QBM117 Business Statistics

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QBM117 Business Statistics

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QBM117 Business Statistics

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Course Code: QBM117
University: Charles Sturt University

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Country: Australia

Question: 

Much of the data that is analysed by businesses is obtained from surveys asking individuals about their spending habits, preferences, etc. It is therefore of interest to know whether the answers obtained from these surveys are likely to be accurate. We therefore consider a case where survey data can be compared with a more objective measure. The data file A6 S2 2018.xls presents 2010 data for all American states of per capita consumption of alcohol based on survey data, and per capita purchases of alcohol based on sales records.
We will consider the sales data as a proxy for actual consumption, and investigate using the survey data as a predictor of sales data.(a) Produce a scatter plot of the data with the predictor variable (survey per capita consumption) on the horizontal axis.
 
(b) Describe the type of relationship that appears to exist between the two variables. Include discussion of any unusual features, for example any data points that do not fit the general pattern. Offer an explanation for why those points do not fit the general pattern.
 
(c) Generate the simple linear regression output for the data. [As a check, make sure that the fourth number from the top of the output, Standard Error, is 0.3619.] Write down an equation for the line of best fit.
 
(d) What are the values of the slope and the intercept for the line of best fit? Interpret these values. Does the intercept have a reliable interpretation? Explain briefly. 
(e) What is the value of the coefficient of determination? Interpret this value. 
(f) Test (by comparing the test statistic to the critical value) whether there is a significant linear relationship between the survey data and the Sales data at the 5% level of significance. Also quote the p-value for this test. 
(g) Find the 95% confidence interval for the slope of the true relationship between survey and sales data. 
(h) Remove the three data points that you judge to be furthest from fitting the general pattern, and repeat the regression analysis to find the slope and the 95% confidence interval for the slope based on this new data set. Comment briefly on the difference between the answers for these two data sets. 
(i) If for certain states you only had the survey data and not the sales data, could you decide which results were likely to yield unreliable predictions? Explain briefly. 

Answer: 
a. 
                               

The graph is a scatter plot indicating the relationship between sales and per capita consumption of alcohol. The independent variable (predictor variable) is the survey’s per capital consumption of alcohol, while the dependent variable is sales.
(b). The graph indicates that there exists a linear relationship between the variables, sales and per capita consumption of alcohol.  The linear relationship has an upward slope indicating that the relationship between the two variables is positive. This means that an increase in the per capita consumption will lead to an increase in the sales.
To understand the strength of the relationship, we look at the R-squareplained by changes in the per capita consumption of alcohol from the survey. Therefore, the linear relationship between the two variables is moderately weak.d of the linear equation of the graph below. R2 = 0.4715, meaning that 47.15% of the variation in sales can be ex
The graph also indicates some outliers in the survey data. These are data points that appear to be unusually far away from the general pattern of the linear trendline. The outliers are seen to be three data points where the sales are extremely high for the given per capital consumption.
(c). The simple linear regression output for the data is given below:
                                     
Let x = Survey per capita consumption
Let y = Sales
 will denote the expected sales when the per capita consumption is x.

Line of best fit:

(d) The value of the slope for the line of best fit is 2.6534; This means if the per capita consumption increases by 1, then the model predicts that the sales will increase by 2.6534. The value of the intercept for the line of best fit is 0.7144; This means if the per capita consumption is 0, then the model predicts the sales will be 0.7144.
The intercept does not have a reliable interpretation because it is unrealistic to have any sales recorded if there is no consumption of alcohol across the states.
(e) The coefficient of determination, R2 is used to assess the ability of the model to predict an outcome. In this case, the value of the coefficient of determination of the simple linear regression model was found to be, R2 = 0.4715. Hence, 47.15% of the variations in Sales (the dependent variable) can be predicted by a simple linear regression and per capita consumption (independent variable).
(f) The test whether there is a significant linear relationship between the survey data and the Sales data, at 5% level of significance, we first formulate the hypothesis as:
H0:  b1 = 0
H1:  b1 ≠ 0
The decision rule is to reject the null hypothesis, if p-value is less than the significance level.
The p-value = 0.0000
Level of significance = 0.05

00 < 0.05, Reject null hypothesis. Hence, there is sufficient evidence to conclude that there is a significant linear relationship between the survey data and the sales data, at 95% confidence level. (g) The 95% confidence interval for the slope of the true relationship between survey and sales data is given by: [1.8380, 3.4687] (h) The regression results after removing the three data points that appear as outliers are given below.                                   The new value of the slope is 2.0320 with a 95% confidence interval of [1.452, 2.610]. The 95% confidence interval of the new data set has a smaller range compared to the confidence interval for the initial data. This is because the removal of the outliers makes the new data set more compact, in that all the data points in the new data set are close and form a general pattern for a linear trendline. (i) If for certain states there was survey data and not the sales data, it is hard to decide which results are likely to yield unreliable predictions. The current data set allows us to model a linear equation that can be used to predict sales data given the survey data. From the linear regression model, we obtain the confidence interval for the slope of the true relationship between survey and sales data. However, there is no way to decide which results are likely to yield unreliable predictions. Free Membership to World's Largest Sample Bank To View this & another 50000+ free samples. Please put your valid email id. E-mail Yes, alert me for offers and important updates Submit  Download Sample Now Earn back the money you have spent on the downloaded sample by uploading a unique assignment/study material/research material you have. After we assess the authenticity of the uploaded content, you will get 100% money back in your wallet within 7 days. UploadUnique Document DocumentUnder Evaluation Get Moneyinto Your Wallet Total 3 pages PAY 2 USD TO DOWNLOAD *The content must not be available online or in our existing Database to qualify as unique. Cite This Work To export a reference to this article please select a referencing stye below: APA MLA Harvard OSCOLA Vancouver My Assignment Help. (2021). Business Statistics. Retrieved from https://myassignmenthelp.com/free-samples/qbm117-business-statistics/model-predicts-the-sales.html. "Business Statistics." My Assignment Help, 2021, https://myassignmenthelp.com/free-samples/qbm117-business-statistics/model-predicts-the-sales.html. My Assignment Help (2021) Business Statistics [Online]. Available from: https://myassignmenthelp.com/free-samples/qbm117-business-statistics/model-predicts-the-sales.html[Accessed 18 December 2021]. My Assignment Help. 'Business Statistics' (My Assignment Help, 2021) accessed 18 December 2021.

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