Glossary

9.3 Probability Distribution Needed for Hypothesis Testing

2 min read•june 25, 2024

distributions are crucial tools in hypothesis testing, helping us make informed decisions about population parameters. From the normal distribution for large samples to the t-distribution for smaller ones, each serves a specific purpose in statistical analysis.

Understanding the assumptions behind these distributions is key to selecting the right test. Whether dealing with means, proportions, or variances, knowing which distribution to use and when ensures accurate results in hypothesis testing and statistical inference.

Probability Distributions for Hypothesis Testing

Probability distributions for hypothesis tests

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  • Normal distribution ()
    • Tests population means with known population or large sample size (n≥30n \geq 30)
    • Tests population proportions with large sample size satisfying conditions
    • Tests population means with unknown population and small sample size (n<30n < 30)
    • Conducts , , and
    • Compares equality of two or more population variances ()
    • Determines overall significance in regression analysis ()
    • Tests population proportions with small sample size not meeting normal approximation conditions

Key assumptions of distribution tests

  • Normal distribution assumptions
    • Data follows normal distribution or sample size is large enough for (n≥30n \geq 30)
    • Observations are independent of each other
    • Population standard deviation is a known value
  • Student's t-distribution assumptions
    • Data follows normal distribution or sample size is large enough for Central Limit Theorem (n≥30n \geq 30)
    • Observations are independent of each other
    • Population standard deviation is an unknown value
  • Binomial distribution assumptions
    • Trials are independent of each other
    • Trials have only two possible outcomes (success or failure)
    • Probability (likelihood of an event occurring) of success is constant across all trials
    • Number of trials is a fixed value

Normal approximation in proportion tests

  • Conditions for using normal approximation () in proportion tests
    1. Independence: taken from population
    2. Sample size: Large enough to satisfy both np≥10np \geq 10 and n(1−p)≥10n(1-p) \geq 10
      • nn represents the sample size
      • pp represents the hypothesized
  • If conditions are met, of sample proportion can be approximated by normal distribution with:
    • Mean: μp^=p\mu_{\hat{p}} = p
    • Standard deviation: σp^=p(1−p)n\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}

Hypothesis Testing Framework

  • Null hypothesis: Initial assumption about a population parameter
  • : Claim to be tested against the null hypothesis
  • : Predetermined threshold for rejecting the null hypothesis
  • : Probability of obtaining test results at least as extreme as the observed results, assuming the null hypothesis is true
  • : Rejecting the null hypothesis when it is actually true
  • : Range of values likely to contain the true population parameter

Key Terms to Review (34)

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.
ANOVA: ANOVA, or Analysis of Variance, is a statistical method used to analyze the differences between two or more group means and determine if they are significantly different from each other. It is a powerful tool for hypothesis testing and is particularly relevant in the context of probability distributions needed for hypothesis testing, the F-distribution, and the F-ratio.
Binomial distribution: A binomial distribution is a discrete probability distribution of the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. It is characterized by parameters $n$ (number of trials) and $p$ (probability of success).
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.
Chi-Square Distribution: The chi-square distribution is a continuous probability distribution that arises when independent standard normal random variables are squared and summed. It is widely used in statistical hypothesis testing, particularly in evaluating the goodness-of-fit of observed data to a theoretical distribution and in testing the independence of two categorical variables.
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.
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.
Error bound for a population mean: The error bound for a population mean is the maximum expected difference between the true population mean and a sample estimate of that mean. It is often referred to as the margin of error in confidence intervals.
F-distribution: The F-distribution is a continuous probability distribution that arises when testing the equality of two population variances. It is a fundamental concept in statistical inference, particularly in hypothesis testing and analysis of variance (ANOVA).
F-test: The F-test is a statistical test used to compare the variances of two or more populations. It is a fundamental concept in hypothesis testing and is particularly relevant in the context of analysis of variance (ANOVA) and comparing the variances of two samples.
Goodness-of-Fit Tests: Goodness-of-fit tests are statistical methods used to determine whether a sample of data fits a particular probability distribution. These tests are employed to assess the appropriateness of a proposed model or distribution for the observed data, ensuring the validity of subsequent statistical analyses.
Hypothesis test: A hypothesis test is a statistical method used to make inferences or draw conclusions about a population based on sample data. It involves comparing observed data with what we expect under the null hypothesis.
Hypothesis Tests: Hypothesis tests are a statistical method used to determine whether a particular claim or hypothesis about a population parameter is supported by the sample data. They provide a systematic approach to making decisions about the validity of a hypothesis based on the evidence provided by the data.
Normal Approximation: The normal approximation is a statistical concept that allows for the use of the normal distribution to approximate other probability distributions, particularly the binomial distribution, when certain conditions are met. This approximation is useful in making inferences about population parameters when dealing with large sample sizes or when the underlying distribution is not known.
Normal approximation to the binomial: Normal approximation to the binomial is a method used to approximate the probabilities of a binomial distribution using the normal distribution when the sample size is large and the probability of success is neither very close to 0 nor 1.
Normally distributed: A normally distributed variable follows a symmetric, bell-shaped curve where most values cluster around the mean. It is characterized by its mean and standard deviation.
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.
Population Mean: The population mean, denoted by the Greek letter μ, is the average or central value of a characteristic or variable within a entire population. It is a fundamental concept in statistics that represents the typical or expected value for a given population.
Population Proportion: The population proportion is the percentage or fraction of a population that possesses a certain characteristic or attribute. It is a fundamental concept in statistics that is used to make inferences about the larger population based on a sample drawn from that population.
Probability: Probability is the measure of the likelihood of an event occurring. It is a fundamental concept in statistics that quantifies the uncertainty associated with random events or outcomes. Probability is central to understanding and analyzing data, making informed decisions, and drawing valid conclusions.
Sampling distribution: A sampling distribution is the probability distribution of a given statistic based on a random sample. It reflects how the statistic would vary if you repeatedly sampled from the same population.
Sampling Distribution: The sampling distribution is the probability distribution of a statistic, such as the sample mean or sample proportion, obtained from repeated sampling of a population. It describes the variability of the statistic and is a crucial concept in statistical inference, allowing for the assessment of the reliability and precision of sample-based estimates of population parameters.
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.
Simple random sample: A simple random sample is a subset of individuals chosen from a larger set, where each individual has an equal probability of being selected. It ensures that every possible sample of the same size has an equal chance of selection.
Standard deviation: Standard deviation is a measure of the dispersion or spread of a set of data points around its mean. It quantifies how much the individual data points deviate from the mean value.
Standard Deviation: Standard deviation is a statistic that measures the dispersion or spread of a set of values around the mean. It helps quantify how much individual data points differ from the average, indicating the extent to which values deviate from the central tendency in a dataset.
Student's t-distribution: The Student's t-distribution is a probability distribution used when the population standard deviation is unknown and the sample size is small. It is a continuous probability distribution that is symmetric and bell-shaped, similar to the normal distribution, but with heavier tails. The Student's t-distribution is particularly important in hypothesis testing and confidence interval estimation when the population standard deviation is unknown.
T-test: The t-test is a statistical hypothesis test that is used to determine if the mean of a population is significantly different from a hypothesized value or the mean of another population. It is commonly used in various statistical analyses, including those related to probability distributions, hypothesis testing, and regression.
Tests of Homogeneity: Tests of homogeneity are statistical tests used to determine whether two or more samples come from populations with the same probability distribution. These tests are particularly relevant in the context of hypothesis testing, as they help assess the underlying assumptions required for various statistical analyses.
Tests of Independence: Tests of independence are statistical methods used to determine whether two categorical variables are related or independent of each other. These tests examine the null hypothesis that the variables are not associated, meaning they are independent, versus the alternative hypothesis that the variables are related or dependent.
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.
Z-distribution: The z-distribution, also known as the standard normal distribution, is a probability distribution that describes the set of standardized normal random variables. It is a fundamental concept in hypothesis testing and statistical inference, as it is used to determine the probability of obtaining a specific value or range of values from a normal distribution.
Z-test: The z-test is a statistical hypothesis test that uses the standard normal distribution to determine if the mean of a population is equal to a specified value. It is commonly used in the context of evaluating null and alternative hypotheses, determining the appropriate probability distribution for hypothesis testing, and conducting hypothesis testing for a single mean or proportion.
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