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Proportions in statistics represent a part of a whole and are often expressed as a fraction or percentage. They help us understand the proportion of interest in a dataset. In hypothesis testing, proportions can tell us if there’s a significant change or difference in sample proportions over time or between groups.
In the given exercise, the proportion of U.S. cotton shipments to Germany that were sent for arbitration was compared between two years, 1964 and 1965. We calculated the proportion for 1964 as 0.40, reflecting 40%, and for 1965 as 0.50, reflecting 50%. This comparison is to determine if the increase in complaints in 1965 might signify a real issue or if it could just be due to chance.
To make comparisons easier, we often calculate a combined proportion, which helps in establishing a uniform proportion level that represents the expected complaint rate if the difference was purely by chance. This is crucial for calculating further statistical metrics such as the z-score, which will test the significance of the observed difference.

The z-score is a statistical metric used to determine the number of standard deviations an element is from the mean. It's a fundamental part of hypothesis testing, as it helps in comparing a sample proportion to a population proportion or another sample proportion.
In this exercise, the z-score was calculated to examine how far the 1965 proportion of complaints deviates from the 1964 proportion. By using the formula:\[ z = \frac{(p_2 - p_1) - 0}{\sqrt{p(1-p)(\frac{1}{n_1}+\frac{1}{n_2})}} \]where \(p_1\) and \(p_2\) refer to the proportions of the different years, and \(n_1\) and \(n_2\) represent the sample sizes, the z-score tells us whether the observed differences could likely be due to statistical variation alone.
In this scenario, the computed z-score came out to approximately 1.49. This value will now be used to compare against the critical values to make an informed decision about the significance of the difference observed.

Critical values are threshold values that define regions where your observed test statistic would lead you to reject the null hypothesis. These values are closely related to confidence levels in hypothesis testing.
To determine if the difference in proportions between 1964 and 1965 is significant, we compare the calculated z-score against the critical value. For a two-tailed test, with a 95% confidence level, the critical values are ±1.96. This means any z-score beyond these thresholds would suggest the difference is statistically significant.
Here, the calculated z-score was 1.49, which lies within the interval of -1.96 to +1.96. Hence, it indicates that the observed difference in proportions does not exceed the critical value range. This suggests that the observed differences might be due to random chance rather than a true deterioration of conditions. The critical value acts as a benchmark, guiding whether the findings in the sample can be generalized to the population.

The significance level, often denoted by α, is a probabilistic metric that helps in hypothesis testing. It represents the risk of concluding that a difference exists when there is none—a Type I error. Commonly set at 0.05 (5%), this level provides a cut-off for decision-making.
In the exercise, a 95% confidence level was assumed, equivalent to a significance level of 0.05, reflecting that there is 5% risk of wrongly claiming that the proportions of complaints in the shipments have significantly increased.
When conducting a hypothesis test, this significance level determines the critical value thresholds. If the z-score exceeds these thresholds, the findings are statistically significant. Otherwise, as in this scenario, when the z-score of 1.49 falls within the critical range, it suggests that there is no statistically significant change, affirming that variations might be due to random sampling variability.
Understanding the significance level allows us to judge the reliability of our statistical conclusions. It outlines how confident we are in our data representations and decisions.