Z-statistic

Definition

The z-statistic is a standardized test statistic used in hypothesis testing to determine the probability of observing a sample statistic given a hypothesized population parameter. It measures the number of standard deviations a sample statistic is from the hypothesized population mean, allowing for the assessment of statistical significance.

5 Must Know Facts For Your Next Test

1. The z-statistic is used to test hypotheses about a single population mean or proportion, as well as the difference between two population means or proportions.
2. The z-statistic follows a standard normal distribution, with a mean of 0 and a standard deviation of 1, allowing for the calculation of p-values and the assessment of statistical significance.
3. The formula for the z-statistic is $z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}$ for a single population mean, and $z = \frac{(\bar{x_1} - \bar{x_2}) - (\mu_1 - \mu_2)}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}$ for the difference between two population means.
4. The z-statistic is used to construct confidence intervals for population parameters, with the level of confidence determined by the chosen significance level (e.g., 95% confidence interval).
5. The choice between using a z-statistic or a t-statistic in hypothesis testing depends on whether the population standard deviation is known (z-statistic) or unknown (t-statistic).

Review Questions

• Explain the purpose of the z-statistic in the context of hypothesis testing for a single population mean.
  • The z-statistic is used in hypothesis testing for a single population mean to determine the probability of observing a sample mean given a hypothesized population mean. By standardizing the sample mean and comparing it to the hypothesized mean, the z-statistic allows researchers to assess the statistical significance of the difference and make inferences about the population parameter. The z-statistic follows a standard normal distribution, which enables the calculation of p-values and the construction of confidence intervals for the population mean.
• Describe how the z-statistic is used in hypothesis testing for the difference between two population means.
  • When testing the difference between two population means, the z-statistic is calculated as the difference between the two sample means, adjusted for the variability in each sample, and divided by the standard error of the difference. This standardized statistic follows a standard normal distribution and allows researchers to determine the probability of observing the sample difference under the null hypothesis that the population means are equal. The z-statistic is then used to calculate the p-value and make a decision about the statistical significance of the difference between the two population means.
• Analyze the role of the z-statistic in the context of hypothesis testing for a single population proportion and the difference between two population proportions.
  • The z-statistic is also used in hypothesis testing for a single population proportion and the difference between two population proportions. In these cases, the z-statistic is calculated by standardizing the sample proportion(s) and comparing them to the hypothesized population proportion(s). For a single proportion, the z-statistic measures the number of standard deviations the sample proportion is from the hypothesized population proportion. For the difference between two proportions, the z-statistic accounts for the variability in each sample proportion and the standard error of the difference. The z-statistic follows a standard normal distribution, allowing for the calculation of p-values and the construction of confidence intervals for the population proportions. This enables researchers to assess the statistical significance of the observed sample proportions and make inferences about the underlying population parameters.

Related terms

Hypothesis Testing:

A statistical method used to determine whether a hypothesis about a population parameter is likely to be true or false based on sample data.

Standard Error:

The standard deviation of the sampling distribution of a statistic, which measures the variability of the statistic across repeated samples.

Confidence Interval:

A range of values, calculated from sample data, that is likely to contain an unknown population parameter with a specified level of confidence.