Question

1. the personality characteristics of business leaders (e.g., ceos) are related to the operations of the businesses that they lead (oreg & berson, 2018). traits like openness to experience are related to positive financial outcomes and other traits are related to negative financial outcomes for their businesses. suppose that a board of directors is interested in evaluating the personality of their leadership. among a sample of $n = 16$ managers, the sample mean of the openness to experiences dimension of personality was $m = 4.50$. assuming that $mu = 4.24$ and $sigma = 1.05$ (cobb - clark & schurer, 2012), use a two - tailed hypothesis test with $alpha =.05$ to test the hypothesis that this company’s business leaders’ openness to experience is different from the population.

Explanation:

Step1: State the hypotheses

The null hypothesis \(H_0:\mu = 4.24\), the alternative hypothesis \(H_1:\mu
eq4.24\)

Step2: Calculate the z - statistic

The formula for the z - statistic in a one - sample z - test is \(z=\frac{M - \mu}{\frac{\sigma}{\sqrt{n}}}\)
Substitute \(M = 4.50\), \(\mu=4.24\), \(\sigma = 1.05\), and \(n = 16\) into the formula:

$$
LATEXBLOCK0
$$

Step3: Determine the critical values

For a two - tailed test with \(\alpha=0.05\), the critical values are \(z_{\alpha/2}=\pm1.96\)

Step4: Make a decision

Since \(|z| = |0.99|\lt1.96\), we fail to reject the null hypothesis.

Answer:

We fail to reject the null hypothesis \(H_0\). There is not enough evidence at the \(\alpha = 0.05\) level to conclude that the company's business leaders' openness to experience is different from the population.