Question

The Transactional Records Access Clearinghouse at Syracuse University reported data showing the odds of an Internal Revenue Service audit. The following table shows the average adjusted gross income reported (in dollars) and the percent of the returns that were audited for 20 selected IRS districts.

District	Adjusted Gross Income ($)	Percent Audited
Los Angeles	36,664	1.3
Sacramento	38,845	1.1
Atlanta	34,886	1.1
Boise	32,512	1.1
Dallas	34,531	1.0
Providence	35,995	1.0
San Jose	37,799	0.9
Cheyenne	33,876	0.9
Fargo	30,513	0.9
New Orleans	30,174	0.9
Oklahoma City	30,060	0.8
Houston	37,153	0.8
Portland	34,918	0.7
Phoenix	33,291	0.7
Augusta	31,504	0.7
Albuquerque	29,199	0.6
Greensboro	33,072	0.6
Columbia	30,859	0.5
Nashville	32,566	0.5
Buffalo	34,296	0.5

(a)

Develop the estimated regression equation that could be used to predict the percent audited given the average adjusted gross income reported (in dollars). (Round your value for the y-intercept to three decimal places and your value for the slope to six decimal places.)

ŷ =

(b)

At the 0.05 level of significance, determine whether the adjusted gross income (in dollars) and the percent audited are related. (Use the F test.)

State the null and alternative hypotheses.

H₀: β₁ ≠ 0
H_a: β₁ = 0

H₀: β₁ = 0
H_a: β₁ ≠ 0   

H₀: β₀ ≠ 0
H_a: β₀ = 0

H₀: β₁ ≥ 0
H_a: β₁ < 0

H₀: β₀ = 0
H_a: β₀ ≠ 0

Find the value of the test statistic. (Round your answer to two decimal places.)

Find the p-value. (Round your answer to three decimal places.)

p-value =

State your conclusion.

Do not reject H₀. We cannot conclude that the relationship between the adjusted gross income (in dollars) and the percent audited is significant.

Do not reject H₀. We conclude that the relationship between the adjusted gross income (in dollars) and the percent audited is significant.

Reject H₀. We conclude that the relationship between the adjusted gross income (in dollars) and the percent audited is significant.

Reject H₀. We cannot conclude that the relationship between the adjusted gross income (in dollars) and the percent audited is significant.

(c)

Did the estimated regression equation provide a good fit? Explain. (Round your answer to three decimal places.)

Since

r² =

is  ---Select--- less than 0.55 at least 0.55 , the estimated regression equation  ---Select--- provided did not provide a good fit.

(d)

Use the estimated regression equation developed in part (a) to calculate a 95% confidence interval for the expected percent audited for districts with an average adjusted gross income of $37,000. (Round your answers to two decimal places.)

% to  %

Answer 1

Let the regression equation is of the form,

y=β0+β1x+e

Where,

α is the shift from the origin and β is the intercept and e is the error

and, y: percent audited

        x: gross adjusted income

a.

          Now, we have to Develop the estimated regression equation that could be used to predict the percent audited given the average adjusted gross income reported (in dollars).

Let the ith observation be,

yi=α+βxi+ei ,here i=1(1)20

            For this we estimated least square estimates of β0 and β1 by minimizing

i=120ei2=i=120(y-β0-β1x)2 =0

By solving the normal equations, we get,

β1=covx,ySx2 and β0=y-β1x

Sxx=1n-1i(xi-x)2 =variance of x

Syy=1n-1i(yi-y)2 =variance of y

covx,y=1nixiyi-xy =covariance b/w x and y.

Sxy=1n-1ixiyi-xy

We have done the analysis in MS Excel,

The required estimated regression equation is,

y(percent audit)=-0.50361+3.196x(gross income)

b.

Hypothesis to be tested:

          Now, we have to test,

H0:ρ=0 ag. H1: ρ≠0

Test Statistic:

          The test statistic for testing H0 is, F=MSRMSE follows Fα;n-1,k-1 ,

under H0

Here, k=2,n=20

And MSR=SSRk-1 where, SSR is sum of square due to regression,

i=120(y-y)2=SSR

And MSE=SSEn-1, SSE=i=120yi-y2,sum of square due to

Residual

Critical region:

          We reject the null H0 if F>Fα;n-1,k-1

Conclusion:

          Let, α=0.05 . Here, F=5.269 and F0.05;20-1,2-1=0.034

Hence, we reject the null H0 at 5% level of significance and conclude on the basis of the given data that Percent audit and average adjusted gross incomes are related.

Hypothesis to be tested for significance of β1 and β0 :

We want to test,

H01:β0=0 ag. H11:β0≠0

H02:β1=0 ag. H12:β1≠0

Test statistic:

The test statistic for testing H01 is,

t1=β0-β0SE(β0) follows t distribution with 18 degrees of freedom, under the null.

SEβ0=MSEi(Xi-X)2i=1nXi220

The test statistic for testing H02 is,

t2=β1-β1SE(β1) follows t distribution with 7-2=5 degrees of freedom, under the null.

SEβ1=MSEi(Xi-X)2

Critical Region:

            Reject the null hypotheses H02 and H01 if |t1 |>t0.05;5 and |t2 |>t0.05;5 at 5 % level of significance.

Conclusion:

          Here, t1=-0.8641 and t2=2.295 and t0.05;18=1.734

Hence, we reject the null hypothesis H01 and accept H02 at 5% level of significance and conclude on the basis of the given sample that Gross Adjusted Income has a significant effect on Percent audit and the intercept has no significant effect.

c.

Here, R^2=0.226<0.55 i.e. only 22.6% variability of the total variability is explained by the regression equation. Hence the model does not provide a good fit.

d.

Now, we have to calculate a 95% confidence interval for the expected percent audited for districts with an average adjusted gross income of $37,000.

The Confidence interval is given by,

y±t0.05;18MSEi(Xi-X)2i=1nXi220

Or, y±SE(β0)

Now, for x=37,000, y=118251.5

The 95% CI is given by,

118251.5±0.582816438

Gross Adjusted Income(x)	Percent($)Audited(y)					SUMMARY OUTPUT
36664	1.3
38845	1.1					Regression Statistics
34886	1.1					Multiple R	0.475868552				R_square	0.22645088
32512	1.1					R Square	0.226450878
34531	1					Adjusted R Square	0.183475927
35995	1					Standard Error	0.207511207
37799	0.9					Observations	20
33876	0.9
30513	0.9					ANOVA
30174	0.9						df	SS	MS	F	Significance F
30060	0.8					Regression	1	0.22690378	0.226904	5.269369	0.03393498
37153	0.8					Residual	18	0.77509622	0.043061
34198	0.7					Total	19	1.002
33291	0.7
31504	0.7						Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
29199	0.6					Intercept	-0.503614543	0.582816438	-0.8641	0.398899	-1.7280664	0.72083736	-1.7280664	0.72083736
33072	0.6					Gross Adjusted Income(x)	3.96913E-05	1.72908E-05	2.295511	0.033935	3.3646E-06	7.6018E-05	3.3646E-06	7.6018E-05
30859	0.5
32566	0.5
34296	0.5
						RESIDUAL OUTPUT				PROBABILITY OUTPUT
						Observation	Predicted Percent($)Audited(y)	Residuals	Standard Residuals		Percentile	Percent($)Audited(y)			1.734064
						1	0.951628104	0.348371896	1.724813		2.5	0.5
						2	1.038194878	0.061805122	0.306001		7.5	0.5
		3.97E-05	3.196			3	0.881056933	0.218943067	1.084002		12.5	0.5
						4	0.786829733	0.313170267	1.550528		17.5	0.6
	y_hat	118251.5				5	0.866966513	0.133033487	0.658658		22.5	0.6
						6	0.925074609	0.074925391	0.370961		27.5	0.7
						7	0.996677755	-0.096677755	-0.47866		32.5	0.7
						8	0.840968697	0.059031303	0.292268		37.5	0.7
						9	0.707486779	0.192513221	0.953146		42.5	0.8
						10	0.694031421	0.205968579	1.019765		47.5	0.8
						11	0.68950661	0.11049339	0.54706		52.5	0.9
						12	0.971037161	-0.171037161	-0.84682		57.5	0.9
						13	0.853749303	-0.153749303	-0.76122		62.5	0.9
						14	0.817749273	-0.117749273	-0.58298		67.5	0.9
						15	0.74682088	-0.04682088	-0.23181		72.5	1
						16	0.655332382	-0.055332382	-0.27395		77.5	1
						17	0.809056874	-0.209056874	-1.03506		82.5	1.1
						18	0.721219977	-0.221219977	-1.09528		87.5	1.1
						19	0.788973065	-0.288973065	-1.43073		92.5	1.1
						20	0.857639052	-0.357639052	-1.7707		97.5	1.3