To solve these problems, we need to use the properties of a binomial distribution, since we are dealing with a fixed number of trials (14 workers) and two possible outcomes (finding the job stressful or not).
(a) Mean Number of Workers Who Find Their Jobs Stressful
The mean \(\mu\) of a binomial distribution can be calculated using the formula:
\[ \mu = n \cdot p \]
where:
- \(n\) is the number of trials (workers selected)
- \(p\) is the probability of success (in this case, the probability that a worker finds their job stressful)
Here, we have:
- \(n = 14\)
- \(p = 0.60\) (60% of respondents find their jobs stressful)
Now we can calculate the mean:
\[ \mu = 14 \cdot 0.60 = 8.4 \]
Answer for (a):
The mean number of workers who find their jobs stressful in a sample of 14 workers is 8.40.
(b) Standard Deviation of the Number Who Find Their Jobs Stressful
The standard deviation \(\sigma\) of a binomial distribution can be calculated using the formula:
\[ \sigma = \sqrt{n \cdot p \cdot (1 - p)} \]
Using the values already provided:
- \(n = 14\)
- \(p = 0.60\)
- \(1 - p = 0.40\)
We can now calculate the standard deviation:
\[ \sigma = \sqrt{14 \cdot 0.60 \cdot 0.40} \]
Calculating inside the square root first:
\[ \sigma = \sqrt{14 \cdot 0.60 \cdot 0.40} = \sqrt{14 \cdot 0.24} = \sqrt{3.36} \]
Now, taking the square root:
\[ \sigma \approx 1.8330 \]
Answer for (b):
The standard deviation of the number who find their jobs stressful in a sample of 14 workers is approximately 1.8330.