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key term - Z*

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Definition

z* is the critical value from the standard normal distribution that corresponds to a given confidence level. It is used in the calculation of confidence intervals for population parameters, such as the population proportion and the population mean.

5 Must Know Facts For Your Next Test

  1. The value of z* depends on the desired confidence level, with higher confidence levels corresponding to larger values of z*.
  2. For a 95% confidence level, the value of z* is approximately 1.96.
  3. The formula for a confidence interval for a population proportion is $\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$.
  4. The formula for a confidence interval for a population mean is $\bar{x} \pm z^* \frac{s}{\sqrt{n}}$.
  5. The value of z* is used in both the population proportion and population mean confidence interval formulas to account for the uncertainty in the sample statistic.

Review Questions

  • Explain the role of z* in the calculation of a confidence interval for a population proportion.
    • The value of z* is a critical component in the formula for calculating a confidence interval for a population proportion. It represents the z-score from the standard normal distribution that corresponds to the desired confidence level, typically 95%. The z* value is multiplied by the standard error of the sample proportion to determine the margin of error, which is then added and subtracted from the sample proportion to obtain the upper and lower bounds of the confidence interval. The z* value accounts for the uncertainty in the sample statistic and ensures that the true population proportion is captured within the calculated interval with the specified level of confidence.
  • Describe how the value of z* changes as the desired confidence level is increased or decreased.
    • The value of z* is directly related to the desired confidence level. As the confidence level increases, the value of z* also increases. For example, at a 90% confidence level, the value of z* is approximately 1.645, while at a 95% confidence level, the value of z* is approximately 1.96, and at a 99% confidence level, the value of z* is approximately 2.576. The higher the confidence level, the more certainty there is that the true parameter value will fall within the calculated confidence interval, but this comes at the cost of a wider interval width. Conversely, as the confidence level decreases, the value of z* also decreases, resulting in a narrower confidence interval but with less certainty that the true parameter value is captured within the interval.
  • Analyze the impact of the sample size on the value of z* and the width of the confidence interval for a population proportion.
    • The value of z* is independent of the sample size, as it is a standardized value from the normal distribution. However, the sample size does have a significant impact on the width of the confidence interval for a population proportion. As the sample size increases, the standard error of the sample proportion decreases, which in turn reduces the margin of error and narrows the width of the confidence interval. This means that larger sample sizes result in more precise estimates of the population proportion, with the true value being more likely to fall within the calculated interval. Conversely, smaller sample sizes lead to wider confidence intervals, as the standard error is larger, and the z* value has a greater influence on the margin of error. Therefore, the sample size is a crucial factor in determining the accuracy and reliability of the confidence interval for a population proportion.

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