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Proportion calculation is a crucial step in understanding any sample data. It tells us how one part of the data compares with the whole. In our scenario about government-sponsored insurance for nursing-home care, we start by calculating the proportion of letters opposing the bill. This is done using the formula:\[ p = \frac{\text{Number of opposing letters}}{\text{Total number of letters}} \]Given the problem, the number of letters that oppose the bill is 871, and the total number of letters is 1128. Therefore, our proportion is:\[ p = \frac{871}{1128} \approx 0.772 \]This calculation shows that approximately 77.2% of the letters were against the bill. This percentage provides a snapshot of the opinion among the sample received, and is pivotal in making any further statistical inference.

The standard error is a measure that tells us how much variation we might expect in our calculated proportion if we were to take different samples from the population. It helps us understand the reliability and precision of our proportion measurement.To calculate the standard error of a proportion, use the formula:\[ SE = \sqrt{\frac{p(1-p)}{n}} \]Where:- \( p \) is the sample proportion (0.772 in our case)- \( n \) is the total number of observations (1128 here)Inserting our values:\[ SE = \sqrt{\frac{0.772 \times (1 - 0.772)}{1128}} \approx 0.0125 \]This standard error of 0.0125 suggests that the sampling variability is low, which indicates our proportion is a fairly precise estimate of the true population proportion.

The Z-score is a statistical measure that helps us understand how far away a sample proportion is from a population proportion under the null hypothesis, expressed in terms of standard deviations. Essentially, it quantifies whether a result is statistically significant.To determine the Z-score in our scenario:\[ Z = \frac{p - 0.5}{SE} \]Here, \( p \) is 0.772 and \( SE \) is 0.0125, \[ Z = \frac{0.772 - 0.5}{0.0125} \approx 21.76 \]A Z-score of 21.76 is exceptionally high, indicating that the sample proportion is far greater than 0.5. By comparing this to a critical value (e.g., 1.96 for 95% confidence level), we can conclude that the proportion opposing the bill is statistically significant, meaning that it truly reflects a majority rather than a random fluctuation.

The null hypothesis is a fundamental part of statistical hypothesis testing, acting as a default or starting assumption that there is no effect or difference. In our specific exercise regarding letters opposing a bill, the null hypothesis posits that there is no preference in public opinion.Statistically, it is stated as:- Null Hypothesis \( H_0 \): The proportion of opposition is 0.5 (50%).This hypothesis assumes that the population is equally divided with no significant preference above or below 50%. When we perform a hypothesis test, we are searching for evidence to reject this null hypothesis.Based on our calculated Z-score of 21.76, which far exceeds typical critical values, the null hypothesis is rejected. This means the evidence strongly suggests more than half of the population opposes the bill, pointing to a skewed preference. However, it's worth noting that a statistical result like this should be interpreted with caution, as the sample may be biased towards those more inclined to express their opinions by writing letters.