Questions: Use the following information to answer the next three exercises: Sixty-five randomly selected car salespersons were asked the number of cars they generally sell in one week. Fourteen people answered that they generally sell three cars; nineteen generally sell four cars; twelve generally sell five cars; nine generally sell six cars; eleven generally sell seven cars. Calculate the following:

Solution Steps

To solve the given problem, we need to calculate the following:

1. The mean number of cars sold per week.
2. The variance of the number of cars sold per week.
3. The standard deviation of the number of cars sold per week.

1. Mean Calculation: The mean is calculated by summing the product of each number of cars sold and the number of salespersons who sold that many cars, then dividing by the total number of salespersons.
2. Variance Calculation: The variance is calculated by finding the average of the squared differences from the mean.
3. Standard Deviation Calculation: The standard deviation is the square root of the variance.

The mean number of cars sold per week is calculated using the formula: \[ \text{Mean} = \frac{\sum (x_i \cdot f_i)}{N} \] where \( x_i \) is the number of cars sold, \( f_i \) is the frequency of salespersons, and \( N \) is the total number of salespersons.

Given: \[ \begin{align_} x &= [3, 4, 5, 6, 7] \\ f &= [14, 19, 12, 9, 11] \\ N &= 65 \end{align_} \]

\[ \text{Mean} = \frac{(3 \cdot 14) + (4 \cdot 19) + (5 \cdot 12) + (6 \cdot 9) + (7 \cdot 11)}{65} = 4.7538 \]

The variance is calculated using the formula: \[ \text{Variance} = \frac{\sum f_i (x_i - \text{Mean})^2}{N} \]

\[ \text{Variance} = \frac{14(3 - 4.7538)^2 + 19(4 - 4.7538)^2 + 12(5 - 4.7538)^2 + 9(6 - 4.7538)^2 + 11(7 - 4.7538)^2}{65} = 1.9086 \]

The standard deviation is the square root of the variance: \[ \text{Standard Deviation} = \sqrt{\text{Variance}} = \sqrt{1.9086} = 1.3815 \]

Final Answer