John has believed for some time that flower sales are related to consumer income. Recently in a trade magazine, he found some interesting data. In a market where average income was $22,000, flower sales were $60,000. In another market where all factors were the same, except that income averaged $29,000, flower sales were $77,500. The article indicated that the relationship between income and flower sales was linear. a) Determine the formula for this relationship. Be careful about which variable is independent and which is dependent. b) Interpret the meaning of the equation's intercept value. c) Interpret the meaning of the slope value. d) If average income in John's market area is $34,000 and he has the only flower shop, how many flowers should he expect to sell? e) Sue is a friend of John's and has a flower shop in another state. She told John that in her market area, a total of 50,000 flowers were sold. What is the average income in that market?
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 ≈ 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 ≈ $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 ≈ $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.