Math / Data & Statistics

QuestionGovernment funding: The following table presents the budget (in millions of dollars) for selected organizations that received U.S. government funding for arts and culture in both 2006 and last year.
Organization | 2006 | Last Year --- | --- | --- Organization 1 | 460 | 440 Organization 2 | 247 | 227 Organization 3 | 142 | 161 Organization 4 | 124 | 156 Organization 5 | 95 | 166 Organization 6 | 18 | 45 Organization 7 | 2 | 3
Part: 0 / 3 Part 1 of 3 Compute the least-squares regression line for predicting last year's budget from the 2006 budget Round the slope and $y$ -intercept to four decimal places as needed. The equation for the least-squares regression line is $\hat{y} =$ .

Studdy Solution

STEP 1

1. We have data for budgets in 2006 and last year for seven organizations.
2. We need to find the least-squares regression line to predict last year's budget from the 2006 budget.

STEP 2

1. Calculate the means of the 2006 and last year budgets.
2. Calculate the slope of the regression line.
3. Calculate the y-intercept of the regression line.
4. Write the equation of the regression line.

STEP 3

Calculate the mean of the 2006 budgets:
$\bar{x} = \frac{460 + 247 + 142 + 124 + 95 + 18 + 2}{7}$
$\bar{x} = \frac{1088}{7} \approx 155.4286$

STEP 4

Calculate the mean of last year's budgets:
$\bar{y} = \frac{440 + 227 + 161 + 156 + 166 + 45 + 3}{7}$
$\bar{y} = \frac{1198}{7} \approx 171.1429$

STEP 5

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 $\sum (x_i - \bar{x})(y_i - \bar{y})$ and $\sum (x_i - \bar{x})^2$ :
\begin{align} \sum (x_i - \bar{x})(y_i - \bar{y}) &= (460 - 155.4286)(440 - 171.1429) + \ldots + (2 - 155.4286)(3 - 171.1429) \\ &= 304.5714 \times 268.8571 + \ldots + (-153.4286) \times (-168.1429) \\ &= 81864.2857 \end{align}
\begin{align} \sum (x_i - \bar{x})^2 &= (460 - 155.4286)^2 + \ldots + (2 - 155.4286)^2 \\ &= 304.5714^2 + \ldots + (-153.4286)^2 \\ &= 119893.4286 \end{align}
$m = \frac{81864.2857}{119893.4286} \approx 0.6826$

STEP 6

Calculate the y-intercept $b$ using the formula:
$b = \bar{y} - m\bar{x}$
$b = 171.1429 - 0.6826 \times 155.4286$
$b \approx 65.3143$

STEP 7

Write the equation of the regression line:
$\hat{y} = 0.6826x + 65.3143$
The equation for the least-squares regression line is:
$\hat{y} = 0.6826x + 65.3143$

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Studdy Solution

STEP 1

1. We have data for budgets in 2006 and last year for seven organizations.
2. We need to find the least-squares regression line to predict last year's budget from the 2006 budget.

STEP 2

1. Calculate the means of the 2006 and last year budgets.
2. Calculate the slope of the regression line.
3. Calculate the y-intercept of the regression line.
4. Write the equation of the regression line.

STEP 3

Calculate the mean of the 2006 budgets:
$\bar{x} = \frac{460 + 247 + 142 + 124 + 95 + 18 + 2}{7}$
$\bar{x} = \frac{1088}{7} \approx 155.4286$

STEP 4

Calculate the mean of last year's budgets:
$\bar{y} = \frac{440 + 227 + 161 + 156 + 166 + 45 + 3}{7}$
$\bar{y} = \frac{1198}{7} \approx 171.1429$

STEP 5

STEP 6

Calculate the y-intercept $b$ using the formula:
$b = \bar{y} - m\bar{x}$
$b = 171.1429 - 0.6826 \times 155.4286$
$b \approx 65.3143$

STEP 7

Write the equation of the regression line:
$\hat{y} = 0.6826x + 65.3143$
The equation for the least-squares regression line is:
$\hat{y} = 0.6826x + 65.3143$

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Studdy Solution

STEP 1

1. We have data for budgets in 2006 and last year for seven organizations.2. We need to find the least-squares regression line to predict last year's budget from the 2006 budget.

STEP 2

1. Calculate the means of the 2006 and last year budgets.2. Calculate the slope of the regression line.3. Calculate the y-intercept of the regression line.4. Write the equation of the regression line.

STEP 3

Calculate the mean of the 2006 budgets:xˉ=460+247+142+124+95+18+27 \bar{x} = \frac{460 + 247 + 142 + 124 + 95 + 18 + 2}{7} xˉ=7460+247+142+124+95+18+2​xˉ=10887≈155.4286 \bar{x} = \frac{1088}{7} \approx 155.4286 xˉ=71088​≈155.4286

STEP 4

Calculate the mean of last year's budgets:yˉ=440+227+161+156+166+45+37 \bar{y} = \frac{440 + 227 + 161 + 156 + 166 + 45 + 3}{7} yˉ​=7440+227+161+156+166+45+3​yˉ=11987≈171.1429 \bar{y} = \frac{1198}{7} \approx 171.1429 yˉ​=71198​≈171.1429

STEP 5

STEP 6

Calculate the y-intercept b b b using the formula:b=yˉ−mxˉ b = \bar{y} - m\bar{x} b=yˉ​−mxˉb=171.1429−0.6826×155.4286 b = 171.1429 - 0.6826 \times 155.4286 b=171.1429−0.6826×155.4286b≈65.3143 b \approx 65.3143 b≈65.3143

STEP 7

Write the equation of the regression line:y^=0.6826x+65.3143 \hat{y} = 0.6826x + 65.3143 y^​=0.6826x+65.3143 The equation for the least-squares regression line is:y^=0.6826x+65.3143 \hat{y} = 0.6826x + 65.3143 y^​=0.6826x+65.3143

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1. We have data for budgets in 2006 and last year for seven organizations.
2. We need to find the least-squares regression line to predict last year's budget from the 2006 budget.

1. Calculate the means of the 2006 and last year budgets.
2. Calculate the slope of the regression line.
3. Calculate the y-intercept of the regression line.
4. Write the equation of the regression line.

Calculate the mean of the 2006 budgets:
$\bar{x} = \frac{460 + 247 + 142 + 124 + 95 + 18 + 2}{7}$
$\bar{x} = \frac{1088}{7} \approx 155.4286$

Calculate the mean of last year's budgets:
$\bar{y} = \frac{440 + 227 + 161 + 156 + 166 + 45 + 3}{7}$
$\bar{y} = \frac{1198}{7} \approx 171.1429$

Calculate the y-intercept $b$ using the formula:
$b = \bar{y} - m\bar{x}$
$b = 171.1429 - 0.6826 \times 155.4286$
$b \approx 65.3143$

Write the equation of the regression line:
$\hat{y} = 0.6826x + 65.3143$
The equation for the least-squares regression line is:
$\hat{y} = 0.6826x + 65.3143$