a) To determine the formula for the relationship between income and flower sales, we need to find the equation of a linear relationship in the form y = mx + b, where y is the dependent variable (flower sales) and x is the independent variable (income).
To find the equation, we need to calculate the slope (m) and the intercept (b) from the given data points.
Let's label the first data point as (x1, y1) = ($22,000, $60,000) and the second data point as (x2, y2) = ($29,000, $77,500).
The formula for the slope (m) is given by:
m = (y2 - y1) / (x2 - x1)
Substituting the values:
m = ($77,500 - $60,000) / ($29,000 - $22,000)
m = $17,500 / $7,000
m β 2.5
Now, to find the intercept (b), we can use either of the data points with the formula:
b = y - mx
Using the first data point, (x1, y1) = ($22,000, $60,000):
b = $60,000 - 2.5 * $22,000
b = $60,000 - $55,000
b β $5,000
Therefore, the formula for the relationship between income (x) and flower sales (y) is:
y = 2.5x + $5,000
b) The intercept value in the equation represents the value of the dependent variable (flower sales) when the independent variable (income) is zero. In this case, it means that when the average income is zero, there would still be an expected flower sales of $5,000. However, in the context of this problem, it is not meaningful to interpret the intercept in this way. It might be more practical to consider the intercept as a fixed cost or baseline sales unrelated to income.
c) The slope value in the equation represents the rate of change in the dependent variable (flower sales) for every unit increase in the independent variable (income). In this case, for every $1,000 increase in average income, there is an expected increase in flower sales of 2.5 * $1,000 = $2,500. So, the slope value of 2.5 indicates that there is a positive relationship between income and flower sales.
d) To estimate the number of flowers John should expect to sell with an average income of $34,000, we can use the formula y = 2.5x + $5,000 and substitute x = $34,000:
y = 2.5 * $34,000 + $5,000
y β $85,000
Therefore, John should expect to sell approximately $85,000 worth of flowers.
e) To determine the average income in Sue's market area when 50,000 flowers were sold, we can rearrange the formula y = 2.5x + $5,000 to solve for x (income).
Substituting y = $50,000:
$50,000 = 2.5x + $5,000
Subtracting $5,000 from both sides:
$45,000 = 2.5x
Dividing by 2.5:
x = $45,000 / 2.5
x = $18,000
Therefore, the average income in Sue's market area is $18,000.