the table shows the height of a plant at each month. Use the table to answer the questions.
number of months | height of plant (in)
1 | 2.3
2 | 3.6
3 | 5.2
4 | 4.9
5 | 7.3
6 | 7.6
7 | 8.0
a) What is the equation of the line of best fit?
b) How tall can we expect the plant to be after 13 months?
c) What is the correlation coefficient? How confident can you be that your predicted value will be reasonably close to the actual value?
a) To find the equation of the line of best fit, we can use the method of least squares. Let's denote the number of months as x and the height of the plant as y. We need to find the equation in the form y = mx + b, where m is the slope and b is the y-intercept.
First, let's calculate the mean values of x and y:
Mean of x: (1 + 2 + 3 + 4 + 5 + 6 + 7) / 7 = 4
Mean of y: (2.3 + 3.6 + 5.2 + 4.9 + 7.3 + 7.6 + 8) / 7 = 5.5286 (rounded to 4 decimal places)
Next, let's calculate the sums of the cross-products of x and y, as well as the sum of the squares of x:
Sum of (x * y): (1 * 2.3) + (2 * 3.6) + (3 * 5.2) + (4 * 4.9) + (5 * 7.3) + (6 * 7.6) + (7 * 8) = 174.7
Sum of (x^2): (1^2) + (2^2) + (3^2) + (4^2) + (5^2) + (6^2) + (7^2) = 84
Now, let's use the formulas to calculate the slope and y-intercept:
m = (Sum of (x * y) - (n * mean of x * mean of y)) / (Sum of (x^2) - (n * mean of x^2))
m = (174.7 - (7 * 4 * 5.5286)) / (84 - (7 * (4^2)))
m = (174.7 - (98.6)) / (84 - (7 * 16))
m = 76.1 / (-8)
m = -9.5125 (rounded to 4 decimal places)
b = mean of y - (m * mean of x)
b = 5.5286 - (-9.5125 * 4)
b = 5.5286 + 38.0500
b = 43.5786 (rounded to 4 decimal places)
Therefore, the equation of the line of best fit is y = -9.5125x + 43.5786.
b) To find the expected height of the plant after 13 months, we can substitute x = 13 into the equation:
y = -9.5125(13) + 43.5786
y = -123.7625 + 43.5786
y = -80.1839 (rounded to 4 decimal places)
Therefore, we can expect the plant to be approximately -80.1839 inches tall after 13 months. However, it's important to note that negative height values do not make sense in this context, so we should interpret this as the plant not being expected to survive or grow in a negative height scenario.
c) The correlation coefficient measures the strength and direction of the linear relationship between two variables. It ranges from -1 to 1, with -1 indicating a perfect negative linear relationship, 0 indicating no linear relationship, and 1 indicating a perfect positive linear relationship.
To calculate the correlation coefficient, we can use the formula:
r = (n * Sum of (x * y) - (Sum of x) * (Sum of y)) / sqrt((n * Sum of (x^2) - (Sum of x)^2) * (n * Sum of (y^2) - (Sum of y)^2))
Let's calculate the necessary components:
n = 7 (number of data points)
Sum of x = 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28
Sum of y = 2.3 + 3.6 + 5.2 + 4.9 + 7.3 + 7.6 + 8 = 38.9
Sum of (x * y) = 174.7
Sum of (x^2) = 84
Sum of (y^2) = (2.3^2) + (3.6^2) + (5.2^2) + (4.9^2) + (7.3^2) + (7.6^2) + (8^2) = 162.98
Now, let's calculate the correlation coefficient:
r = (7 * 174.7 - 28 * 38.9) / sqrt((7 * 84 - 28^2) * (7 * 162.98 - 38.9^2))
r = (1222.9 - 1061.2) / sqrt((588 - 784) * (1138.86 - 1512.21))
r = 161.7 / sqrt((-196) * (-373.35))
r = 161.7 / sqrt(73050.6)
r = 161.7 / 270.0245
r = 0.5991 (rounded to 4 decimal places)
The correlation coefficient is 0.5991. This indicates a moderate positive linear relationship between the number of months and the height of the plant. With a correlation coefficient close to 1, we can be reasonably confident that our predicted value for the plant's height after 13 months will be reasonably close to the actual value.