Plymouth Rock Securities is interested in finding out if there is a relationship between the number of new clients brought into the firm by a broker and the sales performance of the broker. A random sample of 11 brokers' records are reviewed to determine the number of new clients enrolled last year and total sales in millions of dollars: (12pts)
Broker
1
2
3
4
5
6
7
8
9
10
11
Clients
27
11
42
33
15
15
25
36
28
30
17
Sales, $
52
37
64
55
29
34
58
59
44
48
31
a) How closely related is the new client base to sales performance? Draw the scatterplot and compute the correlation and describe the relationship
b) Find the least-squares equation to predict sales from number of clients. Can the least squares equation be used to predict sales?
c) What does the slope represent?
d) What would a new broker who brings in 30 clients sell, on average?
e) How much of the variability in sales is not explained by the number of new clients
To find the answers to these questions, we need to perform some calculations and interpret the results. Let's go step by step:
a) To determine how closely related the new client base is to sales performance, we can start by creating a scatterplot. The x-axis will represent the number of new clients, and the y-axis will represent the sales in millions of dollars. By visualizing the data points, we can get an idea of the relationship between these two variables.
b) To find the least-squares equation to predict sales from the number of clients, we need to perform linear regression analysis. This will help us find the line that best fits the data points. The equation will be of the form:
Sales = a + b * Number of clients
The values of a and b will be determined by the regression analysis.
c) The slope of the least-squares equation represents the change in sales for each additional new client. It tells us how much the sales increase or decrease for every unit increase in the number of clients.
d) To predict the sales for a new broker who brings in 30 clients, we can substitute the given value into the equation obtained from the regression analysis. This will give us an estimate of the average sales for that number of clients.
e) To find out how much of the variability in sales is not explained by the number of new clients, we need to calculate the coefficient of determination (R-squared). This value tells us the proportion of the total variation in sales that can be explained by the number of new clients. The remaining percentage represents the variability that is not explained by this relationship.
Now, let's go ahead and perform the calculations to find the answers to the questions.