government funding: the following table presents the budget (in millions of dollars) for selected…
government funding: the following table presents the budget (in millions of dollars) for selected organizations that received u.s. government funding for arts and culture at two different points in time. use the ti - 84 plus calculator as needed.\n\norganization five years ago last year\ncorporation for public broadcasting 460 444\nnational gallery of art 247 227\ninstitute of museum and library services 142 160\nnational endowment for the humanities 124 163\ncommission of fine arts 95 158\nnational endowment for the arts 18 46\nkennedy center for the performing arts 2 7\n\npart 1 of 3\n(a) compute the least - squares regression line for predicting last years budget from the five - years - ago budget. round the slope and y - intercept to at least four decimal places.\nthe equation for the least - squares regression line is \\(\\hat{y}=\\)\n\npart 2 of 3\n(b) if two institutions had budgets that differed by a million dollars five years ago, by how much would you predict their budgets to differ last year? round the answer to at least two decimal places.\nlast year, their budgets would differ by \\(\\) million dollars.\n\npart 3 of 3\n(c) predict last years budget for an organization whose five - years - ago budget was 120 million dollars. round the answer to at least two decimal places.\nlast years budget is predicted to be \\(\\) million dollars.
Answer
Explanation:
Step1: Define variables
Let $x$ be the budget five - years ago and $y$ be the budget last year. First, calculate the necessary sums: $\sum x$, $\sum y$, $\sum x^2$, $\sum xy$, and $n$ (number of data points). Here $n = 6$.
| Organization | $x$ (Five Years Ago) | $y$ (Last Year) | $x^2$ | $xy$ |
|---|---|---|---|---|
| Corporation for Public Broadcasting | 460 | 444 | 211600 | 204240 |
| National Gallery of Art | 247 | 227 | 61009 | 56069 |
| Institute of Museum and Library Services | 142 | 160 | 20164 | 22720 |
| National Endowment for the Humanities | 124 | 163 | 15376 | 20212 |
| Commission of Fine Arts | 95 | 158 | 9025 | 15010 |
| National Endowment for the Arts | 18 | 46 | 324 | 828 |
| Kennedy Center for the Performing Arts | 2 | 7 | 4 | 14 |
| $\sum$ | $\sum x=1088$ | $\sum y=1205$ | $\sum x^2 = 317502$ | $\sum xy = 319093$ |
Step2: Calculate slope $b_1$
The formula for the slope $b_1$ of the least - squares regression line $\hat{y}=b_0 + b_1x$ is $b_1=\frac{n\sum xy-\sum x\sum y}{n\sum x^2-(\sum x)^2}$. [ \begin{align*} b_1&=\frac{6\times319093 - 1088\times1205}{6\times317502-1088^2}\ &=\frac{1914558-1301040}{1905012 - 1183744}\ &=\frac{613518}{721268}\ &\approx0.8506 \end{align*} ]
Step3: Calculate y - intercept $b_0$
The formula for the y - intercept $b_0$ is $b_0=\bar{y}-b_1\bar{x}$, where $\bar{x}=\frac{\sum x}{n}$ and $\bar{y}=\frac{\sum y}{n}$. $\bar{x}=\frac{1088}{6}\approx181.3333$, $\bar{y}=\frac{1205}{6}\approx200.8333$. [ \begin{align*} b_0&=200.8333-0.8506\times181.3333\ &=200.8333 - 154.2477\ &\approx46.5856 \end{align*} ]
The equation of the least - squares regression line is $\hat{y}=46.5856 + 0.8506x$.
Step4: Answer part (b)
If two organizations had budgets that differed by $A$ million dollars five years ago, the difference in their predicted budgets last year is given by the slope of the regression line. So if the difference in $x$ is $A$, the difference in $\hat{y}$ is $b_1A$. Substituting $A = 1$, the difference in their budgets last year is approximately $0.85$ million dollars.
Step5: Answer part (c)
To predict last year's budget for an organization whose five - years ago budget was $x = 120$ million dollars, substitute $x = 120$ into the regression equation $\hat{y}=46.5856+0.8506x$. [ \begin{align*} \hat{y}&=46.5856+0.8506\times120\ &=46.5856 + 102.072\ &=148.6576\ &\approx148.66 \end{align*} ]
Answer:
Part (a): $\hat{y}=46.5856 + 0.8506x$ Part (b): $0.85$ Part (c): $148.66$