Question

In: Statistics and Probability

An exercise science major wants to try to use body weight to predict how much someone...

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.

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.

Body weight Bench press xy x sq y sq
147 139 20433 21609 19321
127 139 17653 16129 19321
154 129 19866 23716 16641
209 155 32395 43681 24025
201 169 33969 40401 28561
153 135 20655 23409 18225
188 155 29140 35344 24025
174 163 28362 30276 26569
139 126 17514 19321 15876
129 115 14835 16641 13225
167 143 23881 27889 20449
142 124 17608 20164 15376
185 160 29600 34225 25600
161 147 23667 25921 21609
217 161 34937 47089 25921
133 110 14630 17689 12100
180 148 26640 32400 21904
213 159 33867 45369 25281
134 119 15946 17956 14161
135 128 17280 18225 16384
184 155 28520 33856 24025
168 159 26712 28224 25281
209 157 32813 43681 24649
132 139 18348 17424 19321
121 122 14762 14641 14884
179 158 28282 32041 24964
204 162 33048 41616 26244
137 126 17262 18769 15876
148 142 21016 21904 20164
131 139 18209 17161 19321

Solutions

Expert Solution

Body weight Bench press xy x sq y sq
147 139 20433 21609 19321
127 139 17653 16129 19321
154 129 19866 23716 16641
209 155 32395 43681 24025
201 169 33969 40401 28561
153 135 20655 23409 18225
188 155 29140 35344 24025
174 163 28362 30276 26569
139 126 17514 19321 15876
129 115 14835 16641 13225
167 143 23881 27889 20449
142 124 17608 20164 15376
185 160 29600 34225 25600
161 147 23667 25921 21609
217 161 34937 47089 25921
133 110 14630 17689 12100
180 148 26640 32400 21904
213 159 33867 45369 25281
134 119 15946 17956 14161
135 128 17280 18225 16384
184 155 28520 33856 24025
168 159 26712 28224 25281
209 157 32813 43681 24649
132 139 18348 17424 19321
121 122 14762 14641 14884
179 158 28282 32041 24964
204 162 33048 41616 26244
137 126 17262 18769 15876
148 142 21016 21904 20164
131 139 18209 17161 19321
X Y XY X² Y²
total sum 4901 4283 711850 826771 619303

sample size ,   n =   30          
here, x̅ =Σx/n =   163.3667   ,   ȳ = Σy/n =   142.77  
                  
SSxx =    Σx² - (Σx)²/n =   26111          
SSxy=   Σxy - (Σx*Σy)/n =   12151          
SSyy =    Σy²-(Σy)²/n =   7833          
estimated slope , ß1 = SSxy/SSxx =   12150.567   /   26110.967   =   0.46534
                  
intercept,   ß0 = y̅-ß1* x̄ =   66.74506          
                  
so, regression line is   Ŷ =   66.7451   +   0.4653   *x
                  
SSE=   (Sx*Sy - S²xy)/Sx =    2179.1804          
                  
std error ,Se =    √(SSE/(n-2)) =    8.82201          
-------------------

b)

X Value=   150                      
Confidence Level=   95%                      
                          
                          
Sample Size , n=   30                      
Degrees of Freedom,df=n-2 =   28                      
critical t Value=tα/2 =   2.048   [excel function: =t.inv.2t(α/2,df) ]                  
                          
X̅ =    163.37                      
Σ(x-x̅)² =Sxx   26111                      
Standard Error of the Estimate,Se=   8.8220                      
                          
Predicted Y at X=   150   is                  
Ŷ =   66.7451   +   0.4653   *   150   =   136.5466
                          
standard error, S(ŷ)=Se*√(1/n+(X-X̅)²/Sxx) =    1.768                      
margin of error,E=t*Std error=t* S(ŷ) =   2.0484   *   1.7683   =   3.6222      
                          
Confidence Lower Limit=Ŷ +E =    136.547   -   3.622   =   132.92

c)

Confidence Upper Limit=Ŷ +E =   136.547   +   3.622   =   140.17

d)

For Individual Response Y                  
standard error, S(ŷ)=Se*√(1+1/n+(X-X̅)²/Sxx) =   8.9975              
margin of error,E=t*std error=t*S(ŷ)=    2.0484   *   9.00   =   18.4305
                  
Prediction Interval Lower Limit=Ŷ -E =   136.547   -   18.43   =   118.12

e)

Prediction Interval Upper Limit=Ŷ +E =   136.547   +   18.43   =   154.98

orchestra answered 1 year ago
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