation of the ANOVA table is shown below- 

To check the relationship between demand and unit price, the null hypothesis to be taken into consideration is, H0 : β1 = 0 that is, there exists no relationship between demand and unit price against the alternative hypothesis, H1 : β1 ≠ 0, there is a relationship between them. At 5% level of significance, to check whether the demand price and the unit price are related or not, one can check the p-value. If p-value is smaller than the level of significance, α, then the null hypothesis is rejected. Here, α = 0.05 and p-value = 3.78833E-11 which is equivalent to 0. Clearly, p-value < α. Therefore, it indicates that there is a relationship between them (Draper and Smith 2014). The coefficient of determination can be defined as, R2 = 1- (Total Sum of squares/Sum of squares of residual). It gives the measures the proportion of the variation in the dependent variable Y (demand) explained by the independent variable X (unit price). Higher the value of R2, better the fitting between the dependent and the independent variable. Here, the value of R2 = 0.617103338. Thus, the unit price is quite efficient to explain the variation in demand variable. The correlation coefficient r is the square root of R2 which is obtained as r = 0.785559252. Clearly, the value of correlation coefficient shows a positive correlation between demand and the unit price. If one variable increases, then the other will also increase. 3. Response to Question No. 3. The complete ANOVA table is shown below-To check whether the difference among the three treatment is due to chance, the hypotheses can be taken as- H0 : μ1 = μ2 = μ3 = 0 where μ1, μ2 ,& μ3  are the population means of first, second, and third treatments respectively, and the alternative hypothesis H1 : at least one mean is different [μ1 = μ2 = μ3 ≠ 0] From the above ANOVA table, the observed value of F-statistic = Fobserved = 25.89072 and the critical value of F statistic i.e., F2,21 = 3.47 (obtained from the F-TABLE). The null hypothesis is rejected when |Fobserved |> |F2,21|. Also at 5% level of significance, p-value ≈ 0.00 < 0.05. Therefore, it can be concluded that, the null hypothesis is rejected and there is significant difference among the means of the three populations (Fraser 2013). 4. Response to Question No. 4. From the given table, the estimated regression equation can be written as, Y = 0.4977x1 + 0.4733x2 + 0.8051 ; where β1 = 0.4977, β2 = 0.4733. To see whether there is a significant relationship between the dependent variable (y) and the independent variables (x1, x2), the F-test for regression is performed. The null and the alternative hypotheses are given as- H0:   β1 = β2 = 0             against          H1:   βj ≠ 0, for at least one value of j=1, 2.  The ANOVA above table shows that the observed value of the F-statistic is 200.295276 and critical value of F statistic for 1, 5 degrees of freedom that is, F1,5 = 6.61 at 5% level of significance. Clearly, |Fobserved |> |F1,5 | which leads the null hypothesis to be rejected. Therefore, the estimated equation in part (a) represents that there is a significant relationship between x1, x2, and y.

The t-test is performed to check whether β1 and β2 are significantly different from zero. The table below represents the values of t-statistics for the independent variables X1 and X2 and their respective p-values.

The above table shows that when the level of significance = 0.05, the p-value for X1 variable = 0.4716 and 0.4716 > 0.05. Thus, the null hypothesis is accepted, which implies that the value of β1 is not different from zero. Moreover, β1 is zero.

On the other hand, the p-value for X2 variable = 0.0519 which is larger than the level of significance = 0.05. Thus, the null hypothesis is accepted. Thus, β2 is not significantly different from zero.

The value of slope coefficient of X2 is 0.4733 that tells that for one unit change in the value of the X2 variable, the dependent variable Y will increase 0.4733 units. Therefore, one unit increase in the number of advertising spots will increase the number of mobile phones sold per day by 0.4733 units.
The estimated regression equation shown in part (a) is

Y = 0.4977×1 + 0.4733×2 + 0.8051 …. (1) In the problem, the value of X1 = 20,000 & X2 = 10. Now, the number of mobile phones sold per day(y) can be estimated by putting the given values in the regression equation (1).
Therefore y = 0.4977*20000 + 0.4733*10 + 0.8051 = 9954 + 4.733 + 0.8051 = 9959.5381» 9960. Thus, 9960 mobile phones are expected to be sold per day.
References
Draper, N.R. and Smith, H., 2014. Applied regression analysis (Vol. 326). John Wiley & Sons.
Fraser, C., 2013. Business statistics for competitive advantage with Excel 2013: Basics, model building, simulation and cases. Springer Science & Business Media.

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