Fstat Practice Problems Questions 1–6 Are Based On The Follo

Fstat Practice Problemsquestions 1 6 Are Based On The Following Scenar

FStat practice problems Questions 1-6 are based on the following scenario: Question 1: A consumer products company wants to measure the effectiveness of different types of advertising media in the promotion of its products. Specifically, the company is interested in the effectiveness of radio advertising and newspaper advertising (including the cost of discount coupons). A sample of 22 cities with approximately equal populations is selected for study during a test period of one month. Each city is allocated a specific expenditure level both for radio advertising and for newspaper advertising. The sales of the product (in thousands of dollars) and also the levels of the media expenditure (in thousands of dollars) during the test month are recorded, with the following results in Table 1 and corresponding regression output in Table 2.

Question 1: What is the interpretation of the adjusted R-square value for this problem? Why is it a better metric than R-square in Multiple Linear Regression?

Question 2: Predict Sales when radio and newspaper advertising are 27,000 and 32,000 dollars respectively. Options: $890,783.4; 1047.057 ($.; $890.627 ($.; Cannot be determined.

Question 3: The appropriate test statistic and p-value for assessing whether there is evidence that radio advertising aids in predicting sales are: F=40.158, p-value=1.50126E-07; T=1.234, p-value=0; T=7.435, p-value=4.88861E-07; T=5.668, p-value=1.83069E-05.

Question 4: Based on the sample regression coefficient for newspaper advertising, we estimate that: Holding mean radio advertising expenditure constant, as the newspaper advertising expenditure increases by 1 dollar, sales on average increase by 16.795 dollars; or, as the newspaper advertising expenditure increases by 1 ($000), sales on average increase by 16.795 ($000); or, as the newspaper advertising expenditure increases by 1($000), sales on average increase by 16.795 ($); or, as the newspaper advertising expenditure increases by 1 dollar, sales on average increase by 16.795 dollars.

Question 5 & 6: Interpret the following plots (not provided here).

Question 7: A marketing analyst for a shoe manufacturer considers variables X1 (FOREIMP) and X2 (MIDSOLE) to predict durability (Y or LTIMP). With a sample of 20 shoes, complete the ANOVA table and compute R-square. At the 0.01 significance level, assess the overall model significance based on ANOVA results.

Question 8: When to use F-test versus t-test in linear regression? What assumptions are necessary for linear regression?

Question 9: What is multicollinearity? How can it be diagnosed? What are the implications if ignored in modeling?

Question 10: A study measures computer anxiety (scale 20-100) among 200 students in different majors. Complete the ANOVA table and test if mean anxiety differs across majors at 0.05 significance level.

Question 11: Quarterly mortgage application data is visualized in a time series chart. Which exponential smoothing model (simple or adjusted) is appropriate? Use alpha=0.2 and beta=0.3 to forecast applications for Q1-2002.

Paper For Above instruction

The set of practice problems detailed above encompasses a variety of fundamental concepts in regression analysis, time series forecasting, and statistical inference. This comprehensive examination aims to evaluate understanding of model interpretation, hypothesis testing, prediction, and diagnostic procedures integral to data analysis in business contexts. The detailed responses to each question will reinforce key statistical principles and data-driven decision-making methodologies essential for analysts and researchers.

Understanding Adjusted R-squared and Its Significance

The adjusted R-squared statistic provides a measure of the proportion of variance in the dependent variable explained by the independent variables, adjusted for the number of predictors in the model. Unlike R-squared, which can increase simply by adding more variables, the adjusted R-squared penalizes the addition of insignificant predictors, thus offering a more accurate assessment of model fit. It is particularly advantageous in multiple linear regression because it balances model complexity and explanatory power, helping prevent overfitting and ensuring the inclusion of only meaningful predictors (Kutner et al., 2004).

Prediction of Sales Based on Advertising Expenditure

Using the regression model coefficients, predictions can be made for specified levels of advertising expenditure. For example, if the coefficients for radio and newspaper advertising are known from the regression output, substituting the expenditure levels of 27,000 and 32,000 dollars into the equation yields the predicted sales. The precise calculation confirms the correct choice among the options provided. Given typical regression outputs, the predicted sales would be computed as:

Predicted Sales = Intercept + (Coefficient for Radio × 27) + (Coefficient for Newspaper × 32)

The actual numerical prediction from the regression data would determine whether options like $890,783.4 or $1047.057 are the accurate forecast, highlighting the importance of proper model application (Montgomery et al., 2012).

Hypothesis Testing in Regression: F-test and t-test

The F-test assesses the overall significance of the regression model, testing whether at least one predictor variable is significantly related to the response variable. A high F-statistic coupled with a low p-value indicates that the model explains a significant amount of variability in sales. The t-test, on the other hand, evaluates the significance of individual regression coefficients—such as the coefficient for radio advertising—by testing whether they differ significantly from zero. The reported F-statistic (e.g., 40.158, p

Interpreting Regression Coefficients for Advertising Variables

The coefficient for newspaper advertising quantifies the average change in sales associated with a one-unit increase in advertising expenditure, holding other variables constant. If the coefficient is 16.795, then increasing newspaper advertising by $1,000 leads to an average increase of $16,795 in sales, assuming the model's units are in thousands of dollars. This interpretation facilitates managerial decision-making by translating statistical estimates into tangible financial implications (Wooldridge, 2013).

Analyzing Plots and Diagnostic Checks

While the specific plots are not provided here, typical residual plots are used to assess the assumptions of linear regression, such as linearity, homoscedasticity, and normality of residuals. Patterns or systematic deviations in residual plots suggest model inadequacies or violations of assumptions, which may call for model adjustments or transformations to improve validity.

Regression in Context: Durability Prediction of Shoes

In considering the durability of shoes based on forefoot shock absorption (X1) and impact change over time (X2), the ANOVA table enables evaluation of the overall model significance. Calculating R-squared reveals the proportion of variance explained. At the 0.01 significance level, the F-test determines whether the regression model provides a better fit than an intercept-only model, informing whether predictors contribute meaningful information (Draper & Smith, 1998).

F-test Versus t-test in Linear Regression

The F-test is used to evaluate the null hypothesis that all regression coefficients are zero simultaneously, thus assessing the overall model significance. The t-test assesses the significance of individual predictors. Essential assumptions for linear regression include linearity, independence of errors, homoscedasticity (constant variance of errors), and normality of residuals. Violations compromise inference validity and model reliability (Kutner et al., 2004).

Understanding Multicollinearity

Multicollinearity occurs when independent variables are highly correlated, making it difficult to isolate their individual effects on the dependent variable. It can be detected using variance inflation factors (VIFs), condition indices, or correlation matrices. Ignoring multicollinearity can inflate standard errors of coefficients, reduce statistical power, and distort variable significance assessments, leading to unreliable model interpretations and misguided decisions (O'Brien, 2007).

ANOVA in Comparing Computer Anxiety Across Majors

Applying ANOVA to analyze differences in computer anxiety among majors involves partitioning the total variance into between-group and within-group components. The F-statistic tests whether the mean anxiety differs significantly across groups. A significant result at the 0.05 level indicates evidence of disparity, guiding further analysis and targeted interventions.

Time Series Modeling: Exponential Smoothing

Analyzing mortgage application data via a time series chart helps determine the appropriate exponential smoothing model. If data shows a trend or seasonal components, an adjusted ( Holt-Winters) model may be suitable over simple exponential smoothing. Using alpha=0.2 and beta=0.3, one can compute the forecast for Q1-2002, considering the previous data points and smoothing equations. The choice depends on observing the data pattern, with adjusted models better capturing trends and seasonality for accurate forecasting.

Conclusion

This collection of problems underscores the importance of statistical inference, model diagnostics, and forecasting techniques in business analytics. Mastery of these concepts allows analysts to make informed decisions, interpret complex data accurately, and optimize strategies based on empirical evidence.

References

• Draper, N. R., & Smith, H. (1998). Applied Regression Analysis (3rd ed.). Wiley.
• Kutner, M. H., Nachtsheim, C. J., Neter, J., & Li, W. (2004). Applied Linear Statistical Models (4th ed.). McGraw-Hill.
• Montgomery, D. C., Peck, E. A., & Vining, G. G. (2012). Introduction to Linear Regression Analysis. Wiley.
• O'Brien, R. M. (2007). A Caution Regarding Rules of Thumb for Variance Inflation Factors. Quality & Quantity, 41(5), 673-690.
• Wooldridge, J. M. (2013). Introductory Econometrics: A Modern Approach. South-Western Cengage Learning.