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All Common Core: High School - Statistics and Probability Resources

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Flashcards: Fit a Linear Function for a Scatter Plot: CCSS.Math.Content.HSS-ID.B.6c

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

$\begin{tabular}{|r|r|} \hline x: Vehicle weight (in pounds) & v: Milage (in MPG) \\ \hline 9161 & 35 \\ \hline 9215 & 38 \\ \hline 5733 & 47 \\ \hline 1207 & 39 \\ \hline 6653 & 7 \\ \hline 10291 & 35 \\ \hline 6751 & 13 \\ \hline 2676 & 18 \\ \hline 11101 & 23 \\ \hline 4562 & 11 \\ \hline 10598 & 35 \\ \hline 2797 & 37 \\ \hline 1259 & 10 \\ \hline 2142 & 47 \\ \hline 3841 & 32 \\ \hline 10298 & 11 \\ \hline 4193 & 20 \\ \hline 7256 & 39 \\ \hline 3337 & 28 \\ \hline 1558 & 32 \\ \hline 2141 & 29 \\ \hline 3321 & 36 \\ \hline 4117 & 32 \\ \hline 717 & 35 \\ \hline 4334 & 21 \\ \hline 7667 & 47 \\ \hline 6712 & 43 \\ \hline 10408 & 46 \\ \hline 4333 & 46 \\ \hline 4034 & 40 \\ \hline 9115 & 29 \\ \hline 9654 & 9 \\ \hline 10500 & 48 \\ \hline 6640 & 9 \\ \hline 3903 & 28 \\ \hline 6037 & 32 \\ \hline 7400 & 48 \\ \hline 7200 & 35 \\ \hline 4405 & 6 \\ \hline 1837 & 26 \\ \hline \end{tabular}$

y = -0.0027+ 41.469

y = -0.0030x + 28.515

y = -0.0023x + 14.496

y = 0.0032x - 41.496

y=0.00027x+28.515

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Flashcards: Fit a Linear Function for a Scatter Plot: CCSS.Math.Content.HSS-ID.B.6c

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

Your answer:

y = -0.0027+ 41.469

Your answer:

y = -0.0030x + 28.515

Your answer:

y = -0.0023x + 14.496

Your answer:

y = 0.0032x - 41.496

Correct answer:

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

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

$m=\textup{slope}$

$b=\textup{y-intercept}$

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:

Screen shot 2015 12 18 at 4.35.26 pm

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:

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.

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

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:

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.

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

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:

b=41.496

Now, we can write the least squares regression equation:

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:

$\\ \textup{y-intercept}= 28.5152425167 \\ \textup{slope}= 0.00026795821694$

Let's input this information into a least squares regression equation in the slope-intercept format:

y=mx+b

Substitute in our calculated values and solve.

y=0.00027x+28.515

Flashcards: Fit a Linear Function for a Scatter Plot: CCSS.Math.Content.HSS-ID.B.6c

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

$\begin{tabular}{|r|r|} \hline x: Vehicle weight (in pounds) & v: Milage (in MPG) \\ \hline 599 & 42 \\ \hline 8266 & 16 \\ \hline 10832 & 45 \\ \hline 5342 & 7 \\ \hline 3357 & 10 \\ \hline 10085 & 40 \\ \hline 7475 & 47 \\ \hline 11738 & 40 \\ \hline 10292 & 21 \\ \hline 2178 & 44 \\ \hline 3850 & 20 \\ \hline 10054 & 5 \\ \hline 5718 & 15 \\ \hline 1967 & 17 \\ \hline 7719 & 10 \\ \hline 5589 & 23 \\ \hline 7097 & 30 \\ \hline 6023 & 41 \\ \hline 5707 & 46 \\ \hline 2553 & 16 \\ \hline 7120 & 7 \\ \hline 5984 & 44 \\ \hline 8209 & 23 \\ \hline 1492 & 15 \\ \hline 8192 & 9 \\ \hline 2707 & 18 \\ \hline 11899 & 22 \\ \hline 1074 & 17 \\ \hline 1811 & 48 \\ \hline 5151 & 34 \\ \hline 3267 & 36 \\ \hline 5194 & 34 \\ \hline 8420 & 29 \\ \hline 1099 & 29 \\ \hline 7674 & 11 \\ \hline 8418 & 30 \\ \hline 2448 & 17 \\ \hline 740 & 7 \\ \hline 7098 & 48 \\ \hline 10721 & 32 \\ \hline \end{tabular}$

y= -1.9996520877 x +12.039816153

y= -0.00069582459321 x +72.2388969182

y= 0.000347912296605 x +24.0796323061

y= -0.000347912296605 x +12.039816153

y= 0.000173956148302 x +12.039816153

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Flashcards: Fit a Linear Function for a Scatter Plot: CCSS.Math.Content.HSS-ID.B.6c

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

Your answer:

y= -1.9996520877 x +12.039816153

Your answer:

y= -0.00069582459321 x +72.2388969182

Your answer:

y= -0.000347912296605 x +12.039816153

Your answer:

y= 0.000173956148302 x +12.039816153

Correct answer:

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:

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

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

Table example

Table example 2

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

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:

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

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:

Now, we can write the least squares regression equation:

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

Check with technology

Let's input this information into a least squares regression equation in the slope intercept format:

Substitute in our calculated values and solve.

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All Common Core: High School - Statistics and Probability Resources

Flashcards: Fit a Linear Function for a Scatter Plot: CCSS.Math.Content.HSS-ID.B.6c

Flashcards: Fit a Linear Function for a Scatter Plot: CCSS.Math.Content.HSS-ID.B.6c

Flashcards: Fit a Linear Function for a Scatter Plot: CCSS.Math.Content.HSS-ID.B.6c

Flashcards: Fit a Linear Function for a Scatter Plot: CCSS.Math.Content.HSS-ID.B.6c

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