Question

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.

organization five years ago last year
corporation for public broadcasting 460 444
national gallery of art 247 227
institute of museum and library services 142 160
national endowment for the humanities 124 163
commission of fine arts 95 158
national endowment for the arts 18 46
kennedy center for the performing arts 2 7

part 1 of 3
(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.
the equation for the least - squares regression line is \\(\hat{y}=\\)

part 2 of 3
(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.
last year, their budgets would differ by \\(\\) million dollars.

part 3 of 3
(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.
last years budget is predicted to be \\(\\) million dollars.

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)	$x^2$	$xy$
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): $148.66$