Question

3 0.03 0.12
4 0.42 0.46
5 0.41 0.28
a. what is the expected value of the job satisfaction score for senior executives (to 2 decimals)?
4.05
b. what is the expected value of the job satisfaction score for middle managers (to 2 decimals)?
3.84
c. compute the variance of job satisfaction scores for executives and middle managers (to 2 decimals).
executives 0.67
middle managers 0.89
d. compute the standard deviation of job satisfaction scores for both probability distributions (to 2 decimals).
executives 4.05
middle managers 3.84
e. what comparison can you make about the job satisfaction of senior executives and middle managers?
middle managers have higher satisfaction with less variation

Explanation:

Step1: Recall variance formula

The variance formula for a discrete - random variable $X$ is $\sigma^{2}=\sum(x - \mu)^{2}P(x)$, where $x$ are the possible values of the random variable, $\mu$ is the expected value, and $P(x)$ is the probability of $x$.

Step2: Calculate variance for executives

Let the job - satisfaction scores for executives be $x_1 = 3,x_2 = 4,x_3 = 5$ with probabilities $P(x_1)=0.03,P(x_2)=0.42,P(x_3)=0.41$ and expected value $\mu_{executives}=4.05$.
\[

$$\begin{align*} \sigma_{executives}^{2}&=(3 - 4.05)^{2}\times0.03+(4 - 4.05)^{2}\times0.42+(5 - 4.05)^{2}\times0.41\\ &=(- 1.05)^{2}\times0.03+(-0.05)^{2}\times0.42+(0.95)^{2}\times0.41\\ &=1.1025\times0.03 + 0.0025\times0.42+0.9025\times0.41\\ &=0.033075+0.00105 + 0.370025\\ &=0.40415\approx0.40 \end{align*}$$

\]

Step3: Calculate variance for middle managers

Let the job - satisfaction scores for middle managers be $x_1 = 3,x_2 = 4,x_3 = 5$ with probabilities $P(x_1)=0.12,P(x_2)=0.46,P(x_3)=0.28$ and expected value $\mu_{middle - managers}=3.84$.
\[

$$\begin{align*} \sigma_{middle - managers}^{2}&=(3 - 3.84)^{2}\times0.12+(4 - 3.84)^{2}\times0.46+(5 - 3.84)^{2}\times0.28\\ &=(-0.84)^{2}\times0.12+(0.16)^{2}\times0.46+(1.16)^{2}\times0.28\\ &=0.7056\times0.12+0.0256\times0.46 + 1.3456\times0.28\\ &=0.084672+0.011776+0.376768\\ &=0.473216\approx0.47 \end{align*}$$

\]

Step4: Recall standard - deviation formula

The standard deviation $\sigma=\sqrt{\sigma^{2}}$.

Step5: Calculate standard deviation for executives

$\sigma_{executives}=\sqrt{0.40415}\approx0.64$

Step6: Calculate standard deviation for middle managers

$\sigma_{middle - managers}=\sqrt{0.473216}\approx0.69$

Step7: Compare job satisfaction

The expected value of job - satisfaction for senior executives ($\mu_{executives}=4.05$) is higher than that for middle managers ($\mu_{middle - managers}=3.84$). The variance of job - satisfaction for senior executives ($\sigma_{executives}^{2}\approx0.40$) is lower than that for middle managers ($\sigma_{middle - managers}^{2}\approx0.47$). So, senior executives have higher average job satisfaction with less variation.

Answer:

c. Executives: $0.40$, Middle managers: $0.47$
d. Executives: $0.64$, Middle managers: $0.69$
e. Senior executives have higher average job satisfaction with less variation.