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The Z-score is a statistical measure that helps determine how far a single data point is from the mean in terms of standard deviations. In hypothesis testing, it is used to decide whether to reject the null hypothesis. To compute the Z-score, we take the difference between the sample proportion (\( p_{\text{sample}} \) which was 0.13 in our example) and the hypothesized proportion (\( p_0 \) which was 0.125). Then, we divide that difference by the standard error (SE). In our exercise:\[ Z = \frac{0.13 - 0.125}{0.0164} \approx 0.305 \]It tells us how many standard errors the sample proportion is away from the hypothesized proportion. A Z-score close to zero indicates that the sample proportion is very close to the hypothesized value, while a Z-score farther from zero suggests a substantial difference.

The Standard Error (SE) quantifies the amount of variation or "spread" that can be expected from the sample proportion when comparing to the true proportion. It acts as an important component in deriving the Z-score, giving us a reference for the variability within the sample.The formula to calculate the Standard Error of the sample proportion is:\[ SE = \sqrt{\frac{p_0 (1 - p_0)}{n}} \]where:- \( p_0 \) is the hypothesized proportion (0.125 in our problem),- \( n \) is the sample size (400 for our example).Using these values, we calculated:\[ SE = \sqrt{\frac{0.125 \times 0.875}{400}} \approx 0.0164 \]This SE serves as the denominator in our Z-score formula and helps determine how significant the results of the hypothesis test are.

The p-value is a crucial element in hypothesis testing. It measures the probability of observing a sample statistic as extreme or more extreme than the one observed, assuming the null hypothesis is true. In this context, a smaller p-value indicates stronger evidence against the null hypothesis. Our exercise sought a right-tailed test, aiming to prove an increase in union membership. Thus, we found the p-value corresponding to the Z-score of 0.305:Approximately 0.3803.Understanding this value:- A p-value greater than the significance level \( \alpha = 0.05 \) suggests insufficient evidence to reject the null hypothesis.- Conversely, a p-value less than \( \alpha \) would indicate strong evidence to reject it.For our scenario, a p-value of 0.3803 was not significant enough to assert an increase in union membership.

Formulating hypotheses is the first and most critical step in hypothesis testing. It sets the stage for the analysis and what the test aims to prove or disprove.- **Null Hypothesis (\( H_0 \))**: This posits that there is no effect or change in the parameter being tested. In our problem, it was stated as \( H_0: p = 0.125 \). This means that, statistically, the proportion of workers in unions hasn't increased.- **Alternative Hypothesis (\( H_a \))**: This asserts the opposite of the null hypothesis and what we wish to prove. Here, it was \( H_a: p > 0.125 \), indicating the belief that union membership has increased.This framework is crucial because the aim of hypothesis testing is to determine whether there is enough statistical evidence to reject the null hypothesis in favor of the alternative.