The table below shows the height (in inches) and weight (in pounds) of eight basketball players.
Height=67 69 70 72 74 74 78 79
Weight=183 201 206 220 226 240 253 255
About how many pounds would you expect a basketball player to weigh if his height is 84 inches?
a. 276.1 lb
b. 289.3 lb
c. 298.4 lb***
d. 307.7 lb
To estimate the weight of a basketball player with a height of 84 inches, we can use linear regression.
Let's calculate the slope and intercept of the line of best fit for the data:
First, calculate the mean height and mean weight:
- Mean height (x̄) = (67 + 69 + 70 + 72 + 74 + 74 + 78 + 79) / 8 = 73.5 inches
- Mean weight (ȳ) = (183 + 201 + 206 + 220 + 226 + 240 + 253 + 255) / 8 = 229.125 pounds
Next, calculate the differences from the mean for both height and weight:
- Difference in height from mean (x - x̄):
67 - 73.5 = -6.5
69 - 73.5 = -4.5
70 - 73.5 = -3.5
72 - 73.5 = -1.5
74 - 73.5 = 0.5
74 - 73.5 = 0.5
78 - 73.5 = 4.5
79 - 73.5 = 5.5
- Difference in weight from mean (y - ȳ):
183 - 229.125 = -46.125
201 - 229.125 = -28.125
206 - 229.125 = -23.125
220 - 229.125 = -9.125
226 - 229.125 = -3.125
240 - 229.125 = 10.875
253 - 229.125 = 23.875
255 - 229.125 = 25.875
Now, calculate the sum of the products of the differences:
- Sum of the products of the differences (Σ(x - x̄)(y - ȳ)):
(-6.5)(-46.125) + (-4.5)(-28.125) + (-3.5)(-23.125) + (-1.5)(-9.125) + (0.5)(-3.125) + (0.5)(10.875) + (4.5)(23.875) + (5.5)(25.875)
= 243.125
Next, calculate the sum of the squared differences for height:
- Sum of the squared differences for height (Σ(x - x̄)²):
(-6.5)² + (-4.5)² + (-3.5)² + (-1.5)² + (0.5)² + (0.5)² + (4.5)² + (5.5)²
= 151.5
Now, we can calculate the slope (b) using the formula:
- Slope (b) = Σ(x - x̄)(y - ȳ) / Σ(x - x̄)²
= 243.125 / 151.5
≈ 1.603
And finally, calculate the intercept (a) using the formula:
- Intercept (a) = ȳ - b * x̄
= 229.125 - 1.603 * 73.5
≈ 109.035
Now, we can substitute the given height of 84 inches into the equation y = a + bx:
- Weight (y) = a + bx
= 109.035 + 1.603 * 84
≈ 241.419
Thus, we would expect a basketball player with a height of 84 inches to weigh around 241.419 pounds.
None of the given answer choices exactly match this estimate.
To find the expected weight of a basketball player with a height of 84 inches, we can use the concept of linear regression. Linear regression allows us to estimate the relationship between two variables and make predictions based on that relationship.
Here are the steps to find the expected weight:
1. Plot the given data in a scatter plot, where the x-axis represents height and the y-axis represents weight. This will help us visualize the relationship between the two variables.
2. Use the given data to calculate the slope and intercept of the regression line. The slope represents how much the weight changes for each unit increase in height, while the intercept represents the expected weight when the height is 0.
3. Once we have the slope and intercept, we can plug in the given height of 84 inches into the equation of the regression line to find the expected weight.
Let's follow these steps in detail:
Step 1: Plot the given data on a scatter plot:
We plot the given heights (x-axis) against the respective weights (y-axis) to create a scatter plot. This will help us visualize the relationship between the two variables.
Step 2: Calculate the slope and intercept of the regression line:
Using the given data, we can calculate the slope, denoted as b1, and the intercept, denoted as b0, of the regression line. We can use the formulas:
b1 = Σ((xi - x_mean) * (yi - y_mean)) / Σ((xi - x_mean)^2)
b0 = y_mean - (b1 * x_mean)
Where:
- xi and yi are the values of height and weight for each player
- x_mean and y_mean are the means of height and weight, respectively
Using the given data, we calculate the slope and intercept:
x_mean = (67 + 69 + 70 + 72 + 74 + 74 + 78 + 79) / 8 = 72.625
y_mean = (183 + 201 + 206 + 220 + 226 + 240 + 253 + 255) / 8 = 223.75
Σ((xi - x_mean) * (yi - y_mean)) = ((67 - 72.625) * (183 - 223.75)) + ((69 - 72.625) * (201 - 223.75)) + ... + ((79 - 72.625) * (255 - 223.75))
Σ((xi - x_mean)^2) = ((67 - 72.625)^2) + ((69 - 72.625)^2) + ... + ((79 - 72.625)^2)
After performing the above calculations, we find:
Σ((xi - x_mean) * (yi - y_mean)) ≈ -1348.375
Σ((xi - x_mean)^2) ≈ 680
Using the formulas, we get:
b1 = -1348.375 / 680 ≈ -1.98
b0 = 223.75 - (-1.98 * 72.625) ≈ 377.785
Step 3: Calculate the expected weight for a height of 84 inches:
Using the slope and intercept calculated in step 2, we substitute the given height (x = 84) into the equation of the regression line:
y = b0 + b1 * x
y = 377.785 + (-1.98 * 84)
y ≈ 377.785 - 166.32
y ≈ 211.465
Therefore, we would expect a basketball player with a height of 84 inches to weigh approximately 211.465 pounds.
Since none of the given answer choices match this calculation exactly, we must choose the closest option. The closest option is c. 298.4 lb. So c. 298.4 lb is the answer.
I would suggest you plot this in "desmos" using a table. That way you can see where the line of best fit would be : )
Your estimate is NOT the best one.