Cost Based on Employees, Research Paper Example

Analysis of cost of benefits is important to companies because it helps them determine how well they are doing and whether they are really making substantial profit. Employees in a company come with both benefits and costs. Examples of the benefits are that they can bring new ideas and that they lead to increased revenue due to increased labor force.  Examples of costs are training, salaries, benefits and equipment costs. At all times, companies strive to make sure that the benefits outdo the costs.

In this paper, we will estimate a company’s average cost of benefits based on its number of employees. We will also forecast an estimate of a company’s expected average cost of based using its number of employees as a factor. Also, we will use confidence intervals and prediction intervals to estimate the intervals of the true average cost of benefit for companies of 55 employees and the true expected average cost of benefits for a company of 55 employees.

Methods

The data used in this paper contains the number of employees and the health care costs (in thousands) of 500 small to mid-size companies. The number of employees is the explanatory, x, variable whereas the health care costs is the response, y, variable. The number of employees is a discrete variable whereas the health care costs is a continuous variable. The analysis of this data was done using Microsoft Excel.

To estimate a response variable using an explanatory variable, we use regression models. However, to determine whether to use parametric models or nonparametric models, we explored the nature of the relationship between the health care costs and the number of employees. That is, whether the relationship is linear or nonlinear.

We used confidence intervals and prediction intervals to estimate the average cost of benefits and the expected average cost of benefits based on the number of employees. These two intervals were computed using the scenarios of a list of companies that have 55 employees and a company that has 55 employees respectively.

Results

As is visible in Figure 1 above, there is a nonlinear relationship between health care costs and the number of employees. The pattern of the data indicates a curve. Therefore, we will use a nonparametric model. Specifically, we will use polynomial regression to fit a model to our data. Our model will be in the form y? = b₀ + b₁x₁ + b₂x₁² + … + b_nx₁ⁿ

Table 1

Output for the 6^th degree polynomial regression

? = 0.05

The p-value for the regression model (1.4E-161) is less than ? = 0.05, which means that the regression model is statistically significant at the significance level of 0.05. However, looking at the p-values of the estimate coefficients as shown in Table 1, we see that the coefficients for employee^3 (p = 0.876449 > 0.05) and employee^4 (p = 0.120795 > 0.05) are not statistically significant at ? = 0.05. Therefore, we abandon this model for the model with the next best coefficient of determination value, which is the 5^th degree polynomial regression model. Below is a scatterplot of the model:

Running the 5^th order polynomial regression using Analysis Toolpak in Microsoft Excel produces the following output:

Table 2

Output for the 5^th order polynomial regression

Looking at Table 2, the p-value for the regression model (1.6E-161) is less than ? = 0.05, which means that the regression model is statistically significant at the significance level of 0.05. The p-values of the estimate coefficients as shown in are all statistically significant at ? = 0.05, since they are all less than 0.05. Therefore, we concluded that this model is the best model for estimating the average cost of benefits using the number of employees, out of all the models.

The regression equation of the model is:

Cost = 118.4717 – 9.0503employee + 0.319511 employee^2 – 0.00504employee^3 + 3.62E-05employee^4 – 9.5E-08employee^5

The correlation coefficient, r, is -0.8851. This means that there is a strong negative relationship between average cost of benefit and the number of employees in a company. Since the number of employees is the explanatory variable, we concluded that the average cost of benefits decreased as the number of employees increased and vice versa.

Confidence Interval

We need to find the 95% confidence interval for the average cost of benefits for all the companies that have 55 employees. Looking at our data, only 3 companies have 55 employees. The mean of average cost of benefits of these companies is 22.0474 and the standard deviation is 5.2907. The formula for calculating the confidence interval when we don’t know the population standard deviation is . The t critical value for ? = 0.05 and degrees of freedom = n – 1 = 3 – 1 = 2 is 4.303. Therefore, the confidence interval is . This interval means that we are 95% confident that the true average cost of benefits for companies with 55 employees lies between 8.9035 thousand and 35.1913 thousand. That is between 8903.5 and 35191.3.

Prediction interval

We need to find the 95% prediction interval for the average cost of benefits for a company that has 55 employees. Using our 5^th order polynomial regression model, the predicted value of average cost of benefits for a company that has 55 employees is: Cost = 118.4717 – 9.0503(55) + 0.319511(55) ^2 – 0.00504(55) ^3 + 3.62E-05(55) ^4 – 9.5E-08(55) ^5 = 31.5926. The formula for calculating the prediction interval when we don’t know the population standard deviation is . The t critical value for ? = 0.05 and degrees of freedom = n – 2 = 500 – 2 = 498 is 1.965. From Table 2 above, the MSE is 57.2728.  Therefore, the prediction interval is This interval means that we are 95% confident that the true average cost of benefits for a company with 55 employees lies between 16.7068 thousand and 46.4784 thousand. That is between 16706.8 and 46478.4.

Scatterplot

Figure 7. Scatterplot showing the 5^th degree polynomial model for the relationship between health care costs and the number of employees, showing the confidence interval and the prediction interval at x =55 employees.

Conclusion

There is a strong negative relationship between the average cost of benefits and the number of employees such that the average cost of benefits increases with decrease in the number of employees and vice versa. Therefore, companies should aim to have more employees in order to reduce their cost of benefits. However, more research should be done to determine at what threshold the costs equal or exceed the benefits.

Companies with 55 employees generally have average cost of benefits that are between 8903.5 and 35191.3 95% of the time. Similarly, a company with 55 employees should expect, with 95% confidence, an average cost of benefits of between 16706.8 and 46478.4.