Question

The table below shows the shoe size and age of 7 boys

(x) (y)
Tyrone 6 9
Marcel 6 11
Patrick 7 15
Bobby 8 11
Dylan 9 15
Mike 10 16
Jonathan 12 17
Approximately what percent of the boys’ ages is more than 1 year different from
the age predicted by the line of best fit for the data?
A 14%
B 29%
C 43%
D 57%

Answer 1

From the table, we can form the line of best fit with the table below:


`\begin{center} \begin{tabular} {|p{1.8cm}|c|c|c|c|} &X&Y&X^2&XY\\[2ex] Tyrone&6&9&36&54\\ Marcel&6&11&36&66\\ Patrick&7&15&49&105\\ Bobby&8&11&64&88\\ Dylan&9&15&81&135\\ Mike&10&16&100&160\\ Jonathan&12&17&144&204\\[2ex] &\sum X=58&\sum Y=94&\sum X^2=510&\sum XY=812 \end{tabular} \end{center}`

The line of best fit is given by
`y=a+bx`, where:
`a= ((\sum Y)(\sum X^2)-(\sum X)(\sum XY))/(n(\sum X^2)-(\sum X)^2)` and
`b= (n(\sum XY)-(\sum X)(\sum Y))/(n(\sum X^2)-(\sum X)^2)`.

Thus, we have:


`a= ((94)(510)-(58)(812))/(7(510)-(58)^2) \\ \\ = (47,940-47,096)/(3,570-3,364) = (844)/(206) \\ \\ =4.0971`

and


`b= (7(812)-(58)(94))/(7(510)-(58)^2) \\ \\ = (5,684-5,452)/(3,570-3,364) = (232)/(206) \\ \\ =1.1262`

Thus, the line of best fit is
`y=4.0971+1.1262x`

For Tyrone and Marcel with a shoe size of 6, their predicted age will be


`y=4.0971+1.1262(6) \\ \\ =4.0971+6.7572=10.8543\approx11\ years`

Thus Tyrone with an age of 9 is more than 1 year different from his predicted age of 11.

For Patrick with a shoe size of 7, his predicted age will be


`y=4.0971+1.1262(7) \\ \\ =4.0971+7.8834=11.9805\approx12\ years`

Thus Patrick with an age of 15 is more than 1 year different from his predicted age of 12.

For Bobby with a shoe size of 8, his predicted age will be


`y=4.0971+1.1262(8) \\ \\ =4.0971+9.0096=13.1067\approx13\ years`

Thus Bobby with an age of 11 is more than 1 year different from his predicted age of 13.


For Dylan with a shoe size of 9, his predicted age will be


`y=4.0971+1.1262(9) \\ \\ =4.0971+10.1358=14.2329\approx14\ years`

For Mike with a shoe size of 10, his predicted age will be


`y=4.0971+1.1262(10) \\ \\ =4.0971+11.262=15.3591\approx15\ years`

For Jonathan with a shoe size of 12, his predicted age will be


`y=4.0971+1.1262(12) \\ \\ =4.0971+13.5144=17.6115\approx18\ years`

Therefore, Tyrone, Patrick and Dylan which is
`(3)/(7) *100\%=43\%` has their age 1 more than their predicted ages.