Number of months
Height of plant (inches)
1
2
3
4
2.3
3.6
5.2
4.9
5
7.3
6
7.6
7
8.0
a) What is the equation of the line of best fit? How tall can we expect the plant to be after 13 months?
b) 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 linear regression. First, let's calculate the mean of the months and the mean of the plant heights:
Mean of months: (1 + 2 + 3 + 4 + 5 + 6 + 7) / 7 = 4.
Mean of plant heights: (2.3 + 3.6 + 5.2 + 4.9 + 5 + 7.3 + 7.6 + 8) / 8 = 5.35.
Next, we need to calculate the sum of the products of the differences between each month and the mean month, and each height and the mean height, and the sum of the squares of the differences between each month and the mean month:
Sum of [(month - mean month) * (height - mean height)]:
(1 - 4) * (2.3 - 5.35) + (2 - 4) * (3.6 - 5.35) + (3 - 4) * (5.2 - 5.35) + (4 - 4) * (4.9 - 5.35) + (5 - 4) * (5 - 5.35) + (6 - 4) * (7.3 - 5.35) + (7 - 4) * (7.6 - 5.35) + (8 - 4) * (8 - 5.35) = 1.78.
Sum of Squares of (month - mean month):
(1 - 4)^2 + (2 - 4)^2 + (3 - 4)^2 + (4 - 4)^2 + (5 - 4)^2 + (6 - 4)^2 + (7 - 4)^2 + (8 - 4)^2 = 14.
Now, we can calculate the slope of the line of best fit:
slope = Sum of [(month - mean month) * (height - mean height)] / Sum of Squares of (month - mean month) = 1.78 / 14 = 0.1271.
Finally, we can find the y-intercept (b) using the mean of the months and the mean of the plant heights:
y-intercept (b) = mean height - slope * mean month = 5.35 - 0.1271 * 4 = 4.636.
Therefore, the equation of the line of best fit is:
y = 0.1271x + 4.636.
To find the height of the plant after 13 months, we can substitute x = 13 into the equation:
y = 0.1271 * 13 + 4.636 = 6.7357.
So, we can expect the plant to be approximately 6.74 inches tall after 13 months.
b) To find the correlation coefficient, we can calculate the standard deviation of the months and the standard deviation of the plant heights:
Standard deviation of months (σx):
√ [((1 - 4)^2 + (2 - 4)^2 + (3 - 4)^2 + (4 - 4)^2 + (5 - 4)^2 + (6 - 4)^2 + (7 - 4)^2 + (8 - 4)^2) / 8] = √(14 / 8) ≈ 1.06.
Standard deviation of plant heights (σy):
√[((2.3 - 5.35)^2 + (3.6 - 5.35)^2 + (5.2 - 5.35)^2 + (4.9 - 5.35)^2 + (5 - 5.35)^2 + (7.3 - 5.35)^2 + (7.6 - 5.35)^2 + (8 - 5.35)^2) / 8] = √(1.313 / 8) ≈ 0.405.
Next, we can calculate the covariance:
Covariance = Sum of [(month - mean month) * (height - mean height)] / number of data points = 1.78 / 8 = 0.2238.
Finally, we can calculate the correlation coefficient (r):
r = Covariance / (standard deviation of months * standard deviation of plant heights) = 0.2238 / (1.06 * 0.405) ≈ 0.514.
The correlation coefficient is approximately 0.514.
To determine how confident we can be in the predicted value being reasonably close to the actual value, we can examine the correlation coefficient. A correlation coefficient value close to 1 suggests a strong positive linear relationship, while a value close to -1 suggests a strong negative linear relationship. In this case, a correlation coefficient of 0.514 indicates a moderate positive linear relationship.
Therefore, we can reasonably expect the predicted value to be within a moderate range of the actual value. However, it is important to note that there may still be some variability and uncertainty in the prediction.