Inferential Presentation: Statistics

Inferential Presentation: Statistics

This exploratory presentation looks at expense account data of a company’s seven sales representatives over the past four months. The study investigated the relationships between the variables of amounts claimed and amounts reimbursed in relation to legitimate expense claims in order to test the null hypothesis that reimbursement is dependent upon claims fulfilling legitimate expense claim criteria. The value of alpha is equal to a 0.05 level of significance. The inferential analysis follows that outlined by the previous descriptive analysis which demonstrated a strong and dependable relationship between the dependent variable ‘claimed’ and the explanatory variable ‘reimbursed’.

This shows that the data were correctly processed by simply using clustering for the company’s seven sales representatives over the previous four months from the previous data set.

Table 1

	Claimed	Reimbursed
Correlation Table	Data Set #1	Data Set #1
Claimed	1.000	0.857
Reimbursed	0.857	1.000

It shows that the correlation data 0.857 is not correct based on computation using Microsoft excel programs which revealed the value of 0.8768125

	Claimed	Reimbursed
Covariance Table	Data Set #1	Data Set #1
Claimed	78383.46	64843.84
Reimbursed	64843.84	73014.63

This computation for the covariance is absolutely wrong. The actual computations for covariance using Microsoft excel program is

Covariance
	Claimed	Reimburse
Claimed	8330.798469
Reimburse	2468.482143	951.3928571

The above table 1 shows the positive correlation between the variables of claims and reimbursed claims as derived from the StatTools output for calculation and covariance.

This statement is correct because there was a positive correlation with a matrix value of 0.8768125.

Looking further into the relationship using regression analysis we find the following below in table 2:

Table 2

	Multiple	R-Square	Adjusted	Std Err of
Summary	R		R-Square	Estimate
	0.8575	0.7353	0.7330	139.7820051
	Degrees of	Sum of	Mean of	F-Ratio	p-Value
ANOVA Table	Freedom	Squares	Squares
Explained	1	6188762.843	6188762.843	316.7388	<>
Unexplained	114	2227447.019	19539.00894
	Coefficient	Standard	t-Value	p-Value	Lower	Upper
Regression Table		Error			Limit	Limit
Constant	-18.57178804	19.19132541	-0.9677	0.3352	-56.58965587	19.44607978
Claimed	0.82859208	0.046557551	17.7972	<>	0.736361931	0.920822229

The coefficient of correlation of 0.828 for claimed, shows a high correlation and therefore a strong relationship between the dependent variable (reimbursement) and the explanatory variable (claimed).

Using the above the tables (Table 2), the value for the correlation is not 0.828, but, 0.8575. The correlation statement is only quite correct. It should be very high correlation because it is near to 1 and above 0.75.

The confidence limits start at a positive lower limit value of 0.736 at p-value of <0.0001 style="mso-bidi-font-weight: normal">

If the computed data was correct the statement was correct, however, based on computation the regression slope of the confidence interval is very small with 0.060265684 to 0.507232254 which is closer to zero, and has significant relation to the “legitimate” expense claims.

The R² 0.735 at a p-value of <0.0001 style="mso-bidi-font-weight: normal">

If the computed data was correct the statement was correct however based on the computation, the resultant R Square 0.7565 is very close to 1 and it is above 0.75 which means that the correlation is near normal curve distribution, so, it is interpreted as very high positive correlation. Thus, in percentile (%), 75.65 is an indicator of significant relationship between the variables of amounts claimed and amounts reimbursed in relation to “legitimate” expense claims. Finally, the R square of 0.756494365is quite near to the adjusted R square 0.695617957. This means that the regression model approximately fits the data.

From the data above a simple linear regression equation can be calculated thus;

Y= 0.828 X₂ 18.571

(Y= claimed and X₂ = reimbursed)

In this data you can see that the values are inconsistent with those in table 1. This means that the data were absolutely wrong.

The hypothesis test in table 3 below, we look at the alternative hypothesis that submitted claims i.e. those deemed to legitimate are not dependant upon the reimbursement criteria being fulfilled. The results illustrate that at an output at 1% significance it fails to reject the null hypothesis as the p-value is just greater than 0.05 at 0.0519. Indeed, the failure to reject the null is maintained at 5% level of significance.

Table 3

StatTools	(Core Analysis Pack)
Analysis:	Hypothesis Test
Performed By:	Computing Services
Date:	25-Jun-09
Updating:	Live
	Claimed	Reimbursed
Sample Summaries	Data Set #1	Data Set #1
Sample Size	116	116
Sample Mean	$303.66	$233.03
Sample Std Dev	$279.97	$270.53
	Equal	Unequal
Hypothesis Test (Difference of Means)	Variances	Variances
Hypothesized Mean Difference	0	0
Alternative Hypothesis	<> 0	<> 0
Sample Mean Difference	$70.62	$70.62
Standard Error of Difference	36.14719623	36.14719623
Degrees of Freedom	230	229
t-Test Statistic	1.9537	1.9537
p-Value	0.0519	0.0520
Null Hypoth. at 10% Significance	Reject	Reject
Null Hypoth. at 5% Significance	Don't Reject	Don't Reject
Null Hypoth. at 1% Significance	Don't Reject	Don't Reject
Equality of Variances Test
Ratio of Sample Variances	1.0710
p-Value	0.7135

In addition, when we look at the reimbursed outputs in table 4 below, we can see that the coefficient of correlation for reimbursed is a high correlation value at 0.858, again indicating a strong relationship the dependent variable (claimed) and the explanatory variable (reimbursed). The confidence limits begin in the positive 0.736 at <0.0001>2 0.735 at a p-value of <0.0001>

Using the given tables, correlation was not high correlation, but, it is very high correlation because it is above 0.75. The R² 0.735 is very near to the adjusted R 0.733 which means that the regression model approximately fits the data very well.

From the data in table 4 below a simple linear regression equation can be calculated thus;

Y+ 0.887 X₂ 96.847

(Y= reimbursed and X₂ = claimed)

In this data you can see that the values are inconsistent with those in table 1. This means that the data were absolutely wrong.

The hypothesis table 3 shows the alternative hypothesis i.e. that successfully reimbursed claims - those confirmed legitimate are not dependant upon the claims criteria being fulfilled. The results show that the p-value for reimbursed is just above 0.05 at 0.0520 and therefore at a 1% significance fails to reject the null hypothesis. Again, failure to reject the null is maintained at the 5% level of significance.

This statement is correct because the null hypothesis of this study was accepted in the computation of the analysis of variance. The result of the “Analysis of Variance” (ANOVA) shows that the computed F (12.43) is less than the tabular values of F-statistics (230) at 0.05 degree of freedom (1, 4). This means that there is no significant difference between the variables of amounts claimed and amounts reimbursed in relation to “legitimate” expense claims.

Table 4

Multiple

R-Square

Adjusted

StErr of

Summary

R

R-Square

Estimate

0.8575

0.7353

0.7330

144.6618885

Degrees of

Sum of

Mean of

F-Ratio

p-Value

ANOVA Table

Freedom

Squares

Squares

Explained

1

6628413.141

6628413.141

316.7388

<>

Unexplained

114

2385685.066

20927.06198

Coefficient

Standard

t-Value

p-Value

Lower

Upper

Regression Table

Error

Limit

Limit

Constant

96.84747643

17.76052565

5.4530

<>

61.66401192

132.0309409

Reimbursed

0.887455339

0.049865003

17.7972

<>

0.788673152

0.986237527

Conclusion

The data set is limited in that it contains only two numeric variables upon which to undertake regression analysis. The inferential analysis follows that outlined by the previous descriptive analysis which demonstrated a strong and dependable relationship between the dependent variable (claimed) and the explanatory variable (reimbursed).

The regression method was already presented in the tables. The computed value of the regression was 316.7388 while the tabulated value 253 at 0.05 level of significant. This implies that the null hypothesis was rejected because computed value was greater than the tabulated value.

This is to be expected given that it is a truism that claims made would be expected to be reimbursed assuming that they fulfill the claims criteria. We have learned that 23.3% of claims were not reimbursed in the 4 month period due to not being considered “legitimate”. This therefore proves the null hypothesis that the successful reimbursement of expense claims are indeed dependant upon the claim criteria being fulfilled.

This statement is true because there is no significant difference between the variables of amounts claimed and amounts reimbursed in relation to “legitimate” expense claims.

Reference

Albright, S., Winston, W., & Zappe, C. (2006). Data Analysis and Decision Making with Microsoft Excel. Manson, US: Thomson South-Western, USA.