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The sample proportion is a fundamental concept in statistics. It represents the fraction of a sample that has a particular attribute. For this exercise, it’s the fraction of respondents who answered 'Yes' to the survey question about being willing to pay higher taxes for deficit reduction. First, you need the total number of 'Yes' responses, denoted as X. Then, divide X by the total number of responses, denoted as n. Mathematically, it’s represented as \(\hat{p} = \frac{X}{n} \). So, if 150 out of 500 respondents said 'Yes', the sample proportion \(\hat{p}\) would be \(\frac{150}{500} = 0.3\). This value gives you a first insight into what portion of the population might hold this view. Breaking it down into a step-by-step process makes it easier to understand:

• Count the number of 'Yes' responses (X)
• Count the total number of responses (n)
• Divide X by n to get the sample proportion \(\hat{p}\)

Standard error measures the variability or the expected fluctuation in the sample proportion if we were to take multiple samples from the population. It provides insight into how much the sample proportion is likely to differ due to sampling variability. The formula for standard error (SE) is \( \text{SE} = \sqrt{ \frac{ \hat{p} ( 1 - \hat{p} )}{n} } \). For example, if the sample proportion \(\hat{p}\) is 0.3 and the sample size (n) is 500, the standard error would be \(\sqrt{ \frac{ 0.3 \times 0.7 }{ 500 } }\). Simplify the expression under the square root to get your final SE value. This step is essential because it helps in constructing confidence intervals:

• Calculate \(\hat{p} \times ( 1 - \hat{p} ) \)
• Divide by the total number of responses (n)
• Take the square root of the result to get the standard error (SE)

The z-value is a critical value from the standard normal distribution corresponding to the desired confidence level. It tells how many standard deviations away from the mean our value sits. For a 90% confidence interval, the z-value is 1.645. This value comes from z-tables or statistical software. It is used in the final step of constructing the confidence interval, multiplying it by the standard error. Here’s a straightforward way to understand how it's applied:

• Select the desired confidence level (e.g., 90%)
• Refer to the z-table to find the corresponding z-value, here it’s 1.645
• Use this z-value in calculating the confidence interval

In the formula for the confidence interval, \( \text{CI} = \hat{p} \pm ( z \times \text{SE})\), the z-value gives the range around the sample proportion within which we expect the true population proportion to lie with a specific level of confidence.