1) Estimate a multiple regression equation to predict the price of houses in a given community. Employ all available explanatory variables. Is there evident of of multi-collinearity in this model? Explain your response and the associated implications.
2) The owner of a restaurant in Bloomington, Indiana has recorded sales data for the past 19 years. He has also recorded data on potentially relevant variables.
a) Estimate a regression equation for sales as a function of population, advertising in the current year, and advertising in the previous year. Can you expect predictions of sales in future years to be very accurate if they are based on this regression equation? Explain.
b) The company would like to predict sales in the next year (year 20). It doesn't know what the population will be in year 20, so it assumes no change from year 19. It's planned advertising level for year 20 is $30,000. Find a prediction and a 95% prediction interval for sales in year 20.
1) To estimate a multiple regression equation to predict the price of houses in a given community, you will first need a dataset that includes the prices of houses as the dependent variable, and several independent variables that could potential explain the variation in house prices. These independent variables could include variables such as the size of the house, the number of bedrooms and bathrooms, the location or neighborhood, the age of the house, and other relevant factors.
Once you have the dataset, you can use statistical software, such as R, Python, or Excel, to perform a multiple regression analysis. This analysis will help you estimate the coefficients of the independent variables and their significance in explaining the variation in house prices. The estimated regression equation will take the form:
Price = β₀ + β₁X₁ + β₂X₂ + ... + βₙXₙ
where Price is the dependent variable (house prices), β₀ is the intercept, β₁ to βₙ are the estimated coefficients, and X₁ to Xₙ are the independent variables.
To determine if there is evidence of multicollinearity in this model, you can assess the correlation between the independent variables. Multicollinearity occurs when there is a high correlation between two or more independent variables in the regression model. This can lead to unstable and unreliable estimates of the coefficients, as well as inflated standard errors.
You can calculate the correlation coefficients between the independent variables and check if any of them have high correlations (usually above 0.7 or 0.8). Another way to detect multicollinearity is to calculate the variance inflation factor (VIF) for each independent variable. VIF measures how much the variance of an estimated regression coefficient is increased due to multicollinearity. VIF values above 5 or 10 are generally considered to indicate multicollinearity.
If there is evidence of multicollinearity in the model, it can have several implications. Firstly, the estimated coefficients may not be reliable or meaningful because they can change significantly when the model is re-estimated. Secondly, it becomes difficult to interpret the individual effects of the correlated variables on the dependent variable. Lastly, multicollinearity can make it challenging to assess the importance of each independent variable in explaining house prices accurately.
2) a) To estimate a regression equation for sales as a function of population, advertising in the current year, and advertising in the previous year, you will need a dataset that includes the sales data along with the population and advertising expenditure for both the current year and the previous year.
Using the same statistical software mentioned earlier, you can perform a multiple regression analysis to estimate the coefficients of the independent variables. The regression equation will take the form:
Sales = β₀ + β₁Population + β₂Ad_current_year + β₃Ad_previous_year
where Sales is the dependent variable (sales), β₀ is the intercept, β₁ to β₃ are the estimated coefficients, and Population, Ad_current_year, and Ad_previous_year are the independent variables.
However, accurately predicting sales in future years based on this regression equation may not be very reliable for several reasons. Firstly, the relationship between sales and the independent variables might change over time due to external factors or market dynamics. Therefore, assuming that this relationship will remain the same in future years may lead to inaccurate predictions.
In addition, this regression equation assumes that the relationship between sales and the independent variables is linear. However, in reality, the relationship could be non-linear or have other complexities that this equation does not capture.
To improve the accuracy of sales predictions in future years, you may need to consider other relevant variables, such as economic indicators, competition, or seasonal effects. Additionally, incorporating additional regression techniques like time series analysis or forecasting models may be necessary to account for the changing dynamics over time.
b) Assuming that there is no change in population from year 19 to year 20, and a planned advertising level of $30,000 for year 20, you can predict sales in year 20 using the regression equation derived from the previous analysis.
Plug in the values into the equation:
Sales(year 20) = β₀ + β₁Population(year 19) + β₂Ad_current_year(year 20) + β₃Ad_previous_year(year 19)
Since the population value for year 19 is known, you can substitute that value along with the planned advertising level for year 20 into the equation to obtain the predicted sales for year 20.
To calculate the 95% prediction interval for sales in year 20, you can use the predicted value from the regression equation and estimate the standard error of the prediction. The prediction interval will give you a range within which you can expect the actual sales in year 20 to fall with 95% certainty.
The calculation of the prediction interval will require access to the dataset used for the regression analysis, which includes the residuals of the model. With the residuals, you can estimate the standard deviation of the residuals, which is used to calculate the standard error of the prediction. The prediction interval can then be constructed using the predicted value, the standard error, and the appropriate t-value for the desired confidence level.
Note that the prediction interval assumes that the assumptions of the regression model, such as normality and constant variance, hold true.