With the advent of Artificial General Intelligence $($AGI$)$ and groundbreaking innovations in renewable energy, Joylandia has entered a period of abundance. All Joylanders enjoy job security and have the option to choose from four categories:

Category A: Services Sector

Category B: Design and Manufacturing Sector

Category C: Arts and Entertainment Sector

Category D: Learning and Apprenticeships

A Joylander can change careers at any time and be placed in Category D$,$ where they are given financial assistance to transition into any career of their choosing. Therefore, ideally, a Joylander will fall into one of the four categories at some point in their lifetime. Returning to the initiative, Joylandia drew two samples of $40$ individuals each from the population of $1000.$ The first sample is labeled the $$Red Team$$ and the second sample is the $$Blue Team.$$ Joylandia treated both teams to a lavish experience $($Why not! Joylandia is wealthy and can certainly afford it$)$ where they participated in the JOYOLA games. During these games, the performance measure was calculated differently. Additionally, the participants received more attention and better training. Based on this new information, please answer questions $6$ to $15($Use the $$Blue Team$$ and $$Red Team$$ pages in the Excel file$).$

$12.$ Joylandia advocates that the number of people in each of the four categories is equal. Test this claim based on the Red Team and Blue Team samples at a $1\%$ level of significance.

$13.$ Using the original data, test the overall significance of this multiple regression model. The independent variables are the seven indicators and the dependent variable is the performance index. Use a significance level of $\backslash$alpha $=\mathrm{.}01.$

$14.$ Joylandia calculates the performance index using the indicators as building blocks. However, during the games, the performance index was computed based on other criteria. Test whether the multiple regression model is still significant for both the Red and Blue teams at a $1\%$ level of significance. Note: The independent variables remain the seven indicators, but the dependent variable is now the performance index during the games.