Z-Score Formula

Definition

The z-score formula is a standardized measure that expresses the relationship between a data point and the mean of a dataset in terms of the number of standard deviations. It is a fundamental concept in statistics that is widely used in hypothesis testing and other statistical analyses.

5 Must Know Facts For Your Next Test

1. The z-score formula is $z = \frac{x - \mu}{\sigma}$, where $x$ is the data point, $\mu$ is the population mean, and $\sigma$ is the population standard deviation.
2. Z-scores are used to standardize data, allowing for comparisons between different datasets or variables with different units or scales.
3. In hypothesis testing, the z-score is used to calculate the test statistic, which is then compared to a critical value to determine the statistical significance of the results.
4. For a single mean hypothesis test, the z-score is calculated using the sample mean, sample standard deviation, and hypothesized population mean.
5. For a single proportion hypothesis test, the z-score is calculated using the sample proportion, hypothesized population proportion, and the standard error of the proportion.

Review Questions

• Explain the purpose of the z-score formula and how it is used in the context of hypothesis testing of a single mean.
  • The z-score formula is used to standardize data by expressing a data point in terms of the number of standard deviations it is from the mean. In the context of hypothesis testing of a single mean, the z-score is calculated using the sample mean, sample standard deviation, and the hypothesized population mean. The z-score is then used as the test statistic, which is compared to a critical value to determine the statistical significance of the results and make a decision about the null hypothesis.
• Describe how the z-score formula is used in the context of hypothesis testing of a single proportion.
  • For hypothesis testing of a single proportion, the z-score formula is used to calculate the test statistic. In this case, the z-score is calculated using the sample proportion, the hypothesized population proportion, and the standard error of the proportion. The z-score is then compared to a critical value to determine the statistical significance of the results and make a decision about the null hypothesis. The z-score allows for the comparison of the sample proportion to the hypothesized population proportion, taking into account the variability in the sample.
• Analyze the relationship between the z-score, the normal distribution, and the statistical significance of the results in hypothesis testing.
  • The z-score formula is closely linked to the normal distribution, as it expresses a data point in terms of the number of standard deviations from the mean. In hypothesis testing, the z-score is used to calculate the test statistic, which is then compared to a critical value from the standard normal distribution. The closer the z-score is to the critical value, the more likely the null hypothesis is to be true. Conversely, a z-score that is far from the critical value indicates a low probability of the null hypothesis being true, and the results are considered statistically significant. The z-score, normal distribution, and statistical significance are all interconnected concepts that are fundamental to hypothesis testing.

Related terms

Standard Deviation: A measure of the spread or dispersion of a dataset, indicating the average distance of data points from the mean.

Hypothesis Testing: A statistical method used to determine whether a particular claim or hypothesis about a population parameter is likely to be true.

Normal Distribution: A symmetric, bell-shaped probability distribution that is commonly used in statistical analysis due to its properties and widespread occurrence in nature.