The managers of a brokerage firm are interested in finding out if the number of new clients a broker brings into the firm affects the sales generated by the broker. They sample 12 brokers and determine the number of new clients they have enrolled in the last year and their sales amounts in thousands of dollars. These data are presented in the table that follows.

Question

Broker Clients Sales
1 27 52
2 11 37
3 42 64
4 33 55
5 15 29
6 15 34
7 25 58
8 36 59
9 28 44
10 30 48
11 17 31
12 22 38

The Sxy is ?

Answer 1

To calculate \( S_{xy} \), which is the covariance between the number of clients and the sales generated, we first need to find the necessary components, including the mean (average) of the clients and sales. The formula for \( S_{xy} \) is:

\[ S_{xy} = \sum{(X_i - \bar{X})(Y_i - \bar{Y})} \]

where:

• \( X_i \) is the number of clients for each broker,
• \( Y_i \) is the sales for each broker,
• \( \bar{X} \) is the mean number of clients,
• \( \bar{Y} \) is the mean sales.

Step 1: Calculate the means

1. Calculate the mean number of clients (\( \bar{X} \)):

\[ \bar{X} = \frac{\text{Total Clients}}{n} = \frac{27 + 11 + 42 + 33 + 15 + 15 + 25 + 36 + 28 + 30 + 17 + 22}{12} \]

Calculating the total:

\[ \text{Total Clients} = 27 + 11 + 42 + 33 + 15 + 15 + 25 + 36 + 28 + 30 + 17 + 22 = 392 \]

So,

\[ \bar{X} = \frac{392}{12} \approx 32.67 \]

2. Calculate the mean sales (\( \bar{Y} \)):

\[ \bar{Y} = \frac{\text{Total Sales}}{n} = \frac{52 + 37 + 64 + 55 + 29 + 34 + 58 + 59 + 44 + 48 + 31 + 38}{12} \]

Calculating the total:

\[ \text{Total Sales} = 52 + 37 + 64 + 55 + 29 + 34 + 58 + 59 + 44 + 48 + 31 + 38 = 489 \]

So,

\[ \bar{Y} = \frac{489}{12} \approx 40.75 \]

Step 2: Calculate \( S_{xy} \)

Now we calculate \( S_{xy} \):

1. Find \( (X_i - \bar{X}) \) and \( (Y_i - \bar{Y}) \) for each broker, then compute the product:

| Broker | Clients (\(X_i\)) | Sales (\(Y_i\)) | \(X_i - \bar{X}\) | \(Y_i - \bar{Y}\) | \( (X_i - \bar{X})(Y_i - \bar{Y}) \) | |--------|------------------|-----------------|---------------------|---------------------|-------------------------------------| | 1 | 27 | 52 | -5.67 | 11.25 | -63.75 | | 2 | 11 | 37 | -21.67 | -3.75 | 81.26 | | 3 | 42 | 64 | 9.33 | 23.25 | 217.78 | | 4 | 33 | 55 | 0.33 | 14.25 | 4.69 | | 5 | 15 | 29 | -17.67 | -11.75 | 207.64 | | 6 | 15 | 34 | -17.67 | -6.75 | 119.69 | | 7 | 25 | 58 | -7.67 | 17.25 | -132.44 | | 8 | 36 | 59 | 3.33 | 18.25 | 60.83 | | 9 | 28 | 44 | -4.67 | 3.25 | -15.19 | | 10 | 30 | 48 | -2.67 | 7.25 | -19.38 | | 11 | 17 | 31 | -15.67 | -9.75 | 152.14 | | 12 | 22 | 38 | -10.67 | -2.75 | 29.33 |

Calculating the total of the last column:

\[ S_{xy} = -63.75 + 81.26 + 217.78 + 4.69 + 207.64 + 119.69 - 132.44 + 60.83 - 15.19 - 19.38 + 152.14 + 29.33 \] \[ S_{xy} \approx 327.48 \]

Conclusion

Therefore, the value of \( S_{xy} \) is approximately 327.48.