Glossary

10.5 Hypothesis Testing for Two Means and Two Proportions

3 min read•june 25, 2024

Comparing two populations is crucial in statistics. We'll explore methods for testing differences between means and proportions, using independent and paired samples. These techniques help us determine if observed differences are statistically significant.

Interpreting test results is key to drawing meaningful conclusions. We'll learn about p-values, confidence intervals, and how to make decisions based on statistical evidence. We'll also touch on statistical power and , which help us understand the strength of our findings.

Hypothesis Testing for Two Means

Methods for two-population mean tests

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  • used when samples selected independently from separate populations
    • Sample sizes can differ
    • calculated using difference between sample means divided by of difference
      • Standard error calculated using sample variances and sizes
    • approximated using (used when population variances unknown and sample sizes small)
  • used when observations in one sample paired with observations in other sample
    • Sample sizes must be equal
    • calculated using mean difference between paired observations divided by standard error of mean difference
      • Standard error calculated using sample standard deviation of differences and
    • Degrees of freedom equal to sample size minus one

Hypothesis Testing for Two Proportions

Hypothesis tests for population proportions

  • Test statistic calculated using difference between sample proportions divided by standard error of difference
    • Standard error calculated using (total successes in both samples divided by total sample size) and sample sizes
  • Test statistic follows when sample sizes large enough (typically, n1p1^n_1\hat{p_1}, n1(1−p1^)n_1(1-\hat{p_1}), n2p2^n_2\hat{p_2}, and n2(1−p2^)≥10n_2(1-\hat{p_2}) \geq 10)
    • This is an application of the
  • Null hypothesis rejected if test statistic falls in determined by and (left-tailed, right-tailed, or two-tailed)
    • The rejection region is defined by the (e.g., for a given significance level)

Interpreting Results

Interpretation of two-sample test results

  • represents probability of obtaining test statistic as extreme as or more extreme than observed test statistic, assuming null hypothesis true
    • If p-value < significance level, null hypothesis rejected in favor of alternative hypothesis
    • If p-value > significance level, insufficient evidence to reject null hypothesis
  • provides range of values likely to contain true difference between population parameters (means or proportions) with certain level of confidence
    • If confidence interval does not contain null hypothesis value (usually zero), null hypothesis rejected in favor of alternative hypothesis
    • If confidence interval contains null hypothesis value, insufficient evidence to reject null hypothesis
  • When drawing conclusions:
    • If null hypothesis rejected, conclude significant difference between population parameters
    • If null hypothesis not rejected, conclude insufficient evidence to support significant difference between population parameters
    • Exercise caution when concluding no difference between population parameters if null hypothesis not rejected, as this may result from insufficient sample sizes or large data variability

Statistical Power and Effect Size

  • helps determine the sample size needed to detect a significant difference
  • Effect size quantifies the magnitude of the difference between groups or the strength of a relationship
  • Power of a test is influenced by sample size, effect size, and significance level

Key Terms to Review (41)

Alternative Hypothesis: The alternative hypothesis is a statement that suggests a potential outcome or relationship exists in a statistical test, opposing the null hypothesis. It indicates that there is a significant effect or difference that can be detected in the data, which researchers aim to support through evidence gathered during hypothesis testing.
Central Limit Theorem: The Central Limit Theorem states that when a sample of size 'n' is taken from any population with a finite mean and variance, the distribution of the sample means will tend to be normally distributed as 'n' becomes large, regardless of the original population's distribution. This theorem allows for the use of normal probability models in various statistical applications, making it fundamental for inference and hypothesis testing.
Confidence Interval: A confidence interval is a range of values used to estimate the true value of a population parameter, such as a mean or proportion, based on sample data. It provides a measure of uncertainty around the sample estimate, indicating how much confidence we can have that the interval contains the true parameter value.
Critical value: A critical value is a point on the scale of the standard normal distribution that is compared to a test statistic to determine whether to reject the null hypothesis. It separates the region where the null hypothesis is not rejected from the region where it is rejected.
Critical Value: The critical value is a threshold value in statistical analysis that is used to determine whether to reject or fail to reject a null hypothesis. It serves as a benchmark for evaluating the statistical significance of a test statistic and is a crucial concept across various statistical methods and hypothesis testing procedures.
Degrees of Freedom: Degrees of freedom refer to the number of independent values or quantities that can vary in a statistical calculation without breaking any constraints. It plays a crucial role in determining the appropriate statistical tests and distributions used for hypothesis testing, estimation, and data analysis across various contexts.
Effect Size: Effect size is a quantitative measure that indicates the magnitude or strength of the relationship between two variables or the difference between two groups. It provides information about the practical significance of a statistical finding, beyond just the statistical significance.
H0: H0, or the null hypothesis, is a fundamental concept in statistical hypothesis testing. It represents the initial assumption or default position that is tested against the observed data to determine if there is sufficient evidence to reject it in favor of an alternative hypothesis.
Ha: In the context of hypothesis testing, Ha represents the alternative hypothesis, which is the statement that the researcher believes to be true. The alternative hypothesis is the complement of the null hypothesis, and it is the hypothesis that the researcher aims to provide evidence for through the statistical analysis.
Independent Samples Test: The independent samples test is a statistical analysis method used to compare the means or proportions of two independent populations or groups. It is commonly employed in hypothesis testing to determine if there is a significant difference between the characteristics of two distinct samples.
Left-Tailed Test: A left-tailed test is a statistical hypothesis test where the alternative hypothesis specifies that the parameter of interest is less than a certain value. This type of test is used when the researcher is interested in determining if a population parameter, such as a mean or proportion, is significantly lower than a given target value.
N1: n1 is a statistical term that represents the sample size of the first population or group in a comparison of two independent population proportions or means. It is a crucial parameter in various statistical analyses and hypothesis testing procedures.
N2: The term 'n2' refers to the sample size or the number of observations in each of the two independent populations being compared. It is a crucial parameter in the context of hypothesis testing for comparing two population proportions and means.
P-value: The p-value is the probability of obtaining a test statistic at least as extreme as the one actually observed, assuming the null hypothesis is true. It is a crucial concept in hypothesis testing that helps determine the statistical significance of a result.
P̂1: The sample proportion of the first population, which is an estimate of the true population proportion, p1. This term is particularly relevant in the context of comparing two independent population proportions and hypothesis testing for two proportions.
P̂2: The sample proportion squared, $p̂2$, is a statistic used in the context of comparing two independent population proportions and hypothesis testing for two proportions. It represents the estimated proportion of a characteristic or event in a sample, which is then used to make inferences about the corresponding population proportion.
Paired Samples Test: The paired samples test is a statistical method used to compare the means of two related or dependent samples. It is commonly employed in scenarios where the same individuals or subjects are measured under two different conditions or at different time points.
Pareto chart: A Pareto chart is a type of bar graph that represents the frequency or impact of problems or causes in descending order, combined with a cumulative line graph to show the total effect. It helps identify the most significant factors in a dataset.
Pooled Sample Proportion: The pooled sample proportion is a statistical measure used to estimate the common proportion between two populations when conducting hypothesis testing for two proportions. It combines the sample proportions from both groups into a single, weighted average to represent the overall proportion.
Population variance: Population variance is a measure of the dispersion of all values in a population from the population mean. It is calculated as the average of the squared differences between each value and the population mean.
Population Variance: Population variance is a statistical measure that quantifies the spread or dispersion of a population's data around its mean. It represents the average squared deviation of each data point from the population mean, providing insight into the variability within the entire population.
Power Analysis: Power analysis is a statistical concept that helps determine the minimum sample size required to detect an effect of a given size with a desired level of statistical significance and power. It is a crucial tool in experimental design and hypothesis testing across various fields, including statistics, psychology, and medical research.
Rejection Region: The rejection region, also known as the critical region, is a specific range of values for a test statistic that leads to the rejection of the null hypothesis in a hypothesis test. It represents the set of outcomes that are considered statistically significant and unlikely to have occurred by chance if the null hypothesis is true.
Right-Tailed Test: A right-tailed test is a statistical hypothesis test where the alternative hypothesis specifies that the parameter of interest is greater than the value stated in the null hypothesis. This type of test is used when the researcher is interested in determining if a particular characteristic or outcome exceeds a certain threshold or standard.
Sample Size: Sample size refers to the number of observations or data points collected in a statistical study or experiment. It is a crucial factor in determining the reliability and precision of the results, as well as the ability to make inferences about the larger population from the sample data.
Sample Variance: Sample variance is a measure of the spread or dispersion of a set of data points around the sample mean. It represents the average squared deviation from the mean, providing insight into the variability within a sample. This metric is crucial in understanding the characteristics of a sample and making inferences about the corresponding population.
Significance Level: The significance level, denoted as α (alpha), is the probability of rejecting the null hypothesis when it is true. It represents the maximum acceptable probability of making a Type I error, which is the error of rejecting the null hypothesis when it is actually true. The significance level is a crucial concept in hypothesis testing and statistical inference, as it helps determine the strength of evidence required to draw conclusions about a population parameter or the relationship between variables.
Standard Error: Standard error is a statistical term that measures the accuracy with which a sample represents a population. It quantifies the variability of sample means from the true population mean, helping to determine how much sampling error exists when making inferences about the population.
Standard normal distribution: The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. It is used to standardize scores from different normal distributions for comparison.
Standard Normal Distribution: The standard normal distribution, also known as the Z-distribution, is a special case of the normal distribution where the mean is 0 and the standard deviation is 1. It is a fundamental concept in statistics that is used to model and analyze data that follows a normal distribution.
Test statistic: A test statistic is a standardized value that is calculated from sample data during a hypothesis test. It is used to determine whether to reject the null hypothesis.
Test Statistic: A test statistic is a numerical value calculated from sample data that is used to determine whether to reject or fail to reject a null hypothesis in a hypothesis test. It serves as the basis for decision-making in statistical inference, providing a quantitative measure to evaluate the strength of evidence against the null hypothesis.
Two-Tailed Test: A two-tailed test is a statistical hypothesis test in which the critical region is two-sided, meaning that the test statistic can fall in either the upper or lower tail of the distribution. This type of test is used to determine if a parameter is different from a specified value, without specifying the direction of the difference.
Type I error: A Type I error occurs when a true null hypothesis is incorrectly rejected. It is also known as a false positive.
Type I Error: A Type I error, also known as a false positive, occurs when the null hypothesis is true, but it is incorrectly rejected. In other words, it is the error of concluding that a difference exists when, in reality, there is no actual difference.
Type II error: A Type II error occurs when the null hypothesis is not rejected even though it is false. This results in a failure to detect an effect that is actually present.
Type II Error: A type II error, also known as a false negative, occurs when the null hypothesis is true, but it is incorrectly rejected. In other words, the test fails to detect an effect or difference that is actually present in the population. This type of error has important implications in various statistical analyses and hypothesis testing contexts.
Welch-Satterthwaite equation: The Welch-Satterthwaite equation is a method used to determine the appropriate degrees of freedom when conducting hypothesis tests for the difference between two population means with unknown and potentially unequal variances.
Z-score: A z-score represents the number of standard deviations a data point is from the mean. It is used to determine how unusual a particular observation is within a normal distribution.
Z-Score: A z-score is a standardized measure that expresses how many standard deviations a data point is from the mean of a distribution. It allows for the comparison of data points across different distributions by converting them to a common scale.
μ1 - μ2: The difference between two population means, μ1 and μ2, is a key concept in hypothesis testing for two means and two proportions. This term represents the null hypothesis that the two population means are equal, and the alternative hypothesis that they are not equal.
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