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Least Squares RegressionLeast Squares Regression
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Question 1 of 13
1. Question
Solve for the equation of the least squares line of best fit.`barx=12``bary=19``sigma_x=1.8``sigma_y=4``r=0.9`- `y=` (2)`x` (-5)
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The Least Squares Line of Best Fit is a more accurate interpretation of a correlation.Gradient Formula
$$\color{#007DDC}{m}=r\cdot\frac{\color{#9a00c7}{\sigma_y}}{\color{#00880A}{\sigma_x}}$$`y` intercept Formula
$$\color{#e65021}{b}=\bar{y}-\color{#007DDC}{m}\bar{x}$$Gradient-Intercept Form
`y=``m``x+``b`First, use the gradient formula to find the gradient or slope of the line.`m` `=` $$r\cdot\frac{\color{#9a00c7}{\sigma_y}}{\color{#00880A}{\sigma_x}}$$ Gradient Formula `=` $$0.9\cdot\frac{\color{#9a00c7}{4}}{\color{#00880A}{1.8}}$$ Substitute values `m` `=` `2` Next, find the `y` intercept.`b` `=` $$\bar{y}-\color{#007DDC}{m}\bar{x}$$ `y` intercept Formula `=` $$19-(\color{#007DDC}{2})12$$ Substitute values `=` `19-24` Evaluate `b` `=` `-5` Finally, substitute the values into the gradient-intercept form`m=2``b=-5``y` `=` `m``x+``b` Gradient-Intercept Form `y` `=` `2``x-``5` Substitute values `y=2x-5` -
Question 2 of 13
2. Question
The scores of `5` students for their Maths and Science tests were listed in a table. Find the equation of the least squares line of best fit.`\text(Maths)` `x` 81 65 70 52 87 `\text(Science)` `y` 85 72 63 68 91 Round your answer to two decimal places- `S=` (0.69)`M` (-26.81)
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The Least Squares Line of Best Fit is a more accurate interpretation of a correlation.Gradient Formula
$$\color{#007DDC}{m}=r\cdot\frac{\color{#9a00c7}{\sigma_y}}{\color{#00880A}{\sigma_x}}$$`y` intercept Formula
$$\color{#e65021}{b}=\bar{y}-\color{#007DDC}{m}\bar{x}$$Gradient-Intercept Form
`y=``m``x+``b`First, use a calculator and find the following values.Note that the steps may vary depending on the model of your calculator.`(1)` Press on Mode and select Statistics.`(2)` Choose the options for `A+bx` among the mode options.`(3)` Input the `x` values followed by the `=` button on the left column.`(4)` Repeat the same steps for the `y` values on the right column.`(5)` Press AC to clear the values.`(6)` Press Shift`+1+4` and select the option for `barx` to get the mean of `x`.`(7)` Press Shift`+1+4` and select the option for `sigma_x` to get the standard deviation of `x`.`(8)` Press Shift`+1+4` and select the option for `bary` to get the mean of `y`.`(9)` Press Shift`+1+4` and select the option for `sigma_y` to get the standard deviation of `y`.`(10)` Press Shift`+1+5` to get the correlation coefficient `(r)`.Following the steps above, the given values would be the following:`barx` `=` `71 \text((Maths))` `bary` `=` `75.8 \text((Science))` `sigma_x` `=` `12.28` `sigma_y` `=` `10.53` `r` `=` `0.8` Next, use the gradient formula to find the gradient or slope of the line.`m` `=` $$r\cdot\frac{\color{#9a00c7}{\sigma_y}}{\color{#00880A}{\sigma_x}}$$ Gradient Formula `=` $$0.8\cdot\frac{\color{#9a00c7}{10.53}}{\color{#00880A}{12.28}}$$ Substitute values `m` `=` `0.69` Next, find the `y` intercept.`b` `=` $$\bar{y}-\color{#007DDC}{m}\bar{x}$$ `y` intercept Formula `=` $$75.68-(\color{#007DDC}{0.69})71$$ Substitute values `b` `=` `26.81` Finally, substitute the values into the gradient-intercept form`m=0.69``b=26.81``y` `=` `m``x+``b` Gradient-Intercept Form `y` `=` `0.69``x+``26.81` Substitute values `S` `=` `0.69M+26.81` Change `y` and `x` to `S` and `M` respectively `S=0.69M+26.81` -
Question 3 of 13
3. Question
The scores of `5` students for their Maths and Science tests were listed in a table. Draw the least square line of best fit given that its equation is `S=0.69M+26.81`.
Hint
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The Least Squares Line of Best Fit is a more accurate interpretation of a correlation.Gradient Formula
$$\color{#007DDC}{m}=r\cdot\frac{\color{#9a00c7}{\sigma_y}}{\color{#00880A}{\sigma_x}}$$`y` intercept Formula
$$\color{#e65021}{b}=\bar{y}-\color{#007DDC}{m}\bar{x}$$Gradient-Intercept Form
`y=``m``x+``b`First, create a table of values illustrating the values of Maths `(x)` and Science `(y)`We can use `0`, `50` and `100` as test values of `x`.`x` `0` `50` `100` `y` Next, recall that `S=y` and `M=x`Substitute the values of `x` to `M` to solve for the `y` values.`x=0``S` `=` `0.69M+26.81` `S` `=` `0.69(0)+26.81` Substitute `x=0` `S` `=` `26.81` `x=50``S` `=` `0.69M+26.81` `S` `=` `0.69(50)+26.81` Substitute `x=50` `S` `=` `61.3` `x=100``S` `=` `0.69M+26.81` `S` `=` `0.69(100)+26.81` Substitute `x=100` `S` `=` `95.8` `x` `0` `50` `100` `y` `26.81` `61.3` `95.8` Finally, plot these three points and draw a line connecting them.`S=0.69M+26.81`
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Question 4 of 13
4. Question
The weekly exercise routine and pulse rate of `10` students were recorded on this table of values. Given that `r=-0.99`, find the equation of the least squares line of best fit.`\text(Exercise)` 0.5 1 2 3 4 5 6 7 8 9 `\text(BPM)` 80 75 70 67 62 61 55 52 48 46 Round your answer to two decimal places- `B=` (-3.84)`E+` (79.07)
Hint
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Nice Job!
Incorrect
The Least Squares Line of Best Fit is a more accurate interpretation of a correlation.Gradient Formula
$$\color{#007DDC}{m}=r\cdot\frac{\color{#9a00c7}{\sigma_y}}{\color{#00880A}{\sigma_x}}$$`y` intercept Formula
$$\color{#e65021}{b}=\bar{y}-\color{#007DDC}{m}\bar{x}$$Gradient-Intercept Form
`y=``m``x+``b`First, use a calculator and find the following values.Note that the steps may vary depending on the model of your calculator.`(1)` Press on Mode and select Statistics.`(2)` Choose the options for `A+bx` among the mode options.`(3)` Input the `x` values followed by the `=` button on the left column.`(4)` Repeat the same steps for the `y` values on the right column.`(5)` Press AC to clear the values.`(6)` Press Shift`+1+4` and select the option for `barx` to get the mean of `x`.`(7)` Press Shift`+1+4` and select the option for `sigma_x` to get the standard deviation of `x`.`(8)` Press Shift`+1+4` and select the option for `bary` to get the mean of `y`.`(9)` Press Shift`+1+4` and select the option for `sigma_y` to get the standard deviation of `y`.Following the steps above, the given values would be the following:`barx` `=` `4.55 \text((Exercise))` `bary` `=` `61.6 \text((BPM))` `sigma_x` `=` `2.80` `sigma_y` `=` `10.87` Next, use the gradient formula to find the gradient or slope of the line.`r` `=` `-0.99` `m` `=` $$r\cdot\frac{\color{#9a00c7}{\sigma_y}}{\color{#00880A}{\sigma_x}}$$ Gradient Formula `=` $$-0.99\cdot\frac{\color{#9a00c7}{10.87}}{\color{#00880A}{2.80}}$$ Substitute values `m` `=` `-3.84` Next, find the `y` intercept.`b` `=` $$\bar{y}-\color{#007DDC}{m}\bar{x}$$ `y` intercept Formula `=` $$61.6-(\color{#007DDC}{-3.84})4.55$$ Substitute values `b` `=` `79.07` Finally, substitute the values into the gradient-intercept form`m=-3.84``b=79.07``y` `=` `m``x+``b` Gradient-Intercept Form `y` `=` `-3.84``x+``79.07` Substitute values `B` `=` `-3.84E+79.07` Change `y` and `x` to `B` and `E` respectively `B=-3.84E+79.07` -
Question 5 of 13
5. Question
The weekly exercise routine and pulse rate of `10` students were recorded to this table of values. Draw the least square line of best fit given that its equation is `B=-3.84E+79.07`.
Hint
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The Least Squares Line of Best Fit is a more accurate interpretation of a correlation.Gradient Formula
$$\color{#007DDC}{m}=r\cdot\frac{\color{#9a00c7}{\sigma_y}}{\color{#00880A}{\sigma_x}}$$`y` intercept Formula
$$\color{#e65021}{b}=\bar{y}-\color{#007DDC}{m}\bar{x}$$Gradient-Intercept Form
`y=``m``x+``b`First, create a table of values illustrating the values of Exercise `(x)` and BPM `(y)`We can use `0`, `5` and `10` as test values of `x`.`x` `0` `5` `10` `y` Next, recall that `B=y` and `E=x`Substitute the values of `x` to `E` to solve for the `y` values.`x=0``B` `=` `-3.84E+79.07` `B` `=` `-3.84(0)+79.07` Substitute `x=0` `B` `=` `79.07` `x=5``B` `=` `-3.84E+79.07` `B` `=` `-3.84(5)+79.07` Substitute `x=5` `B` `=` `59.87` `x=10``B` `=` `-3.84E+79.07` `B` `=` `-3.84(10)+79.07` Substitute `x=10` `B` `=` `40.67` `x` `0` `5` `10` `y` `79.07` `59.87` `40.67` Finally, plot these three points and draw a line connecting them.`B=-3.84E+79.07`
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Question 6 of 13
6. Question
The height and weight of `10` girls were listed in a table of values. Find the equation of the least squares line of best fit.`\text(Height)` 162 167 179 162 149 154 144 157 172 160 `\text(Weight)` 74 65 87 54 53 61 57 71 79 58 Round your answer to two decimal places- `W=` (0.86)`H` (-72.22)
Hint
Help VideoCorrect
Well Done!
Incorrect
The Least Squares Line of Best Fit is a more accurate interpretation of a correlation.Gradient Formula
$$\color{#007DDC}{m}=r\cdot\frac{\color{#9a00c7}{\sigma_y}}{\color{#00880A}{\sigma_x}}$$`y` intercept Formula
$$\color{#e65021}{b}=\bar{y}-\color{#007DDC}{m}\bar{x}$$Gradient-Intercept Form
`y=``m``x+``b`First, use a calculator and find the following values.Note that the steps may vary depending on the model of your calculator.`(1)` Press on Mode and select Statistics.`(2)` Choose the options for `A+bx` among the mode options.`(3)` Input the `x` values followed by the `=` button on the left column.`(4)` Repeat the same steps for the `y` values on the right column.`(5)` Press AC to clear the values.`(6)` Press Shift`+1+4` and select the option for `barx` to get the mean of `x`.`(7)` Press Shift`+1+4` and select the option for `sigma_x` to get the standard deviation of `x`.`(8)` Press Shift`+1+4` and select the option for `bary` to get the mean of `y`.`(9)` Press Shift`+1+4` and select the option for `sigma_y` to get the standard deviation of `y`.`(10)` Press Shift`+1+5` to get the correlation coefficient `(r)`.Following the steps above, the given values would be the following:`barx` `=` `160.6 \text((Height))` `bary` `=` `65.9 \text((Weight))` `sigma_x` `=` `9.90` `sigma_y` `=` `10.88` `r` `=` `0.78` Next, use the gradient formula to find the gradient or slope of the line.`m` `=` $$r\cdot\frac{\color{#9a00c7}{\sigma_y}}{\color{#00880A}{\sigma_x}}$$ Gradient Formula `=` $$0.78\cdot\frac{\color{#9a00c7}{10.88}}{\color{#00880A}{9.90}}$$ Substitute values `m` `=` `0.86` Next, find the `y` intercept.`b` `=` $$\bar{y}-\color{#007DDC}{m}\bar{x}$$ `y` intercept Formula `=` $$65.9-(\color{#007DDC}{0.86})160.6$$ Substitute values `b` `=` `-72.22` Finally, substitute the values into the gradient-intercept form`m=0.86``b=-72.22``y` `=` `m``x+``b` Gradient-Intercept Form `y` `=` `0.86``x``-72.22` Substitute values `W` `=` `0.86H-72.22` Change `y` and `x` to `W` and `H` respectively `W=0.86H-72.22` -
Question 7 of 13
7. Question
The height and weight of `10` girls were listed in a table of values. Given that the equation of the least square line of best fit is `W=0.86H-72.22`, find the weight of a girl with a height of `175 \text(cm)`.Round your answer to two decimal places- `W=` (77.73) `\text(kg)`
Hint
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Correct!
Incorrect
The Least Squares Line of Best Fit is a more accurate interpretation of a correlation.Gradient Formula
$$\color{#007DDC}{m}=r\cdot\frac{\color{#9a00c7}{\sigma_y}}{\color{#00880A}{\sigma_x}}$$`y` intercept Formula
$$\color{#e65021}{b}=\bar{y}-\color{#007DDC}{m}\bar{x}$$Gradient-Intercept Form
`y=``m``x+``b`Since we are given the value `H=175`, we can substitute it to the formula and solve for the weight `(W)`.`W` `=` `0.86H-72.22` `W` `=` `0.86(175)-72.22` Substitute `H=175` `W` `=` `77.73` `W=77.73 \text(kg)` -
Question 8 of 13
8. Question
The height and arm span of `8` female swimmers were listed in a table of values. Find the means, standard deviation and the correlation coefficient of the two set of values.`\text(Height)` 189 190 197 186 195 185 192 196 `\text(Arm Span)` 190 191 199 187 198 186 194 200 Round your answer to three decimal places-
`(i) barx=` (191.25)`(ii) bary=` (190.125)`(iii) sigma_x=` (4.235)`(iv) sigma_y=` (5.110)`(v) r=` (0.99)
Hint
Help VideoCorrect
Great Work!
Incorrect
The Least Squares Line of Best Fit is a more accurate interpretation of a correlation.Gradient Formula
$$\color{#007DDC}{m}=r\cdot\frac{\color{#9a00c7}{\sigma_y}}{\color{#00880A}{\sigma_x}}$$`y` intercept Formula
$$\color{#e65021}{b}=\bar{y}-\color{#007DDC}{m}\bar{x}$$Gradient-Intercept Form
`y=``m``x+``b`Let `x` represent `text(Height) = H`Let `y` represent `text(Arm Span) = S`First, use a calculator and find the following values.Note that the steps may vary depending on the model of your calculator.`(1)` Press on Mode and select Statistics.`(2)` Choose the options for `A+bx` among the mode options.`(3)` Input the `x` values followed by the `=` button on the left column.`(4)` Repeat the same steps for the `y` values on the right column.`(5)` Press AC to clear the values.`(6)` Press Shift`+1+4` and select the option for `barx` to get the mean of `x`.`(7)` Press Shift`+1+4` and select the option for `sigma_x` to get the standard deviation of `x`.`(8)` Press Shift`+1+4` and select the option for `bary` to get the mean of `y`.`(9)` Press Shift`+1+4` and select the option for `sigma_y` to get the standard deviation of `y`.`(10)` Press Shift`+1+5` to get the correlation coefficient `(r)`.Following the steps above, the given values would be the following:`barx` `=` `191.25 \text((Height))` `bary` `=` `190.125 \text((Arm Span))` `sigma_x` `=` `4.235` `sigma_y` `=` `5.110` `r` `=` `0.99` `barx=191.25``bary=190.125``sigma_x=4.235``sigma_y=5.110``r=0.99` -
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Question 9 of 13
9. Question
The height and arm span of `8` female swimmers were listed in a table of values. Find the equation of the least squares line of best fit.`\text(Height)` 189 190 197 186 195 185 192 196 `\text(Arm Span)` 190 191 199 187 198 186 194 200 Round your answer to three decimal places- `S=` (1.195)`H` (-35.419)
Hint
Help VideoCorrect
Great Work!
Incorrect
The Least Squares Line of Best Fit is a more accurate interpretation of a correlation.Gradient Formula
$$\color{#007DDC}{m}=r\cdot\frac{\color{#9a00c7}{\sigma_y}}{\color{#00880A}{\sigma_x}}$$`y` intercept Formula
$$\color{#e65021}{b}=\bar{y}-\color{#007DDC}{m}\bar{x}$$Gradient-Intercept Form
`y=``m``x+``b`From the previous question, the given values would be the following:`barx` `=` `191.25 \text((Height))` `bary` `=` `190.125 \text((Arm Span))` `sigma_x` `=` `4.235` `sigma_y` `=` `5.110` `r` `=` `0.99` Use the gradient formula to find the gradient or slope of the line.`m` `=` $$r\cdot\frac{\color{#9a00c7}{\sigma_y}}{\color{#00880A}{\sigma_x}}$$ Gradient Formula `=` $$0.99\cdot\frac{\color{#9a00c7}{5.11}}{\color{#00880A}{4.235}}$$ Substitute values `m` `=` `1.195` Next, find the `y` intercept.`b` `=` $$\bar{y}-\color{#007DDC}{m}\bar{x}$$ `y` intercept Formula `=` $$190.125-(\color{#007DDC}{1.195})191.25$$ Substitute values `b` `=` `-35.419` Finally, substitute the values into the gradient-intercept form`m=1.195``b=-35.419``y` `=` `m``x+``b` Gradient-Intercept Form `y` `=` `1.195``x``-35.419` Substitute values `S` `=` `1.195H-35.419` Change `y` and `x` to `S` and `H` respectively `S=1.195H-35.419` -
Question 10 of 13
10. Question
The height and arm span of `8` female swimmers were listed in a table of values. Given that the equation of the least square line of best fit is `S=1.195H-35.419`, find the arm span of a girl with a height of `201 \text(cm)`.Round your answer to three decimal places- `S=` (204.776) `\text(cm)`
Hint
Help VideoCorrect
Well Done!
Incorrect
The Least Squares Line of Best Fit is a more accurate interpretation of a correlation.Gradient Formula
$$\color{#007DDC}{m}=r\cdot\frac{\color{#9a00c7}{\sigma_y}}{\color{#00880A}{\sigma_x}}$$`y` intercept Formula
$$\color{#e65021}{b}=\bar{y}-\color{#007DDC}{m}\bar{x}$$Gradient-Intercept Form
`y=``m``x+``b`Since we are given the value `H=201`, we can substitute it to the formula and solve for the arm span `(S)`.`S` `=` `1.195H-35.419` `S` `=` `1.195(201)-35.419` Substitute `H=201` `S` `=` `204.776` `S=204.776 \text(cm)` -
Question 11 of 13
11. Question
The height and weight of `10` girls were listed in a table of values. Draw the least square line of best fit given that its equation is `W=0.86H-72.22`.
Hint
Help VideoCorrect
Nice Job!
Incorrect
The Least Squares Line of Best Fit is a more accurate interpretation of a correlation.Gradient Formula
$$\color{#007DDC}{m}=r\cdot\frac{\color{#9a00c7}{\sigma_y}}{\color{#00880A}{\sigma_x}}$$`y` intercept Formula
$$\color{#e65021}{b}=\bar{y}-\color{#007DDC}{m}\bar{x}$$Gradient-Intercept Form
`y=``m``x+``b`First, create a table of values illustrating the values of Height `(x)` and Weight `(y)`We can use `145`, `160` and `180` as test values of `x`.`x` `145` `160` `180` `y` Next, recall that `W=y` and `H=x`Substitute the values of `x` to `H` to solve for the `y` values.`x=145``W` `=` `0.86H-72.22` `W` `=` `0.86(145)-72.22` Substitute `x=145` `W` `=` `52.48` `x=160``W` `=` `0.86H-72.22` `W` `=` `0.86(160)-72.22` Substitute `x=160` `W` `=` `65.30` `x=180``W` `=` `0.86H-72.22` `W` `=` `0.86(180)-72.22` Substitute `x=180` `W` `=` `82.58` `x` `145` `160` `180` `y` `52.48` `65.30` `82.58` Finally, plot these three points and draw a line connecting them.`W=0.86H-72.22`
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Question 12 of 13
12. Question
Solve for the gradient given the following values.`barx=28.5``bary=53.4``b=-3.6`- `m=` (2)
Hint
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Excellent!
Incorrect
The Least Squares Line of Best Fit is a more accurate interpretation of a correlation.Gradient Formula
$$\color{#007DDC}{m}=r\cdot\frac{\color{#9a00c7}{\sigma_y}}{\color{#00880A}{\sigma_x}}$$`y` intercept Formula
$$\color{#e65021}{b}=\bar{y}-\color{#007DDC}{m}\bar{x}$$Gradient-Intercept Form
`y=``m``x+``b`Use the gradient-intercept form and substitute the known values to solve for `m`Use `bary` and `barx` as substitute for `y` and `x`, respectively.`b=-3.6``barx=28.5``bary=53.4``bary` `=` `m``barx+``b` Gradient-Intercept Form `53.4` `=` `(``m``)28.5-``3.6` Substitute values `53.4` `+3.6` `=` `28.5m-3.6` `+3.6` Add `3.6` to both sides `57``divide28.5` `=` `28.5m``divide28.5` Divide both sides by `28.5` `2` `=` `m` `m` `=` `2` `m=2` -
Question 13 of 13
13. Question
Solve for the `y` intercept given the following values.`barx=40.5``bary=51.2``m=2/5`- `b=` (35)
Hint
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Incorrect
The Least Squares Line of Best Fit is a more accurate interpretation of a correlation.Gradient Formula
$$\color{#007DDC}{m}=r\cdot\frac{\color{#9a00c7}{\sigma_y}}{\color{#00880A}{\sigma_x}}$$`y` intercept Formula
$$\color{#e65021}{b}=\bar{y}-\color{#007DDC}{m}\bar{x}$$Gradient-Intercept Form
`y=``m``x+``b`Substitute the known values to the `y` intercept formula.`m=2/5``barx=40.5``bary=51.2``b` `=` $$\bar{y}-\color{#007DDC}{m}\bar{x}$$ `y` intercept Formula `=` $$51.2-\left(\color{#007DDC}{\frac{2}{5}}\right)40.5$$ Substitute values `=` `51.2-16.2` Evaluate `b` `=` `35` `b=35`