As a domain expert in statistics, I often encounter questions regarding the calculation of z-scores for confidence intervals. The z-score is a crucial component in determining the confidence interval for a population parameter based on a sample statistic. It is a measure of how many standard deviations an element is from the mean. When we talk about the z-score of a confidence interval, we're essentially looking for the point estimate plus or minus a certain number of standard deviations that will cover the desired confidence level. Let's break down the process step by step: ### Step 1: Determine the Confidence Level The first step in finding the z-score is to identify the confidence level you want for your interval. The confidence level represents the percentage of possible confidence intervals that will contain the true population parameter. Common confidence levels are 90%, 95%, and 99%. ### Step 2: Calculate the z-score Once you have the confidence level, you need to calculate the z-score that corresponds to the tail ends of the distribution outside the confidence interval. To do this, you divide the (1 - confidence level) by 2. For instance, if you want a 95% confidence interval, you would calculate (1 - 0.95) / 2 = 0.025. ### Step 3: Use the Z-Table The next step is to look up the calculated value from Step 2 in the standard normal (z) table, which provides the area to the left of the z-score. This will give you the z-value that corresponds to the desired confidence level. ### Step 4: Calculate the Sample Proportion If you're working with proportions, you'll need to calculate the sample proportion (\(\hat{p}\)) by dividing the number of successes (events) by the total number of trials or observations. ### Step 5: Calculate the Standard Error The standard error (SE) for a proportion is calculated using the formula SE = \(\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\), where \(n\) is the sample size. ### Step 6: Determine the Margin of Error Finally, the margin of error (ME) is calculated by multiplying the z-score from Step 3 by the standard error from Step 5: ME = z * SE. ### Step 7: Construct the Confidence Interval The confidence interval is then constructed as the point estimate (sample proportion or mean) plus or minus the margin of error. Now, let's address the provided reference content: - Step 1: The reference suggests dividing the confidence level by 2, which is incorrect. The correct approach is to find the area in each tail outside the confidence interval, which is (1 - confidence level) / 2. - Step 2: Looking up the value in the z-table is correct, but the calculation provided in Step 1 is incorrect based on the reference content. - Step 3: Calculating the sample proportion (\(\hat{p}\)) is correctly described. To correct the reference content: - For a 95% confidence level, you would calculate (1 - 0.95) / 2 = 0.025, not 0.475. - The z-value corresponding to an area of 0.025 in each tail (for a two-tailed test) is approximately 1.96. - The calculation of \(\hat{p}\) as 24/160 = 0.15 is correct if those are your sample data. Now, let's proceed with the translation into Chinese. read more >>

Step 1: Divide your confidence level by 2: .95/2 = 0.475. Step 2: Look up the value you calculated in Step 1 in the z-table and find the corresponding z-value. The z-value that has an area of .475 is 1.96. Step 3: Divide the number of events by the number of trials to get the --P-hat-- value: 24/160 = 0.15.Mar 6, 2018read more >>