attempting to analyze the relationship between costs and sales, the owner of a fashion shop recorded the…

attempting to analyze the relationship between costs and sales, the owner of a fashion shop recorded the monthly costs ($ thousands) and the sales ($ millions) for a sample of 15 months. and completed a regression model for sales vs costs (i.e. y=sales, x=costs)\nthe coefficient of determination for this data is $r^{2}=0.5009$ and the regression equation is: $hat{y}=-0.9702 - 0.9397x$.\nfor part (a), express your answer in percent form (i.e. 3.00% instead of 0.03) and give at least 2 decimal places.\n(a) what percent of the variation in sales cannot be explained due to the cost factors? \nfor parts (b) and (c), give at least 4 decimal places.\n(b) calculate the correlation coefficient $r = \n(c) what is the slope of the equation? \nfor part (d), give at least 1 decimal place.\n(d) what would be the best predicted sales for the costs of $62,000? (note x is in thousands) ($ millions)

attempting to analyze the relationship between costs and sales, the owner of a fashion shop recorded the monthly costs ($ thousands) and the sales ($ millions) for a sample of 15 months. and completed a regression model for sales vs costs (i.e. y=sales, x=costs)\nthe coefficient of determination for this data is $r^{2}=0.5009$ and the regression equation is: $hat{y}=-0.9702 - 0.9397x$.\nfor part (a), express your answer in percent form (i.e. 3.00% instead of 0.03) and give at least 2 decimal places.\n(a) what percent of the variation in sales cannot be explained due to the cost factors? \nfor parts (b) and (c), give at least 4 decimal places.\n(b) calculate the correlation coefficient $r = \n(c) what is the slope of the equation? \nfor part (d), give at least 1 decimal place.\n(d) what would be the best predicted sales for the costs of $62,000? (note x is in thousands) ($ millions)

Answer

Explanation:

Step1: Find the unexplained variation in sales

The coefficient of determination $r^{2}$ represents the proportion of the variation in the dependent - variable (sales) that is explained by the independent variable (costs). The proportion of the variation that is not explained is $1 - r^{2}$. Given $r^{2}=0.5909$, the unexplained proportion is $1 - 0.5909 = 0.4091$. To convert to a percentage, we multiply by 100: $0.4091\times100 = 40.91%$.

Step2: Calculate the correlation coefficient $r$

We know that $r^{2}=0.5909$, so $r=\pm\sqrt{0.5909}$. Since the slope of the regression line $\hat{y}=-0.9702 - 0.9397x$ is negative, the correlation coefficient $r$ is negative. So $r=-\sqrt{0.5909}\approx - 0.7688$.

Step3: Identify the slope of the equation

The general form of a simple - linear regression equation is $\hat{y}=b_{0}+b_{1}x$, where $b_{1}$ is the slope. For the equation $\hat{y}=-0.9702 - 0.9397x$, the slope is $-0.9397$.

Step4: Predict sales for a given cost

The cost $x = 62$ (since $x$ is in thousands). Substitute $x = 62$ into the regression equation $\hat{y}=-0.9702-0.9397x$. Then $\hat{y}=-0.9702-0.9397\times62=-0.9702 - 58.2614=-59.2316\approx - 59.2$ (in millions).

Answer:

(a) 40.91% (b) -0.7688 (c) -0.9397 (d) -59.2