In: Statistics and Probability
An exercise science major wants to try to use body weight to predict how much someone can bench press. He collects the data shown below on 30 male students. Both quantities are measured in pounds.
Body Weight | Bench Press |
147 | 134 |
134 | 131 |
141 | 125 |
129 | 135 |
152 | 147 |
176 | 142 |
196 | 171 |
200 | 158 |
132 | 134 |
176 | 153 |
204 | 153 |
194 | 156 |
211 | 164 |
145 | 129 |
180 | 141 |
201 | 162 |
124 | 118 |
145 | 151 |
172 | 143 |
145 | 137 |
137 | 121 |
153 | 143 |
191 | 170 |
177 | 160 |
150 | 138 |
202 | 157 |
144 | 135 |
151 | 153 |
202 | 157 |
216 | 169 |
a) What type of association does there appear to be between
these two variables?
Solved Answer: positive association
b) Compute a 95% confidence interval for the average bench press of 150 pound males. What is the lower limit? Give your answer to two decimal places.
c) Compute a 95% confidence interval for the average bench press of 150 pound males. What is the upper limit? Give your answer to two decimal places.
d) Compute a 95% prediction interval for the bench press of a 150 pound male. What is the lower limit? Give your answer to two decimal places.
e) Compute a 95% prediction interval for the bench press of a 150 pound male. What is the upper limit? Give your answer to two decimal places.
Please provide formulas used for the confidence intervals and prediction intervals and standard error. Thank you!
Body Weight, X | Bench Press, Y | XY | X² | Y² |
147 | 134 | 19698 | 21609 | 17956 |
134 | 131 | 17554 | 17956 | 17161 |
141 | 125 | 17625 | 19881 | 15625 |
129 | 135 | 17415 | 16641 | 18225 |
152 | 147 | 22344 | 23104 | 21609 |
176 | 142 | 24992 | 30976 | 20164 |
196 | 171 | 33516 | 38416 | 29241 |
200 | 158 | 31600 | 40000 | 24964 |
132 | 134 | 17688 | 17424 | 17956 |
176 | 153 | 26928 | 30976 | 23409 |
204 | 153 | 31212 | 41616 | 23409 |
194 | 156 | 30264 | 37636 | 24336 |
211 | 164 | 34604 | 44521 | 26896 |
145 | 129 | 18705 | 21025 | 16641 |
180 | 141 | 25380 | 32400 | 19881 |
201 | 162 | 32562 | 40401 | 26244 |
124 | 118 | 14632 | 15376 | 13924 |
145 | 151 | 21895 | 21025 | 22801 |
172 | 143 | 24596 | 29584 | 20449 |
145 | 137 | 19865 | 21025 | 18769 |
137 | 121 | 16577 | 18769 | 14641 |
153 | 143 | 21879 | 23409 | 20449 |
191 | 170 | 32470 | 36481 | 28900 |
177 | 160 | 28320 | 31329 | 25600 |
150 | 138 | 20700 | 22500 | 19044 |
202 | 157 | 31714 | 40804 | 24649 |
144 | 135 | 19440 | 20736 | 18225 |
151 | 153 | 23103 | 22801 | 23409 |
202 | 157 | 31714 | 40804 | 24649 |
216 | 169 | 36504 | 46656 | 28561 |
Ʃx = | Ʃy = | Ʃxy = | Ʃx² = | Ʃy² = |
5027 | 4387 | 745496 | 865881 | 647787 |
Sample size, n = | 30 |
x̅ = Ʃx/n = | 167.567 |
y̅ = Ʃy/n = | 146.233 |
SSxx = Ʃx² - (Ʃx)²/n = | 23523.4 |
SSyy = Ʃy² - (Ʃy)²/n = | 6261.37 |
SSxy = Ʃxy - (Ʃx)(Ʃy)/n = | 10381 |
a) There appear to be a positive association between these two variables.
b) Slope, b = SSxy/SSxx = 0.441307296
y-intercept, a = y̅ -b* x̅ = 72.28494079
Regression equation :
ŷ = 72.2849 + 0.4413 x
Predicted value of y at X = 150
ŷ = 72.2849 + 0.4413 * 150 = 138.481
Sum of Square error, SSE = SSyy -SSxy²/SSxx = 1680.140918
Standard error, se = √(SSE/(n-2)) = 7.74629
Critical value, t_c = T.INV.2T(0.05, 28) = 2.0484
95% Confidence interval :
c) 95% Confidence interval :
d) 95% prediction interval:
e) 95% prediction interval: