Final answer:
The p-value in this context is the probability of observing a sample proportion of union membership as extreme as 0.14 or more if the true population proportion is 0.114. The test statistic is calculated using a standard formula for a single proportion z-test. The exact p-value would be found using the calculated z-value and a z-table or statistical software.
Step-by-step explanation:
To determine the p-value for the hypothesis test regarding union membership, we start by stating the null hypothesis that the proportion of workers in unions is equal to the reported proportion of 11.4%. The alternative hypothesis would be that the proportion has increased. The sample proportion (p') from the 300 workers is 42/300 = 0.14. The test statistic for a proportion is calculated using the formula:
z = (p' - p0) / sqrt(p0(1-p0)/n)
Where:
- p0 = 0.114 (reported proportion)
- n = 300 (sample size)
Substituting the values:
z = (0.14 - 0.114) / sqrt(0.114(1-0.114)/300)
Upon calculation, the test statistic z-value is obtained. To find the p-value, we use the z-value and with technology or a z-table, we can determine the probability of observing a sample proportion as extreme as 0.14, if the true proportion is indeed 0.114. This p-value is then compared to the significance level (which was not provided in the question), and if the p-value is less than the significance level, we reject the null hypothesis. The p-value is the probability of observing a test statistic as extreme as the sample's, assuming the null hypothesis is true. For this specific case, because the actual numbers are not given, the p-value would be compared against the predetermined significance level to make a decision regarding the null hypothesis.