a) 
The appropriate numerical descriptive measure to determine the strength and direction of the linear relationship between two variables is the correlation coefficient. The correlation coefficient is a value that ranges from -1 to 1, where -1 indicates a perfect negative linear relationship, 0 indicates no linear relationship, and 1 indicates a perfect positive linear relationship. 
For the relationship between weekly gross revenue and newspaper advertising: 
The correlation coefficient is 0.8929. This indicates a strong positive linear relationship between weekly gross revenue and newspaper advertising. As newspaper advertising increases, the weekly gross revenue tends to increase.
For the relationship between weekly gross revenue and television advertising:
The correlation coefficient is 0.7451. This suggests a moderate positive linear relationship between weekly gross revenue and television advertising. As television advertising increases, the weekly gross revenue tends to increase, but the strength of the relationship is slightly weaker compared to the newspaper advertising relationship.
In summary, the data suggests a stronger positive linear relationship between weekly gross revenue and newspaper advertising compared to weekly gross revenue and television advertising.

b)
 Based on the provided information from the Excel output, we can estimate a simple linear regression model to quantify the relationship between weekly gross revenue and television advertising expenditure. The model can be represented as follows:

Weekly Gross Revenue = -45.432 + 40.064 * Television Advertising ($100s)

This equation indicates that the intercept term (constant) is -45.432, and the coefficient for television advertising expenditure is 40.064.

The coefficient represents the expected change in weekly gross revenue for every $100 increase in television advertising expenditure. In this case, for every $100 increase in television advertising expenditure, we expect the weekly gross revenue to increase by $40.064.

Additionally, the Excel output provides the statistical information for the regression model. The R-squared value of 0.5552 indicates that approximately 55.52% of the variation in weekly gross revenue can be explained by the linear relationship with television advertising expenditure.

The p-value associated with the coefficient for television advertising expenditure is 0.0339, which is less than the conventional significance level of 0.05. This suggests that the coefficient is statistically significant, indicating a significant relationship between weekly gross revenue and television advertising expenditure.

Overall, the estimated regression model provides insights into the relationship between weekly gross revenue and television advertising expenditure, indicating a positive impact of television advertising on weekly gross revenue.
c) 
The goodness of fit of a regression model can be assessed using the R-squared value, which indicates the proportion of the variance in the dependent variable that is explained by the independent variable(s) in the model.

In this case, the R-squared value is 0.5552, which means that approximately 55.52% of the variation in weekly gross revenue can be explained by the linear relationship with television advertising expenditure.

This indicates a moderate level of goodness of fit. While the model explains a significant portion of the variation in weekly gross revenue, there is still a substantial amount of unexplained variability. Other factors not included in the model may contribute to the remaining variance.

It's important to note that the interpretation of the goodness of fit also depends on the specific context and the available data. A higher R-squared value would indicate a stronger fit, while a lower value would suggest a weaker fit. However, it is always advisable to consider additional diagnostic measures and evaluate the model's assumptions to ensure its reliability and appropriateness for making accurate predictions or drawing meaningful conclusions.
d)
 To improve the goodness of fit of the estimated model, several steps can be taken:

Consider additional variables: Including other relevant independent variables that may influence weekly gross revenue could improve the model's fit. For example, factors like online advertising expenditure, seasonality, or competitor advertising could be incorporated if they are available.

Non-linear transformations: It's possible that the relationship between weekly gross revenue and television advertising expenditure is not strictly linear. Exploring non-linear transformations of the variables, such as quadratic or logarithmic transformations, may capture a more accurate representation of the underlying relationship.

Interaction terms: Interaction terms can be introduced to account for the possibility of combined effects between variables. For example, the interaction between television advertising expenditure and another variable could improve the model's fit by capturing synergistic or dampening effects.

Outliers and influential observations: Carefully examining the data for outliers or influential observations and considering their impact on the model's fit is important. These data points may disproportionately affect the model's estimates and overall goodness of fit. Removing or appropriately addressing outliers can help improve the model's performance.

Assumption checking: Assessing the model's assumptions, such as linearity, independence, normality of residuals, and homoscedasticity, is crucial. If any assumptions are violated, appropriate transformations, data adjustments, or alternative modelling techniques may be necessary.
Data quality and sample size: Ensuring high-quality data and a sufficiently large sample size can contribute to a more reliable and accurate model. Collecting more comprehensive and precise data can enhance the goodness of fit and the model's predictive capabilities.
e) 
Based on the provided information from the Excel output, we can estimate a multiple linear regression model to quantify the relationship between weekly gross revenue and television and newspaper advertising expenditure. The model can be represented as follows:

Weekly Gross Revenue = -42.570 +22.402 * Television Advertising + 19.499 * Newspaper Advertising

This equation indicates that the intercept term (constant) is -42.570, the coefficient for television advertising expenditure is 22.402, and the coefficient for newspaper advertising expenditure is 19.499.

The coefficients represent the expected change in weekly gross revenue for a unit increase in the corresponding advertising expenditure, while holding other variables constant. In this case, for every additional unit increase in television advertising expenditure, we expect the weekly gross revenue to increase by 22.402. Conversely, for every additional unit increase in newspaper advertising expenditure, we expect the weekly gross revenue to increase by 19.499.

Additionally, the Excel output provides the statistical information for the regression model. The R-squared value of 0.9322 indicates that approximately 93.22% of the variation in weekly gross revenue can be explained by the linear relationship with television and newspaper advertising expenditures.

The p-values associated with the coefficients for television and newspaper advertising expenditures are both less than the conventional significance level of 0.05. This suggests that both coefficients are statistically significant, indicating significant relationships between weekly gross revenue and both advertising expenditures.

Overall, the estimated multiple linear regression model provides insights into the relationship between weekly gross revenue and television and newspaper advertising expenditures, indicating a significant impact of both advertising mediums on weekly gross revenue.
f) 
The estimated regression coefficients provide insights into the relationship between weekly gross revenue and television and newspaper advertising expenditure. Let's examine each coefficient:
Newspaper Advertising Expenditure: The coefficient estimate for newspaper advertising expenditure is 19.499. This positive coefficient indicates a positive relationship between newspaper advertising expenditure and weekly gross revenue. As newspaper advertising expenditure increases, weekly gross revenue tends to increase as well.
Television Advertising Expenditure: The coefficient estimate for television advertising expenditure is 22.402. This positive coefficient indicates a direct positive relationship between television advertising expenditure and weekly gross revenue. As television advertising expenditure increases, weekly gross revenue tends to increase as well. This suggests that investing more in television advertising can potentially lead to higher weekly gross revenue.
g) 
To test the overall validity/utility of the estimated multiple regression model, we can perform an overall F-test. The null hypothesis (H0) assumes that the regression model has no explanatory power and that all regression coefficients are zero. The alternative hypothesis (Ha) suggests that at least one of the regression coefficients is non-zero, indicating the model's validity.
In this case, the obtained F-value is 34.389, and the associated p-value is 0.0012. Since the p-value is less than the chosen significance level of 0.05, we can conclude that the overall F-test is statistically significant.
Therefore, we reject the null hypothesis, providing evidence that the estimated multiple regression model has overall validity/utility at the 5% level of significance. The regression model as a whole demonstrates a significant ability to explain the variation in the weekly gross revenue based on the included independent variables (television and newspaper advertising expenditure).
(h) 
Test whether linear relationship exists between weekly gross revenue and individual independent variables in part (e) at the 5% level of significance. (2 marks) 
To test whether a linear relationship exists between weekly gross revenue and individual independent variables at the 5% level of significance, we can perform simple linear regression for each independent variable separately and look at their p-values. If the p-value is less than 0.05, we can conclude that there is a significant linear relationship between the independent variable and the weekly gross revenue.
From the given multiple linear regression output, we already have the p-values for each independent variable:
X Variable 1: p-value = 0.0252
X Variable 2: p-value = 0.0033
Since both p-values are less than 0.05, we can conclude that there is a significant linear relationship between each independent variable (X Variable 1 and X Variable 2) and the weekly gross revenue at the 5% level of significance.

(I) 
Will your conclusion in part (h) change if the level of significance changes to 1% level of significance? (1 mark) 
If the level of significance changes to 1%, we will compare the p-values of the independent variables to 0.01 instead of 0.05. Let's revisit the p-values:

1. X Variable 1: p-value = 0.0252
2. X Variable 2: p-value = 0.0033

At the 1% level of significance:
- For X Variable 1, the p-value (0.0252) is greater than 0.01, so we cannot conclude that there is a significant linear relationship between X Variable 1 and the weekly gross revenue.
- For X Variable 2, the p-value (0.0033) is less than 0.01, so we can still conclude that there is a significant linear relationship between X Variable 2 and the weekly gross revenue.

In summary, our conclusion changes for X Variable 1 when the level of significance is changed to 1%. We can no longer conclude that there is a significant linear relationship between X Variable 1 and the weekly gross revenue at the 1% level of significance. However, the conclusion for X Variable 2 remains the same.


(j)
Compare the fitness of the multiple linear regression model with that of the simple linear regression model. (1 mark)
To compare the fitness of the multiple linear regression model with that of the simple linear regression models, we can look at the R Square and Adjusted R Square values for each model.

Multiple Linear Regression (from part h):
- R Square: 0.932229532
- Adjusted R Square: 0.905121345

Simple Linear Regression (X Variable 1):
- R Square: 0.555185363
- Adjusted R Square: 0.48104959

Simple Linear Regression (X Variable 2):
- R Square: 0.797265397
- Adjusted R Square: 0.763476296

The R Square value represents the proportion of the variance in the dependent variable (weekly gross revenue) that is predictable from the independent variables. The Adjusted R Square value adjusts the R Square value based on the number of independent variables and the sample size, penalizing the addition of irrelevant variables.
Comparing the models, we can see that the Multiple Linear Regression model has higher R Square and Adjusted R Square values than both Simple Linear Regression models. This indicates that the Multiple Linear Regression model provides a better fit to the data and explains more of the variance in the weekly gross revenue compared to the Simple Linear Regression models, 
(K) Based on your answers to (a) to (j) above, write a research report (of maximum 300 words) for the head of research division of your company, summarizing your findings and highlighting the managerial implications of these results?