Question

A sports medicine major wanted to conduct anexperiment to determine if there is a correlationbetween the members of the soccer teams legstrength and the time it takes to sprint 40 yards.He sets up the following test and records thedata.Every day for a week he counts how many timeseach player can leg press 350 pounds. Thefollowing week, he has each player sprint 40yards every day. The table shows the averagenumber of leg-press repetitions and the average40-yard dash time (in seconds) for sevenrandomly selected players. What is the equationof the line of best fit? How many seconds shouldbe expect a player to take to run 40 yards if thatplayer can do 22 leg-press repetitions? Roundany values to the nearest tenth, if necessary.

Answer 1

Given

The data,

To find:

1) The equation of the line of best fit.

2) The time when the player can do 22 leg press repetitions.

Step-by-step explanation:

It is given that,

That implies,

The equation of the line of best fit is given by,

`T=kL+b`

Then,

`k=\frac{\bar{LT}-\bar{L}\bar{T}}{\bar{L^2}-(\bar{L})^2},\text{ }b=\bar{T}-k\bar{L}`

Therefore,

`\begin{gathered} \bar{L}=(12+32+11+7+23+28+15)/(7) \\ =(128)/(7) \\ =18.3 \end{gathered}`

Also,

`\begin{gathered} \bar{T}=(8.6+14.6+7.1+8.3+11.9+13.4+9.5)/(7) \\ =(73.4)/(7) \\ =10.5 \end{gathered}`

Also,

`\begin{gathered} \bar{L^2}=(12^2+32^2+7^2+11^2+23^2+28^2+15^2)/(7) \\ =(2876)/(7) \\ =410.9 \end{gathered}`

Also,

`\begin{gathered} \bar{LT}=(12*8.6+32*14.6+7*7.1+11*8.3+23*11.9+28*13.4+15*9.5)/(7) \\ =(1502.8)/(7) \\ =214.7 \end{gathered}`

That implies,

`\begin{gathered} k=(214.7-(18.3*10.5))/(410.9-(18.3)^2) \\ =(214.7-192.15)/(410.9-334.89) \\ =(22.55)/(76.01) \\ =0.3 \end{gathered}`

And,

`\begin{gathered} b=\bar{T}-k\bar{L} \\ =10.5-(0.3*18.3) \\ =10.5-5.49 \\ =5 \end{gathered}`

Hence, the equation of the best line of fit is,

`T=0.3L+5`

And, the time taken when L=22 is,

`\begin{gathered} T=0.3(22)+5 \\ =6.6+5 \\ =11.6 \end{gathered}`

Hence, the time taken to complete 22 Leg press repetition is 11.6.