Math / Data & Statistics

QuestionFind the equation of the regression line for the given data. Then construct a scatter plot of the data and draw the regression line. (The pair of variables have a significant correlation.) Then use the regression equation to predict the value of $y$ for each of the given $x$ -values, if meaningful. The table below shows the heights (in feet) and the number of stories of six notable buildings in a city. \begin{tabular}{|l|c|c|c|c|c|c|} \hline Height, $\mathbf{x}$ & 775 & 619 & 519 & 508 & 491 & 474 \\ \hline Stories, $\mathbf{y}$ & 53 & 47 & 44 & 42 & 37 & 36 \\ \hline \end{tabular} (a) $x=500$ feet (b) $x=649$ feet (c) $x=321$ feet (d) $x=732$ feet
Find the regression equation. $\hat{y}=\square x+(\square)$ (Round the slope to three decimal places as needed. Round the $y$ -intercept to two decimal places as needed.)

Studdy Solution

STEP 1

1. We assume a linear relationship between the height of the buildings ( $x$ ) and the number of stories ( $y$ ).
2. The regression line is of the form $\hat{y} = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.
3. We will use the least squares method to find the best-fitting line.

STEP 2

1. Calculate the slope ( $m$ ) of the regression line.
2. Calculate the y-intercept ( $b$ ) of the regression line.
3. Write the equation of the regression line.
4. Construct a scatter plot and draw the regression line.
5. Use the regression equation to predict $y$ for given $x$ -values.

STEP 3

Calculate the mean of $x$ and $y$ .
$\bar{x} = \frac{775 + 619 + 519 + 508 + 491 + 474}{6}$
$\bar{y} = \frac{53 + 47 + 44 + 42 + 37 + 36}{6}$

STEP 4

Calculate the slope $m$ using the formula:
$m = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}}$

STEP 5

Calculate the y-intercept $b$ using the formula:
$b = \bar{y} - m\bar{x}$

STEP 6

Write the equation of the regression line:
$\hat{y} = mx + b$

STEP 7

Construct a scatter plot of the data points $(x_i, y_i)$ .

STEP 8

Draw the regression line on the scatter plot using the equation $\hat{y} = mx + b$ .

STEP 9

Use the regression equation to predict $y$ for each given $x$ -value: (a) $x = 500$ (b) $x = 649$ (c) $x = 321$ (d) $x = 732$
Calculate $\hat{y}$ for each $x$ .
The regression equation is:
$\hat{y} = mx + b$
Predicted values: (a) $\hat{y}$ for $x = 500$ (b) $\hat{y}$ for $x = 649$ (c) $\hat{y}$ for $x = 321$ (d) $\hat{y}$ for $x = 732$

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Studdy Solution

STEP 1

1. We assume a linear relationship between the height of the buildings ( $x$ ) and the number of stories ( $y$ ).
2. The regression line is of the form $\hat{y} = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.
3. We will use the least squares method to find the best-fitting line.

STEP 2

1. Calculate the slope ( $m$ ) of the regression line.
2. Calculate the y-intercept ( $b$ ) of the regression line.
3. Write the equation of the regression line.
4. Construct a scatter plot and draw the regression line.
5. Use the regression equation to predict $y$ for given $x$ -values.

STEP 3

Calculate the mean of $x$ and $y$ .
$\bar{x} = \frac{775 + 619 + 519 + 508 + 491 + 474}{6}$
$\bar{y} = \frac{53 + 47 + 44 + 42 + 37 + 36}{6}$

STEP 4

Calculate the slope $m$ using the formula:
$m = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}}$

STEP 5

Calculate the y-intercept $b$ using the formula:
$b = \bar{y} - m\bar{x}$

STEP 6

Write the equation of the regression line:
$\hat{y} = mx + b$

STEP 7

Construct a scatter plot of the data points $(x_i, y_i)$ .

STEP 8

Draw the regression line on the scatter plot using the equation $\hat{y} = mx + b$ .

STEP 9

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Studdy Solution

STEP 1

STEP 2

1. Calculate the slope (mmm) of the regression line.2. Calculate the y-intercept (bbb) of the regression line.3. Write the equation of the regression line.4. Construct a scatter plot and draw the regression line.5. Use the regression equation to predict yyy for given xxx-values.

STEP 3

Calculate the mean of xxx and yyy.xˉ=775+619+519+508+491+4746\bar{x} = \frac{775 + 619 + 519 + 508 + 491 + 474}{6}xˉ=6775+619+519+508+491+474​yˉ=53+47+44+42+37+366\bar{y} = \frac{53 + 47 + 44 + 42 + 37 + 36}{6}yˉ​=653+47+44+42+37+36​

STEP 4

Calculate the slope mmm using the formula:m=∑(xi−xˉ)(yi−yˉ)∑(xi−xˉ)2m = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}}m=∑(xi​−xˉ)2∑(xi​−xˉ)(yi​−yˉ​)​

STEP 5

Calculate the y-intercept bbb using the formula:b=yˉ−mxˉb = \bar{y} - m\bar{x}b=yˉ​−mxˉ

STEP 6

Write the equation of the regression line:y^=mx+b\hat{y} = mx + by^​=mx+b

STEP 7

Construct a scatter plot of the data points (xi,yi)(x_i, y_i)(xi​,yi​).

STEP 8

Draw the regression line on the scatter plot using the equation y^=mx+b\hat{y} = mx + by^​=mx+b.

STEP 9

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1. Calculate the slope ( $m$ ) of the regression line.
2. Calculate the y-intercept ( $b$ ) of the regression line.
3. Write the equation of the regression line.
4. Construct a scatter plot and draw the regression line.
5. Use the regression equation to predict $y$ for given $x$ -values.

Calculate the mean of $x$ and $y$ .
$\bar{x} = \frac{775 + 619 + 519 + 508 + 491 + 474}{6}$
$\bar{y} = \frac{53 + 47 + 44 + 42 + 37 + 36}{6}$

Calculate the slope $m$ using the formula:
$m = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}}$

Calculate the y-intercept $b$ using the formula:
$b = \bar{y} - m\bar{x}$

Write the equation of the regression line:
$\hat{y} = mx + b$

Construct a scatter plot of the data points $(x_i, y_i)$ .

Draw the regression line on the scatter plot using the equation $\hat{y} = mx + b$ .