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In a poll about work, 45% of respondents said that their jobs were sometimes or always stressful. Twelve workers are chosen at random.

(a) What is the mean number who find their jobs stressful in a sample of 12 workers? Round the answer to two decimal places.
(b) What is the standard deviation of the number who find their jobs stressful in a sample of 12 workers? Round the answer to four decimal places.

3 answers

To find the mean and standard deviation of the number of workers who find their jobs stressful, we can model this situation using a binomial distribution. In this case, the parameters are defined as follows:

  • \( n = 12 \) (the number of trials or workers)
  • \( p = 0.45 \) (the probability of success, which in this case refers to finding the job stressful)

(a) Mean Calculation

The mean \( \mu \) of a binomial distribution is given by the formula:

\[ \mu = n \cdot p \]

Now, plugging in the values:

\[ \mu = 12 \cdot 0.45 = 5.4 \]

Thus, the mean number of workers who find their jobs stressful in a sample of 12 workers is 5.40 (rounded to two decimal places).

(b) Standard Deviation Calculation

The standard deviation \( \sigma \) of a binomial distribution is given by the formula:

\[ \sigma = \sqrt{n \cdot p \cdot (1 - p)} \]

Now, calculate \( \sigma \) by substituting the values:

\[ \sigma = \sqrt{12 \cdot 0.45 \cdot (1 - 0.45)} = \sqrt{12 \cdot 0.45 \cdot 0.55} \]

Calculating inside the square root:

\[ = \sqrt{12 \cdot 0.45 \cdot 0.55} = \sqrt{2.97} \approx 1.724 \]

Thus, the standard deviation of the number of workers who find their jobs stressful in a sample of 12 workers is approximately 1.7240 (rounded to four decimal places).

To summarize the results:

  • (a) Mean: 5.40
  • (b) Standard Deviation: 1.7240
B is incorrect

Let’s recalculate the standard deviation to ensure accuracy.

The standard deviation \( \sigma \) for a binomial distribution is calculated using the formula:

\[ \sigma = \sqrt{n \cdot p \cdot (1 - p)} \]

Given:

  • \( n = 12 \)
  • \( p = 0.45 \)
  • Thus, \( 1 - p = 0.55 \)

Plugging in the values:

  1. Calculate \( n \cdot p \cdot (1 - p) \):

\[ n \cdot p \cdot (1 - p) = 12 \cdot 0.45 \cdot 0.55 \]

Calculating step-by-step:

  • First, calculate \( 0.45 \cdot 0.55 \):

\[ 0.45 \cdot 0.55 = 0.2475 \]

  • Now multiply by \( n = 12 \):

\[ 12 \cdot 0.2475 = 2.97 \]

  1. Now, take the square root to find the standard deviation:

\[ \sigma = \sqrt{2.97} \]

Calculating \( \sqrt{2.97} \):

\[ \sigma \approx 1.7240 \]

Thus, my previous answer for the standard deviation was indeed evaluated correctly.

So the results are:

  • (a) Mean: 5.40
  • (b) Standard Deviation: 1.7240

If you believe there is still an error, please confirm the calculations or parameters provided so we can address any discrepancies correctly.