Z*

Definition

The z* value, also known as the critical z-value, is a standardized score that represents the boundary between the acceptance and rejection regions in a hypothesis test or the construction of a confidence interval. It is a key parameter used in statistical inference to determine the probability of observing a particular sample statistic under the null hypothesis.

5 Must Know Facts For Your Next Test

1. The z* value is used to determine the critical value(s) for a hypothesis test or the margin of error in a confidence interval.
2. The specific value of z* depends on the desired level of significance (α) and the direction of the hypothesis test (one-tailed or two-tailed).
3. In a 95% confidence interval, the z* value is approximately 1.96, which corresponds to a significance level of α = 0.05.
4. The z* value is a standardized normal distribution value that represents the probability of observing a sample statistic under the null hypothesis.
5. The z* value is a crucial parameter in the construction of confidence intervals for population parameters, such as the population mean or proportion.

Review Questions

• Explain the role of the z* value in a hypothesis test.
  • In a hypothesis test, the z* value represents the critical value that separates the acceptance and rejection regions. The z* value is determined based on the desired significance level (α) and the direction of the test (one-tailed or two-tailed). The sample test statistic is compared to the z* value to decide whether to reject or fail to reject the null hypothesis. The z* value helps to control the probability of making a Type I error, which is the error of rejecting the null hypothesis when it is true.
• Describe how the z* value is used in the construction of a confidence interval.
  • The z* value is a crucial parameter in the construction of a confidence interval for a population parameter, such as the mean or proportion. The z* value is used to calculate the margin of error, which is added and subtracted from the sample statistic to create the confidence interval. The specific z* value used depends on the desired level of confidence, typically 90%, 95%, or 99%. For example, in a 95% confidence interval, the z* value is approximately 1.96, which corresponds to a significance level of α = 0.05.
• Analyze the relationship between the z* value, the significance level, and the probability of observing a sample statistic under the null hypothesis.
  • The z* value is directly related to the significance level (α) and the probability of observing a sample statistic under the null hypothesis. As the significance level decreases (e.g., from 0.10 to 0.05 to 0.01), the z* value increases, indicating a more extreme value is required to reject the null hypothesis. This is because a lower significance level corresponds to a lower probability of making a Type I error, or rejecting the null hypothesis when it is true. Conversely, as the significance level increases, the z* value decreases, and the probability of observing a sample statistic under the null hypothesis increases. The z* value, therefore, is a critical parameter that balances the trade-off between the risk of a Type I error and the power of the statistical test.

Related terms

Hypothesis Test: A statistical procedure used to determine whether a claim or hypothesis about a population parameter is supported by sample data.

Confidence Interval:

A range of values that is likely to contain an unknown population parameter with a specified level of confidence.

Standardized Score: A measure of the distance between a data point and the mean of a distribution, expressed in terms of the standard deviation.