Create a linear regression equation that can be used to model the amount of money in the account (the balance), y, based on the time (in weeks), x. Round all values to the nearest hundredth.
Use the model to approximate the amount in the account after 15 weeks. Round to the nearest hundredth.
Use the model to approximate how long it would take to have $50.00 in the account.
To create a linear regression equation, we need to first gather some data points. These data points will consist of the time in weeks (x) and the corresponding amount of money in the account (y). Once we have enough data points, we can use these to find the equation of a straight line that best represents the relationship between x and y.
Let's assume we have the following data points:
x | y
-----
1 | 10.50
3 | 23.70
5 | 36.90
7 | 50.10
9 | 63.30
To find the equation of the line, we can use the least squares method. This method minimizes the sum of the squared differences between the actual y-values and the predicted y-values.
Step 1: Calculate the mean values of x and y.
Mean of x (x̄) = (1+3+5+7+9)/5 = 5
Mean of y (ȳ) = (10.50+23.70+36.90+50.10+63.30)/5 = 36.30
Step 2: Calculate the differences between each x-value and the mean of x, and each y-value and the mean of y.
differences of x (x - x̄): -4, -2, 0, 2, 4
differences of y (y - ȳ): -25.8, -12.6, 0.6, 13.8, 27
Step 3: Calculate the squared differences for each pair of differences.
squared differences (x - x̄)^2: 16, 4, 0, 4, 16
squared differences (y - ȳ)^2: 667.24, 158.76, 0.36, 190.44, 729
Step 4: Calculate the product of the differences (x - x̄) and (y - ȳ).
product of differences (x - x̄)(y - ȳ): 103.2, 31.8, 0, 27.6, 108
Step 5: Calculate the sum of the squared differences and the sum of the product of differences.
sum of squared differences: 40
sum of product of differences: 271.6
Step 6: Calculate the slope (m) of the regression line.
m = sum of product of differences / sum of squared differences
m = 271.6 / 40
m = 6.79
Step 7: Calculate the y-intercept (b) of the regression line using the mean values.
b = ȳ - m * x̄
b = 36.30 - 6.79 * 5
b = 36.30 - 33.95
b = 2.35
Therefore, the linear regression equation that models the amount of money in the account (the balance) based on the time (in weeks) is:
y = 6.79x + 2.35
To approximate the amount in the account after 15 weeks, we can substitute x = 15 into the equation and solve for y:
y = 6.79 * 15 + 2.35
y = 101.85 + 2.35
y ≈ 104.20
After 15 weeks, the approximate amount in the account is $104.20.
To approximate how long it would take to have $50.00 in the account, we can substitute y = 50 into the equation and solve for x:
50 = 6.79x + 2.35
6.79x = 50 - 2.35
6.79x ≈ 47.65
x ≈ 7
So, it would take approximately 7 weeks to have $50.00 in the account.
If you copied and pasted data here, that does not work on this site.
Anyway Google linear regression. I bet you even find a calculator :)