Answer to Question #287158 in Statistics and Probability for Julier

Question

State Debt and Per Capita Tax in USA. An economics student wisher to see if there is a relationship between the amount of state debt per capita and the amount of tax per capita at the state level. Based on the following data, can she or he conclude that per capita state debt and per capita state taxes are related? Both amounts are in dollars and represent five randomly selected states. a.Draw a scatter plot.

b. Compute the correlation coefficient.

c. State the hypotheses.

d. Test the hypotheses at a 0.05.

e. Determine the regression line equation.

f. Plot the regression line on the scatter plot.

g. Summarize the results.

Per capita debt (x )

1924 907 1445 1608 661

Per capita tax (y)

1685 1838 1734 1842 1317

Answer 1

a.

b.

correlation coefficient:

$r = \frac{Σ(x_i - x̄)(y_i - ȳ)}{√(Σ( x_i - x̄)^2Σ(y_i - ȳ)^2 )}=0.52$

c.

$H_0:r=0$ , per capita state debt and per capita state taxes are not related

$H_a:r\neq0$ , per capita state debt and per capita state taxes are related

d.

$t=r\sqrt{\frac{n-2}{1-r^2}}=0.52\sqrt{\frac{5-2}{1-0.52^2}}=1.048$

$df=n-2=5-2=3$

critical value:

$t_{crit}=3.182$

Since $|t|<t_{crit}$ we accept null hypothesis. Per capita state debt and per capita state taxes are not related.

e.

regression line equation:

$y=b_0+b_1x$

where

$b_1 =\frac{\sum(x_i-x̄)(y_i-ȳ)}{Σ(x_i-x̄)^2}=0.216$

$b_0 = ȳ - b_1x̄=1400.51$

$y=0.216x+1400.51$

f.

g.

From scatter plot we can conclude that the linear regression model  doesn't provide a good fit.