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Common Core: High School - Statistics and Probability : Fit a Linear Function for a Scatter Plot: CCSS.Math.Content.HSS-ID.B.6c

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Example Questions

Example Question #151 : High School: Statistics & Probability

Use technology to find the least squares regression equation for the following data set.

$\begin{tabular}{|r|r|} \hline x: Vehicle weight (in pounds) & v: Milage (in MPG) \\ \hline 11144 & 20 \\ \hline 1855 & 10 \\ \hline 3944 & 42 \\ \hline 4130 & 25 \\ \hline 6980 & 46 \\ \hline 1253 & 8 \\ \hline 882 & 22 \\ \hline 7179 & 5 \\ \hline 6441 & 36 \\ \hline 7517 & 46 \\ \hline 2851 & 46 \\ \hline 7141 & 19 \\ \hline 10278 & 42 \\ \hline 11216 & 21 \\ \hline 2401 & 37 \\ \hline 8199 & 47 \\ \hline 3486 & 34 \\ \hline 1365 & 6 \\ \hline 9592 & 38 \\ \hline 7283 & 35 \\ \hline 5121 & 11 \\ \hline 10096 & 13 \\ \hline 4884 & 21 \\ \hline 1020 & 16 \\ \hline 8909 & 47 \\ \hline 11949 & 40 \\ \hline 6921 & 11 \\ \hline 1720 & 16 \\ \hline 9961 & 9 \\ \hline 3943 & 34 \\ \hline 2200 & 25 \\ \hline 11200 & 11 \\ \hline 2355 & 22 \\ \hline 8015 & 40 \\ \hline 11842 & 48 \\ \hline 1381 & 16 \\ \hline 9604 & 45 \\ \hline 2075 & 17 \\ \hline 4849 & 34 \\ \hline 9335 & 12 \\ \hline \end{tabular}$ $\displaystyle \begin{tabular}{|r|r|} \hline x: Vehicle weight (in pounds) & v: Milage (in MPG) \\ \hline 11144 & 20 \\ \hline 1855 & 10 \\ \hline 3944 & 42 \\ \hline 4130 & 25 \\ \hline 6980 & 46 \\ \hline 1253 & 8 \\ \hline 882 & 22 \\ \hline 7179 & 5 \\ \hline 6441 & 36 \\ \hline 7517 & 46 \\ \hline 2851 & 46 \\ \hline 7141 & 19 \\ \hline 10278 & 42 \\ \hline 11216 & 21 \\ \hline 2401 & 37 \\ \hline 8199 & 47 \\ \hline 3486 & 34 \\ \hline 1365 & 6 \\ \hline 9592 & 38 \\ \hline 7283 & 35 \\ \hline 5121 & 11 \\ \hline 10096 & 13 \\ \hline 4884 & 21 \\ \hline 1020 & 16 \\ \hline 8909 & 47 \\ \hline 11949 & 40 \\ \hline 6921 & 11 \\ \hline 1720 & 16 \\ \hline 9961 & 9 \\ \hline 3943 & 34 \\ \hline 2200 & 25 \\ \hline 11200 & 11 \\ \hline 2355 & 22 \\ \hline 8015 & 40 \\ \hline 11842 & 48 \\ \hline 1381 & 16 \\ \hline 9604 & 45 \\ \hline 2075 & 17 \\ \hline 4849 & 34 \\ \hline 9335 & 12 \\ \hline \end{tabular}$

Possible Answers:

$\displaystyle y= -0.00107126275369 x +10.1650071345$

$\displaystyle y= 0.000535631376846 x +10.1650071345$

$\displaystyle y= 0.00107126275369 x + 20.3300142691$

$\displaystyle y= -1.99892873725 x +10.1650071345$

$\displaystyle y= -0.00214252550738 x+ 60.9900428072$

Correct answer:

$\displaystyle y= 0.00107126275369 x + 20.3300142691$

Explanation:

In order to solve this problem, we need to define what is meant by the phrase 'least squares regression equation.' Essentially, the least squares regression equation is the equation of the best-fit line of the data if it were graphed on a scatter plot. Trend lines for data that possess a linear relationship are typically written in the slope-intercept form. This form is commonly written using the equation:

y=mx+b $\displaystyle y=mx+b$

In this equation, the variables are represented in the following manner:

$\\ \\ m=\textup{slope} \\ b=\textup{y-intercept}$ $\displaystyle \\ \\ m=\textup{slope} \\ b=\textup{y-intercept}$

Problems associated with this standard require us to use technology such as statistical programs or calculators to find the least squares regression equation. Essentially, we must use technology to calculate the slope and y-intercept of a group of data points. Let's first use an example to illustrate this process. Consider the following data set:

Table example

In order to solve this example and calculate the least squares regression equation, we need to insert the data into a spreadsheet program either using copy/paste techniques or manual entry. The data should look like the following figure:

Table example 2

After the data has been entered into the program, we can use formulas to calculate the slope and y-intercept needed for the least squares regression equation. The slope is calculated using the formula outlined in the red box on the following figure. In this equation, the range of cells in column B represents the y-values while cells from column A denote the x-values.

$=\textup(INDEX(LINEST(B2:B41,A2:A41),1))$ $\displaystyle =\textup(INDEX(LINEST(B2:B41,A2:A41),1))$

The number one at the end of the equation designates the specific statistic that we wish to calculate. In this case, one means the slope.

Slope with technology

We can see that the slope of the least squares regression equation is the following:

$\displaystyle m=-0.0032$

Now, let's calculate the y-intercept. It is calculated by using the formula outlined in the red box of the following figure. Again, column B represents the y-values while cells from column A denote the x-values.

$=\textup(INDEX(LINEST(B2:B41,A2:A41),2))$ $\displaystyle =\textup(INDEX(LINEST(B2:B41,A2:A41),2))$

The number two at the end of the equation designates the specific statistic that we wish to calculate. In this case, two means the y-intercept.

Y intercept with technology

We can see that the y-intercept of the least squares regression equation is the following:

$\displaystyle b=41.496$

Now, we can write the least squares regression equation:

$\displaystyle y=-0.0032x+41.496$

We can check this answer by graphing the points on a scatter plot and fitting it with a trendline.

Check with technology

We can see that the answers are the same. Now, we have learned how to find the least squares regression equation using technology. Let's use this information to solve the given problem. When we insert the data into a statistical program we can calculate the following information:

y-intercept= $\displaystyle 20.3300142691$
slope= $\displaystyle 0.00107126275369$
Let's input this information into a least squares regression equation in the slope intercept format:

y=mx+b $\displaystyle y=mx+b$

Substitute in our calculated values and solve.

$\displaystyle y= 0.00107126275369 x + 20.3300142691$

Report an Error

Example Question #152 : High School: Statistics & Probability

Use technology to find the least squares regression equation for the following data set.

$\begin{tabular}{|r|r|} \hline x: Vehicle weight (in pounds) & v: Milage (in MPG) \\ \hline 5415 & 11 \\ \hline 3576 & 5 \\ \hline 3431 & 27 \\ \hline 2393 & 36 \\ \hline 9839 & 41 \\ \hline 10786 & 34 \\ \hline 1839 & 14 \\ \hline 4888 & 18 \\ \hline 3055 & 21 \\ \hline 9752 & 49 \\ \hline 10301 & 33 \\ \hline 10341 & 7 \\ \hline 1593 & 38 \\ \hline 1668 & 5 \\ \hline 8072 & 28 \\ \hline 947 & 31 \\ \hline 2182 & 11 \\ \hline 4504 & 38 \\ \hline 2213 & 9 \\ \hline 645 & 20 \\ \hline 6896 & 13 \\ \hline 4978 & 46 \\ \hline 7352 & 26 \\ \hline 3807 & 17 \\ \hline 5898 & 30 \\ \hline 1329 & 41 \\ \hline 5190 & 38 \\ \hline 10103 & 23 \\ \hline 2246 & 36 \\ \hline 11124 & 10 \\ \hline 1062 & 35 \\ \hline 5795 & 8 \\ \hline 4652 & 6 \\ \hline 7243 & 44 \\ \hline 8222 & 44 \\ \hline 3687 & 28 \\ \hline 7119 & 19 \\ \hline 6051 & 41 \\ \hline 4858 & 14 \\ \hline 9312 & 33 \\ \hline \end{tabular}$ $\displaystyle \begin{tabular}{|r|r|} \hline x: Vehicle weight (in pounds) & v: Milage (in MPG) \\ \hline 5415 & 11 \\ \hline 3576 & 5 \\ \hline 3431 & 27 \\ \hline 2393 & 36 \\ \hline 9839 & 41 \\ \hline 10786 & 34 \\ \hline 1839 & 14 \\ \hline 4888 & 18 \\ \hline 3055 & 21 \\ \hline 9752 & 49 \\ \hline 10301 & 33 \\ \hline 10341 & 7 \\ \hline 1593 & 38 \\ \hline 1668 & 5 \\ \hline 8072 & 28 \\ \hline 947 & 31 \\ \hline 2182 & 11 \\ \hline 4504 & 38 \\ \hline 2213 & 9 \\ \hline 645 & 20 \\ \hline 6896 & 13 \\ \hline 4978 & 46 \\ \hline 7352 & 26 \\ \hline 3807 & 17 \\ \hline 5898 & 30 \\ \hline 1329 & 41 \\ \hline 5190 & 38 \\ \hline 10103 & 23 \\ \hline 2246 & 36 \\ \hline 11124 & 10 \\ \hline 1062 & 35 \\ \hline 5795 & 8 \\ \hline 4652 & 6 \\ \hline 7243 & 44 \\ \hline 8222 & 44 \\ \hline 3687 & 28 \\ \hline 7119 & 19 \\ \hline 6051 & 41 \\ \hline 4858 & 14 \\ \hline 9312 & 33 \\ \hline \end{tabular}$

Possible Answers:

$\displaystyle y= -0.000556560509801 x +11.358668286$

$\displaystyle y= -1.99944343949 x +11.358668286$

$\displaystyle y= -0.0011131210196 x +68.1520097158$

$\displaystyle y= 0.000556560509801 x + 22.7173365719$

$\displaystyle y= 0.000278280254901 x +11.358668286$

Correct answer:

$\displaystyle y= 0.000556560509801 x + 22.7173365719$

Explanation:

y=mx+b $\displaystyle y=mx+b$

In this equation, the variables are represented in the following manner:

$\\ \\ m=\textup{slope} \\ b=\textup{y-intercept}$ $\displaystyle \\ \\ m=\textup{slope} \\ b=\textup{y-intercept}$

Table example

Table example 2

$=\textup(INDEX(LINEST(B2:B41,A2:A41),1))$ $\displaystyle =\textup(INDEX(LINEST(B2:B41,A2:A41),1))$

The number one at the end of the equation designates the specific statistic that we wish to calculate. In this case, one means the slope.

Slope with technology

We can see that the slope of the least squares regression equation is the following:

$\displaystyle m=-0.0032$

$=\textup(INDEX(LINEST(B2:B41,A2:A41),2))$ $\displaystyle =\textup(INDEX(LINEST(B2:B41,A2:A41),2))$

The number two at the end of the equation designates the specific statistic that we wish to calculate. In this case, two means the y-intercept.

Y intercept with technology

We can see that the y-intercept of the least squares regression equation is the following:

$\displaystyle b=41.496$

Now, we can write the least squares regression equation:

$\displaystyle y=-0.0032x+41.496$

We can check this answer by graphing the points on a scatter plot and fitting it with a trendline.

Check with technology

y-intercept= $\displaystyle 22.7173365719$
slope= $\displaystyle 0.000556560509801$
Let's input this information into a least squares regression equation in the slope intercept format:

y=mx+b $\displaystyle y=mx+b$

Substitute in our calculated values and solve.

$\displaystyle y= 0.000556560509801 x + 22.7173365719$

Report an Error

Example Question #153 : High School: Statistics & Probability

Use technology to find the least squares regression equation for the following data set.

$\begin{tabular}{|r|r|} \hline x: Vehicle weight (in pounds) & v: Milage (in MPG) \\ \hline 2651 & 43 \\ \hline 8351 & 46 \\ \hline 11009 & 29 \\ \hline 4599 & 28 \\ \hline 4142 & 33 \\ \hline 10683 & 28 \\ \hline 6367 & 14 \\ \hline 6227 & 49 \\ \hline 2626 & 35 \\ \hline 8047 & 38 \\ \hline 4850 & 34 \\ \hline 4153 & 41 \\ \hline 10773 & 33 \\ \hline 1335 & 41 \\ \hline 599 & 43 \\ \hline 11213 & 26 \\ \hline 4710 & 6 \\ \hline 9313 & 36 \\ \hline 2369 & 47 \\ \hline 6975 & 47 \\ \hline 1966 & 32 \\ \hline 5773 & 47 \\ \hline 4882 & 49 \\ \hline 4563 & 15 \\ \hline 1318 & 13 \\ \hline 6092 & 49 \\ \hline 11295 & 29 \\ \hline 881 & 12 \\ \hline 7323 & 46 \\ \hline 5380 & 33 \\ \hline 7290 & 10 \\ \hline 7518 & 27 \\ \hline 1902 & 21 \\ \hline 11147 & 29 \\ \hline 5096 & 41 \\ \hline 11686 & 39 \\ \hline 3539 & 36 \\ \hline 3415 & 6 \\ \hline 7761 & 44 \\ \hline 11231 & 45 \\ \hline \end{tabular}$ $\displaystyle \begin{tabular}{|r|r|} \hline x: Vehicle weight (in pounds) & v: Milage (in MPG) \\ \hline 2651 & 43 \\ \hline 8351 & 46 \\ \hline 11009 & 29 \\ \hline 4599 & 28 \\ \hline 4142 & 33 \\ \hline 10683 & 28 \\ \hline 6367 & 14 \\ \hline 6227 & 49 \\ \hline 2626 & 35 \\ \hline 8047 & 38 \\ \hline 4850 & 34 \\ \hline 4153 & 41 \\ \hline 10773 & 33 \\ \hline 1335 & 41 \\ \hline 599 & 43 \\ \hline 11213 & 26 \\ \hline 4710 & 6 \\ \hline 9313 & 36 \\ \hline 2369 & 47 \\ \hline 6975 & 47 \\ \hline 1966 & 32 \\ \hline 5773 & 47 \\ \hline 4882 & 49 \\ \hline 4563 & 15 \\ \hline 1318 & 13 \\ \hline 6092 & 49 \\ \hline 11295 & 29 \\ \hline 881 & 12 \\ \hline 7323 & 46 \\ \hline 5380 & 33 \\ \hline 7290 & 10 \\ \hline 7518 & 27 \\ \hline 1902 & 21 \\ \hline 11147 & 29 \\ \hline 5096 & 41 \\ \hline 11686 & 39 \\ \hline 3539 & 36 \\ \hline 3415 & 6 \\ \hline 7761 & 44 \\ \hline 11231 & 45 \\ \hline \end{tabular}$

Possible Answers:

$\displaystyle y= -0.000881028673391 x 91.0360514355$

$\displaystyle y= 0.000440514336696 x + 30.3453504785$

$\displaystyle y= -0.000440514336696 x +15.1726752392$

$\displaystyle y= 0.000220257168348 x +15.1726752392$

$\displaystyle y= -1.99955948566 x +15.1726752392$

Correct answer:

$\displaystyle y= 0.000440514336696 x + 30.3453504785$

Explanation:

y=mx+b $\displaystyle y=mx+b$

In this equation, the variables are represented in the following manner:

$\\ \\ m=\textup{slope} \\ b=\textup{y-intercept}$ $\displaystyle \\ \\ m=\textup{slope} \\ b=\textup{y-intercept}$

Table example

Table example 2

$=\textup(INDEX(LINEST(B2:B41,A2:A41),1))$ $\displaystyle =\textup(INDEX(LINEST(B2:B41,A2:A41),1))$

The number one at the end of the equation designates the specific statistic that we wish to calculate. In this case, one means the slope.

Slope with technology

We can see that the slope of the least squares regression equation is the following:

$\displaystyle m=-0.0032$

$=\textup(INDEX(LINEST(B2:B41,A2:A41),2))$ $\displaystyle =\textup(INDEX(LINEST(B2:B41,A2:A41),2))$

The number two at the end of the equation designates the specific statistic that we wish to calculate. In this case, two means the y-intercept.

Y intercept with technology

We can see that the y-intercept of the least squares regression equation is the following:

$\displaystyle b=41.496$

Now, we can write the least squares regression equation:

$\displaystyle y=-0.0032x+41.496$

We can check this answer by graphing the points on a scatter plot and fitting it with a trendline.

Check with technology

y-intercept= $\displaystyle 30.3453504785$
slope= $\displaystyle 0.000440514336696$
Let's input this information into a least squares regression equation in the slope intercept format:

y=mx+b $\displaystyle y=mx+b$

Substitute in our calculated values and solve.

$\displaystyle y= 0.000440514336696 x + 30.3453504785$

Report an Error

Example Question #151 : Interpreting Categorical & Quantitative Data

Use technology to find the least squares regression equation for the following data set.

Screen shot 2015 12 18 at 4.35.26 pm

Possible Answers:

$\displaystyle y = -0.0023x + 14.496$

$\displaystyle y = -0.0030x + 41.946$

$\displaystyle y = 0.0032x - 41.496$

$\displaystyle y = -0.0302x + 41.469$

$\displaystyle y = -0.0032x + 41.496$

Correct answer:

$\displaystyle y = -0.0032x + 41.496$

Explanation:

In order to solve this problem, we need to define what is meant by the phrase "least squares regression equation." Essentially, the least squares regression equation is the equation of the best-fit line of the data if it were graphed on a scatter plot. Trend lines for data that possess a linear relationship are typically written in the slope intercept form. This form is commonly written using the equation:

y=mx+b $\displaystyle y=mx+b$

In this equation, the variables are represented in the following manner:

$m=\textup{slope}$ $\displaystyle m=\textup{slope}$

$b=\textup{y-intercept}$ $\displaystyle b=\textup{y-intercept}$

In order to solve this question and calculate the least squares regression equation, we need to insert the data into a spreadsheet program either using copy/paste techniques or manual entry. The data should look like the following figure.

Screen shot 2016 01 12 at 3.46.09 pm

After the data has been entered into the program, we can use formulas to calculate the slope and y-intercept needed for the least squares regression equation. The slope is calculated using the formula outlined in the red box on the following figure. In this equation the range of cells in column B represents the y-values while cells from column A denote the x-values.

$\textup{=INDEX(LINEST(B2:B41,A2:A41),1)}$ $\displaystyle \textup{=INDEX(LINEST(B2:B41,A2:A41),1)}$

The number one at the end of the equation designates the specific statistic that we wish to calculate. In this case, one means the slope.

Slope with technology

We can see that the slope of the least squares regression equation is the following:

$\displaystyle m=-0.0032$

$\textup{=INDEX(LINEST(B2:B41,A2:A41),2)}$ $\displaystyle \textup{=INDEX(LINEST(B2:B41,A2:A41),2)}$

The number two at the end of the equation designates the specific statistic that we wish to calculate. In this case, one means the y-intercept.

Y intercept with technology

We can see that the y-intercept of the least squares regression equation is the following:

$\displaystyle b=41.496$

Now, we can write the least squares regression equation:

$\displaystyle y = -0.0032x + 41.496$

We can check this answer by graphing the points on a scatter plot and fitting it with a trendline.

Check with technology

We can see that the answers are the same. Now, we have calculated how to find the least squares regression equation using technology.

Report an Error

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Common Core: High School - Statistics and Probability : Fit a Linear Function for a Scatter Plot: CCSS.Math.Content.HSS-ID.B.6c

Study concepts, example questions & explanations for Common Core: High School - Statistics and Probability

All Common Core: High School - Statistics and Probability Resources

Example Questions

Example Question #151 : High School: Statistics & Probability

Example Question #152 : High School: Statistics & Probability

Example Question #153 : High School: Statistics & Probability

Example Question #151 : Interpreting Categorical & Quantitative Data

All Common Core: High School - Statistics and Probability Resources

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