Question

Taller basketball players have a theoretical shooting advantage because it’s harder to block them. But can a player’s height determine how well they shoot free throws, where there is no defender?

Player Height (cm)	Free Throw Shooting Percentage
188	74%
208	80%
186	75%
190	76%
211	85%
205	90%
214	88%
200	80%

a) Determine the coefficient of determination and interpret its value.

b) What is the equation of the regression line? Keep three decimal places for calculated values.

c) Estimate the percentage of free throws a 200cm tall player will make.

Answer 1

a)

These are the data that have been provided for the dependent and independent variable:

Obs.	Player Height (cm)	Free Throw Shooting Percentage
1	188	74
2	208	80
3	186	75
4	190	76
5	211	85
6	205	90
7	214	88
8	200	80

We need to compute the coefficient of determination, which is computed by squaring the correlation coefficient, which needs to be computed first.

Now, with the provided sample data, we need to construct the following table to compute the correlation coefficient:

Obs.	Player Height (cm)	Free Throw Shooting Percentage	X_i^2​	Y_i^2	X_i Y_i​
1	188	74	35344	5476	13912
2	208	80	43264	6400	16640
3	186	75	34596	5625	13950
4	190	76	36100	5776	14440
5	211	85	44521	7225	17935
6	205	90	42025	8100	18450
7	214	88	45796	7744	18832
8	200	80	40000	6400	16000
Sum =	1602	648	321646	52746	130159

Based on the table above, we compute the following sum of squares that will be used in the calculation of the correlation coefficient:

Now, the correlation coefficient is computed using the following expression::

Then, the coefficient of determination, or R-Squared coefficient (R^2) , is computed by simply squaring the correlation coefficient that was found above. So we get:

Therefore, based on the sample data provided, it is found that the coefficient of determination is R^2 = 0.7225 . This implies that approximately 72.25% of variation in the dependent variable is explained by the independent variable.

Based on the above table, the following is calculated:

Therefore, based on the above calculations, the regression coefficients (the slope m, and the y-intercept n) are obtained as follows:

Therefore, we find that the regression equation is:

Free Throw Shooting Percentage = -13.0263 + 0.4695 Player Height (cm)

Player Height (cm) = 200

Free Throw Shooting Percentage = -13.0263 + 0.4695*200

Free Throw Shooting Percentage = 80.8737

Free Throw Shooting Percentage = 80.1