Question

The idea of tax efficiency is based on how much tax is due to capital gains stock or mutual fund investors pay of their investments. You become interested in this in one of your economic development classes. You collect data on the variables (percentage of investments in energy and tax efficiency). The basic data for the variables is presented below. What is the predictor (independent x) variable? What is the response (dependent y) variable? Compute the value of r. Based on the Pearson’s r table is the correlation statistically significant at the .05 significance level? (Use a one tail left test.) What portion of the change in the response variable is associated with the change in the predictor variable? (R squared) Values needed for computation are below. You might want to use the formula for r below as the data for the formula is directly below. (Note: you have the complete value for the numerator of that formula.) Standard deviation of x = 2.42, Standard deviation of y = 10.39, Mean of x = 5.59, Mean of y = 81.74, n = 10, Sum of (x-x mean) times (y- y mean) = -220.33 r equals fraction numerator sum for blank of open parentheses x minus top enclose x close parentheses open parentheses y minus top enclose y close parentheses over denominator open parentheses n minus 1 close parentheses s subscript y s subscript x end fraction

Answer 1

	x	y
Mean	5.59	81.74
Sd	2.42	10.39
n	10

= -220.33

We want to know how much tax efficieny is there based on capital gains that is the investments. Since the tax is dependent on gains, tax : dependent variable and gain: independent variable

• What is the predictor (independent x) variable? What is the response (dependent y) variable?

predictor (independent x) variable: Tax efficiency

response (dependent y) variable: Investments in energy

Compute the value of r.

Assuming that the data is a sample and not a population

Cov (x,Y) =

Cov (x,y) = -24.4811

r = (-24.4811)/ (2.42 * 10.39)

r = -0.9736

• Based on the Pearson’s r table is the correlation statistically significant at the .05 significance level? (Use a one tail left test.)

Null: = 0

Alternative: < 0

We reject if the |r| > |C.V.|

= 0.05

C.V. = r_0.05,8 .........where df = n-2 = 8 and we check under 1 tailed values

C.V. = -0.549 ..............negative because left tailed

Since |-0.9736|> |-0.549|

We reject the null hypothesis at 0.05. There is sufficient evidence to conclude that there is negative correlation between the tax efficiency and investment in energy.

• What portion of the change in the response variable is associated with the change in the predictor variable?

R-squared = r²

R-squared = 0.948

=94.8%

So 94.8% of variation in tax efficiency is expalined by the gains in energy.

since r-sqaured > 75%, regression model is good fit for the data.