Questions: In a poll about work, 71% of respondents said that their jobs were sometimes or always stressful. 12 workers are chosen at random. Using a cutoff of 0.05, would it be unusual if fewer than 5 of them find their jobs stressful?

In a poll about work, 71% of respondents said that their jobs were sometimes or always stressful. 12 workers are chosen at random. Using a cutoff of 0.05, would it be unusual if fewer than 5 of them find their jobs stressful?

Transcript text: in a poll about work, $71 \%$ of respondents said that their jobs were sometimes or always stressful. 12 workers are chosen at random. using a cutoff of 0.05 , would it be unusual if fewer that 5 of them find their jobs stressful?

Solution

Solution Steps

Step 1: Define the Problem

In a poll, $71\%$ of respondents indicated that their jobs were sometimes or always stressful. We randomly select $n = 12$ workers. We want to determine if it is unusual for fewer than $5$ of them to find their jobs stressful, using a significance level of $\alpha = 0.05$.

Step 2: Calculate the Probability

We need to calculate the probability of $X$, the number of workers finding their jobs stressful, being less than $5$. This can be expressed as:

\[ P(X < 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) \]

Using the binomial probability formula:

\[ P(X = x) = \binom{n}{x} \cdot p^x \cdot q^{n-x} \]

where $p = 0.71$ and $q = 1 - p = 0.29$.

The calculated probabilities for each case are: