Z-scores

Definition

A z-score, also known as a standard score, is a statistical measure that represents the number of standard deviations a data point is from the mean of a dataset. It is used to standardize and compare data points across different distributions.

5 Must Know Facts For Your Next Test

1. Z-scores are calculated by subtracting the mean from a data point and dividing the result by the standard deviation.
2. Z-scores allow you to compare data points from different distributions by converting them to a common scale.
3. In a normal distribution, approximately 68% of data points fall within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations.
4. Z-scores are used to determine the probability of a data point occurring within a normal distribution, which is essential for hypothesis testing.
5. Negative z-scores indicate data points that fall below the mean, while positive z-scores indicate data points that fall above the mean.

Review Questions

• Explain how z-scores are calculated and their purpose in the context of 8.2 A Single Population Mean Using the Student's t-Distribution.
  • In the context of 8.2 A Single Population Mean Using the Student's t-Distribution, z-scores are used to standardize the sample mean by converting it to a common scale. The z-score is calculated by subtracting the population mean from the sample mean and dividing the result by the population standard deviation. This allows for the comparison of the sample mean to the hypothesized population mean and the determination of the probability of the sample mean occurring under the null hypothesis, which is essential for conducting a hypothesis test on a single population mean.
• Describe the relationship between z-scores, the normal distribution, and the assumptions required for using the Student's t-Distribution in 8.2 A Single Population Mean.
  • The Student's t-Distribution is used when the population standard deviation is unknown, and the sample size is small (n < 30). In this case, z-scores cannot be directly calculated, as the population standard deviation is not known. Instead, the t-statistic is used, which follows the Student's t-Distribution. The assumptions for using the Student's t-Distribution in 8.2 A Single Population Mean include that the population is normally distributed and the sample is randomly selected. The z-scores are still important, as they provide a standardized measure to compare the sample mean to the hypothesized population mean and determine the probability of the observed result.
• Analyze how z-scores are utilized in the decision-making process for hypothesis testing on a single population mean in the context of 8.2 A Single Population Mean Using the Student's t-Distribution.
  • In the context of 8.2 A Single Population Mean Using the Student's t-Distribution, z-scores are used to determine the p-value, which is the probability of obtaining a test statistic as extreme or more extreme than the observed value, given that the null hypothesis is true. The p-value is then compared to the chosen significance level to make a decision about the null hypothesis. If the p-value is less than the significance level, the null hypothesis is rejected, indicating that the sample mean is significantly different from the hypothesized population mean. The z-score provides a standardized measure to quantify the strength of the evidence against the null hypothesis, which is crucial for the decision-making process in hypothesis testing.

Related terms

Standard Deviation: A measure of the amount of variation or dispersion of a set of values from the mean value.

Normal Distribution: A symmetric, bell-shaped probability distribution where most values cluster around 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.