Glossary

8.2 One-Sample Z-Test and T-Test for Means

3 min read•july 23, 2024

One-sample tests for means are crucial tools in statistics. They help us determine if a sample's average differs significantly from a known or hypothesized population . These tests come in two flavors: z-tests for large samples or known population standard deviations, and t-tests for smaller samples with unknown standard deviations.

Understanding when to use each test and how to interpret the results is key. By calculating test statistics and comparing them to critical values or p-values, we can make informed decisions about our hypotheses. This process allows us to draw meaningful conclusions from our data in various real-world scenarios.

One-Sample Tests for Means

Z-test vs t-test for means

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  • Use a z-test when the population (σ\sigma) is known or the is large (n ≥ 30), even if the population standard deviation is unknown
  • Use a t-test when the population standard deviation (σ\sigma) is unknown and the sample size is small (n < 30)
  • Example: Testing the mean weight of a product with a known standard deviation from historical data (z-test) vs testing the mean height of students in a small class (t-test)

One-sample z-test for means

  • State the null and alternative hypotheses
    • H0:μ=μ0H_0: \mu = \mu_0 population mean equals the hypothesized value
    • H1:μ≠μ0H_1: \mu \neq \mu_0 two-tailed, μ>μ0\mu > \mu_0 right-tailed, or μ<μ0\mu < \mu_0 left-tailed
  • Calculate the using the formula z=xˉ−μ0σ/nz = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}
    • xˉ\bar{x} represents the sample mean
    • μ0\mu_0 represents the hypothesized population mean
    • σ\sigma represents the population standard deviation
    • nn represents the sample size
  • Compare the calculated z-score to the critical z-value or use the p-value to make a decision
    • Reject H0H_0 if ∣z∣>zα/2|z| > z_{\alpha/2} two-tailed, z>zαz > z_{\alpha} right-tailed, or z<−zαz < -z_{\alpha} left-tailed
    • α\alpha represents the significance level (commonly 0.05)
  • Example: Testing if the mean weight of a product differs from the advertised weight

One-sample t-test for means

  • State the null and alternative hypotheses (same as z-test)
  • Calculate the t-score using the formula t=xˉ−μ0s/nt = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}
    • ss represents the sample standard deviation
  • Determine the degrees of freedom df=n−1df = n - 1
  • Compare the calculated t-score to the critical or use the p-value to make a decision
    • Reject H0H_0 if ∣t∣>tα/2|t| > t_{\alpha/2} two-tailed, t>tαt > t_{\alpha} right-tailed, or t<−tαt < -t_{\alpha} left-tailed
  • Example: Testing if the mean height of students in a small class differs from the national average

P-values in mean testing

  • The p-value represents the probability of obtaining a test statistic as extreme as or more extreme than the observed value, assuming the is true
  • Calculate the p-value using the z-score or t-score and the corresponding distribution (standard normal or t-distribution)
  • Interpret the p-value
    1. Reject H0H_0 (statistically significant result) if p-value ≤ α\alpha
    2. Fail to reject H0H_0 (statistically insignificant result) if p-value > α\alpha
  • A smaller p-value provides stronger evidence against the null hypothesis
  • Example: A p-value of 0.01 indicates strong evidence against the null hypothesis, while a p-value of 0.25 suggests weak evidence against the null hypothesis

Key Terms to Review (18)

Alternative Hypothesis: The alternative hypothesis is a statement that contradicts the null hypothesis, suggesting that there is an effect, a difference, or a relationship in the population. It serves as the focus of research, aiming to provide evidence that supports its claim over the null hypothesis through statistical testing and analysis.
Confidence Level: The confidence level is a statistical measure that reflects the degree of certainty in an estimate, typically expressed as a percentage. It indicates the proportion of times that a statistical procedure will produce an interval that contains the true parameter if the procedure were repeated numerous times. This concept is vital in constructing confidence intervals, conducting hypothesis tests, determining sample sizes, and understanding errors in statistical analysis.
Independence: Independence refers to the condition where two or more events or variables do not influence each other. In statistics, it is a crucial concept that indicates that the occurrence of one event does not affect the probability of another event happening. This idea is foundational in many statistical analyses, including hypothesis testing, regression analysis, and various non-parametric methods.
Margin of error: The margin of error is a statistic that expresses the amount of random sampling error in a survey's results. It provides an estimate of the uncertainty around a sample statistic, helping to convey how much the results may differ from the true population value. This concept is crucial when interpreting data, as it indicates the range within which the true value is likely to fall and connects closely to confidence levels and sample size.
Market Research: Market research is the process of gathering, analyzing, and interpreting information about a market, including information about the target audience, competitors, and overall industry trends. It helps businesses understand their customers' needs and preferences, enabling them to make informed decisions regarding product development, marketing strategies, and sales approaches.
Mean: The mean, often referred to as the average, is a measure of central tendency that is calculated by summing all values in a dataset and dividing by the total number of values. This concept is crucial for making informed decisions based on data analysis, as it provides a single value that represents the overall trend in a dataset.
Normal distribution: Normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. This characteristic forms a bell-shaped curve, which is significant in various statistical methods and analyses.
Null hypothesis: The null hypothesis is a statement that assumes there is no effect or no difference in a given situation, serving as a default position that researchers aim to test against. It acts as a baseline to compare with the alternative hypothesis, which posits that there is an effect or a difference. This concept is foundational in statistical analysis and hypothesis testing, guiding researchers in determining whether observed data can be attributed to chance or if they suggest significant effects.
One-Sample T-Test: A one-sample t-test is a statistical method used to determine if the mean of a single sample significantly differs from a known population mean. This test is especially useful when the sample size is small (typically less than 30) and the population standard deviation is unknown, making it crucial for situations where data is limited or hard to obtain.
One-sample z-test: A one-sample z-test is a statistical method used to determine whether the mean of a single sample differs significantly from a known population mean when the population variance is known. This test is especially useful when dealing with large sample sizes, typically n > 30, as it assumes that the sampling distribution of the sample mean is approximately normal due to the Central Limit Theorem. The z-test provides a way to make inferences about population parameters based on sample data.
Quality Control: Quality control refers to the processes and measures implemented to ensure that products or services meet specified quality standards and requirements. This concept is crucial in maintaining consistency, minimizing defects, and enhancing customer satisfaction through statistical methods and inspections.
Random Sampling: Random sampling is a statistical technique used to select a subset of individuals from a larger population in such a way that every individual has an equal chance of being chosen. This method ensures that the sample accurately represents the population, minimizing bias and allowing for reliable inferences to be made about the larger group.
Sample size: Sample size refers to the number of observations or data points included in a statistical sample, which is crucial for ensuring the reliability and validity of the results. A larger sample size can lead to more accurate estimates and stronger statistical power, while a smaller sample size may result in less reliable outcomes. Understanding the appropriate sample size is essential for various analyses, as it affects the confidence intervals, error rates, and the ability to detect significant differences or relationships within data.
Standard Deviation: Standard deviation is a statistical measure that quantifies the amount of variation or dispersion of a set of values. It indicates how much individual data points deviate from the mean (average) of the data set, helping to understand the spread and reliability of the data in business contexts.
T-value: The t-value is a statistic that measures the size of the difference relative to the variation in your sample data. It is used in hypothesis testing to determine if there is a significant difference between the means of a sample and a known population mean. In essence, the t-value helps assess how far the sample mean deviates from the population mean, taking into account the sample size and variability.
Type I Error: A Type I error occurs when a null hypothesis is incorrectly rejected when it is actually true, leading to a false positive conclusion. This concept is crucial in statistical hypothesis testing, as it relates to the risk of finding an effect or difference that does not exist. Understanding the implications of Type I errors helps in areas like confidence intervals, model assumptions, and the interpretation of various statistical tests.
Type II Error: A Type II Error occurs when a statistical test fails to reject a false null hypothesis. This means that the test concludes there is no effect or difference when, in reality, one exists. Understanding Type II Errors is crucial for interpreting results in hypothesis testing, as they relate to the power of a test and the implications of failing to detect a true effect.
Z-score: A z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It indicates how many standard deviations an element is from the mean, allowing for comparison between different datasets and understanding the relative position of a value within a distribution.
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