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. \begin{tabular}{llc} \hline Organization & 2006\mathbf{2 0 0 6} & Last Year \\ \hline Organization 1 & 460 & 450 \\ Organization 2 & 247 & 229 \\ Organization 3 & 142 & 154 \\ Organization 4 & 124 & 165 \\ Organization 5 & 95 & 156 \\ Organization 6 & 18 & 42 \\ Organization 7 & 2 & 3 \\ \hline \end{tabular}
Part: 0/30 / 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 yy-intercept to four decimal places as needed.
The equation for the least-squares regression line is yundefined=\widehat{y}=\square. \square

Studdy Solution

STEP 1

What is this asking? We need to find the line that *best* predicts an organization's budget last year based on its budget in 2006. Watch out! Don't mix up which year is xx and which is yy!
Also, keep track of those decimals!

STEP 2

1. Calculate the means of the 2006 budgets (xx) and last year's budgets (yy).
2. Calculate the slope (bb) of the regression line.
3. Calculate the y-intercept (aa) of the regression line.
4. Assemble the equation.

STEP 3

We **add** all the 2006 budgets together: 460+247+142+124+95+18+2=1088460 + 247 + 142 + 124 + 95 + 18 + 2 = 1088.
This is the **total budget** across all organizations in 2006.

STEP 4

We **divide** the **total budget** by the number of organizations, which is 7, to get the **mean** 2006 budget: 10887155.4286\frac{1088}{7} \approx 155.4286.

STEP 5

We **add** all of last year's budgets: 450+229+154+165+156+42+3=1199450 + 229 + 154 + 165 + 156 + 42 + 3 = 1199.
This is the **total budget** across all organizations last year.

STEP 6

We **divide** the **total budget** by the number of organizations (7) to get the **mean** of last year's budget: 11997171.2857\frac{1199}{7} \approx 171.2857.

STEP 7

For each organization, we multiply the difference between its 2006 budget and the **mean** 2006 budget by the difference between its last year's budget and the **mean** last year's budget.
Then we **add** these products together: (460155.4286)(450171.2857)+(247155.4286)(229171.2857)+...+(2155.4286)(3171.2857)46195.4284(460-155.4286)(450-171.2857) + (247-155.4286)(229-171.2857) + ... + (2-155.4286)(3-171.2857) \approx 46195.4284

STEP 8

For each organization, we **square** the difference between its 2006 budget and the **mean** 2006 budget.
Then we **add** these squared differences: (460155.4286)2+(247155.4286)2+...+(2155.4286)2154267.8572(460-155.4286)^2 + (247-155.4286)^2 + ... + (2-155.4286)^2 \approx 154267.8572

STEP 9

We **divide** the numerator from step 2.2.1. by the denominator from step 2.2.2. to get the slope (bb): 46195.4284154267.85720.2994\frac{46195.4284}{154267.8572} \approx 0.2994.

STEP 10

The y-intercept (aa) is calculated using the formula a=yˉbxˉa = \bar{y} - b\bar{x}, where yˉ\bar{y} is the **mean** of last year's budgets, xˉ\bar{x} is the **mean** of the 2006 budgets, and bb is the slope we just calculated. a=171.2857(0.2994)(155.4286)124.6037a = 171.2857 - (0.2994)(155.4286) \approx 124.6037

STEP 11

The least-squares regression line is given by y^=a+bx\hat{y} = a + bx.
Substituting the values we calculated, we get y^=124.6037+0.2994x\hat{y} = 124.6037 + 0.2994x.

STEP 12

The equation for the least-squares regression line is y^=124.6037+0.2994x\hat{y} = 124.6037 + 0.2994x.

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