Questions: The workers' union at a certain university is quite strong. About 96% of all workers employed by the university belong to the workers' union. Recently, the workers went on strike, and now a local TV station plans to interview a sample of 10 workers, chosen at random, to get their opinions on the strike. Answer the following. (a) Estimate the number of workers in the sample who are union members by giving the mean of the relevant distribution (that is, the expectation of the relevant random variable). Do not round your response. (b) Quantify the uncertainty of your estimate by giving the standard deviation of the distribution. Round your response to at least three decimal places.

Solution Steps

To estimate the number of workers in the sample who are union members, we calculate the mean of the relevant distribution. The mean (\(\mu\)) of a binomial distribution is given by the formula:

\[ \mu = n \times p \]

where:

• \(n = 10\) (the number of workers in the sample)
• \(p = 0.96\) (the probability that a worker is a union member)

Substituting the values, we find:

\[ \mu = 10 \times 0.96 = 9.6 \]

Next, we quantify the uncertainty of our estimate by calculating the standard deviation (\(\sigma\)) of the distribution. The standard deviation of a binomial distribution is given by the formula:

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

Substituting the values, we have:

\[ \sigma = \sqrt{10 \times 0.96 \times (1 - 0.96)} = \sqrt{10 \times 0.96 \times 0.04} \approx 0.620 \]

Final Answer