Question

Accounting and Machiavellianism. Refer to the Behavioral Research in Accounting (January 2008) study of Machiavellian traits in accountants, Exercise 1.33 (p. 52). Recall that Machiavellian describes negative character traits that include manipulation, cunning, duplicity, deception, and bad faith. A Machiavellian (“Mach”) rating score was determined for each in a sample of accounting alumni of a large southwestern university. Scores range from a low of 40 to a high of 160, with the theoretical neutral Mach rating score of 100. The 122 purchasing managers in the sample had a mean Mach rating score of 99.6, with a standard deviation of 12.6.

a. From the sample, estimate the true mean Mach rating score of all purchasing managers.

b. Form a 95% confidence interval for the estimate, part b.

c. Give a practical interpretation of the interval, part c.

d. A director of purchasing at a major firm claims that the true mean Mach rating score of all purchasing managers is 85. Is there evidence to dispute this claim? From the given information we have 122 purchasing managers in the sample had a mean Mach rating score of 99.6, with a standard deviation of 12.6.

Short answer

a.The true mean Mach rating score of all purchasing managers is 99.6.

b. The 95% confidence interval for true mean is (97.364 ,101.839).

c. It can conclude that it is 95% confident that the true mean Mach rating score of all purchasing managers will lie between 97.364 and 101.839.

d. Because Mach rating score belong to the 95% confidence interval for the true mean Mach rating score of all purchasing managers.

Step-by-step solution

Given information

It is given that 122 purchasing managers in the sample had a mean Mach rating score of 99.6, with a standard deviation of 12.6. i.e.$n=122,\overline{x}=99.6,\sigma =12.6$

Estimating the true mean Mach rating score of all purchasing managers

a.

Since, sample mean is unbiased estimator of population mean.

So, the true mean Mach rating score of all purchasing managers is

$\begin{array}{rcl}\mu & =& \overline{x}\\ & =& 99.6\\ & & \end{array}$

Calculating the 95% confidence interval

b.

The 95% confidence interval for population mean is given by

$\left(\overline{x}-E<\mu <\overline{x}+E\right)$

The above figure shows that the area in each of the red-shaded tails is$\frac{\alpha }{2}=0.025$

Now, find $z_{\frac{\alpha }{2}}=1.96$by noting that the cumulative area to its left must be 1-0.025 or 0.975.

Now, refer to the standard normal table values to find the area of 0.975 corresponds to$z=1.96$

For the 95% confidence level, the critical value is therefore$z_{\frac{\alpha }{2}}=1.96$

To find the critical z score for the 95% confidence level, look up 0.975 in the body of the table.

Now, the margin of error is calculated by using the following formula:

$E=z_{\frac{\alpha }{2}}\times \frac{\sigma }{\sqrt{n}}$with $z_{\frac{\alpha }{2}}=1.96$

$\begin{array}{rcl}E& =& 1.96\times \frac{12.6}{\sqrt{122}}\\ & =& 1.96\times 1.1408\\ & =& 2.236\\ & & \end{array}$

Simply substitute and E in the general format for confidence interval is as follows:

$$\begin{gathered} \overline{x}-E<\mu <\overline{x}+E \\ 99.6-2.2359<\mu <99.6+2.2359 \\ 97.364<\mu <101.839 \end{gathered}$$

Interpretation

c.

It can conclude that it is 95% confident that the true mean Mach rating score of all purchasing managers will lie between 97.364 and 101.839.

Checking the evidence 

d.

Given that a director of purchasing at a major firm claims that the true mean Mach rating score of all purchasing managers is 85.

The evidence that the true mean Mach rating score of all purchasing managers is 85.

Because Mach rating score belong to the 95% confidence interval for the true mean Mach rating score of all purchasing managers.