Quantitative Business Analysis

1 Quantitative Business Analysis The selected scenarios evaluate the effectiveness of a lie detector program for university staff.

Parameters P = the population proportion of university staff accurately detected as lying by the lie detector program.

The example considers a scenario where a university has implemented a new lie detector program for staff members during sensitive investigations, such as academic misconduct or financial irregularities. The university wants to assess whether the lie detector program effectively detects lies.

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Hypotheses H0: p = 0.50 (Null Hypothesis) Ha: p > 0.50 (Right-tailed Alternative Hypothesis) In this scenario, the null hypothesis (H0) states that the proportion of university staff accurately detected as lying by the lie detector program is equal to 50%. The alternative hypothesis (Ha) states that the proportion of staff accurately detected as lying is greater than 50%, indicating the effectiveness of the lie detector program.

Test Statistic Formula Since the test deals with the population, mean (μ), and the sample size is typically large, a z-test can be appropriate for hypothesis testing (Lind et al., 2021). The hypothesis testing would use the z-test formula: Z = (x̅ – μ0) / (σ /√n). However, instead of the mean x̅, there would be the use of the population so that the actual formula is a z-test for proportions as the scenario fits.

Thus, Z = (P-P0)/( √(P0(1-P0)/n). In this formula, the denominator, the standard error used as an estimate of the standard deviation of the sampling distribution of the sample proportion for measuring the variability of sample proportions that we would expect to observe if we repeated the sampling process multiple times.

Where: 

P is the sample population of university staff accurately detected as lying



P0 is the hypothesized value of population proportion under the null hypothesis



n is the sample size.

3 Suppose in our sample of 9 staff members out of a population of 45, 6 were accurately detected as lying by the lie detector program. Therefore, P=6/9=0.67(approximately). An assumption is set for the significance level at α=0.05 (5%).

Calculations: SE=(0.50(1−0.50))/9 ≈0.1581 Z=(0.67−0.50)/0.1581 ≈1.0759 For a right-tailed test at α=0.05, the critical z-value is approximately 1.645. Since the calculated z-value of 1.0759 is less than the critical z-value of 1.645, we fail to reject the null hypothesis (H0). There is not enough evidence to suggest that the proportion of university staff accurately detected as lying by the lie detector program is greater than 50%. The lie detector program's effectiveness is not statistically significant at the 5% significance level.

The Tail of the Test: The hypothesis test is right-tailed because we want to determine whether the proportion of staff accurately detected as lying exceeds the assumed baseline proportion of 50%.

Type I and Type II Errors In the context of the lie detector program, a Type I Error would occur if we reject the null hypothesis (H0) when it is true. This would mean concluding that the lie detector program effectively detects lies when, in reality, it is not, potentially leading to unjust consequences for staff members.

Type II Error will occur if the results fail to reject the null hypothesis (H0) when it is false. This would mean not recognizing the effectiveness of the lie detector program, and the

4 university might continue to use an ineffective program, missing out on an opportunity to improve investigative accuracy.

Level of Significance The level of significance, denoted by the Greek symbol alpha (α), represents the probability of making a Type I Error. It is the threshold at which a decision to reject the null hypothesis occurs. Commonly used values for α are 0.05 or 0.01, corresponding to a 5% and 1% probability of making a Type I Error, respectively.

The choice of the level of significance depends on the consequences of making Type I and Type II Errors and the level of certainty required in the decision-making process. In this scenario, there can be a selection of α =0.05 since it is a common choice for many applications and balances the risk of Type I and Type II Errors.

5 Reference Lind, D. A., Marchal, W. G., & Wathen, S. A. (2021). Statistical techniques in business and economics with Connect (18th ed.). McGraw Hill Education.