All Study Guides Statistical Inference Unit 7
🎣 Statistical Inference Unit 7 – Hypothesis Testing: Principles & Single TestsHypothesis testing is a statistical method used to make decisions about populations based on sample data. It involves formulating null and alternative hypotheses, collecting data, and calculating test statistics to determine whether to reject or fail to reject the null hypothesis.
Key concepts include p-values, significance levels, and types of errors. The process involves stating hypotheses, choosing a test statistic, collecting data, determining p-values, and interpreting results. Various types of tests are used depending on the research question and data characteristics.
What's the Big Idea?
Hypothesis testing is a statistical method used to make decisions or draw conclusions about a population based on sample data
Involves formulating a null hypothesis (H 0 H_0 H 0 ) and an alternative hypothesis (H a H_a H a ) about a population parameter
Collect sample data and calculate a test statistic to determine whether to reject or fail to reject the null hypothesis
The decision is based on the probability (p-value) of observing the sample data assuming the null hypothesis is true
Hypothesis testing allows researchers to make evidence-based decisions in various fields (psychology, medicine, business)
The significance level (α \alpha α ) is the probability of rejecting the null hypothesis when it is actually true (Type I error)
Commonly set at 0.05, meaning a 5% chance of making a Type I error
The power of a test is the probability of rejecting the null hypothesis when the alternative hypothesis is true (1 - Type II error)
Key Concepts You Need to Know
Null hypothesis (H 0 H_0 H 0 ): A statement of no effect or no difference, assumed to be true unless evidence suggests otherwise
Alternative hypothesis (H a H_a H a ): A statement that contradicts the null hypothesis, representing the researcher's claim or theory
Test statistic: A value calculated from the sample data used to determine whether to reject the null hypothesis (e.g., z-score, t-score, chi-square)
P-value: The probability of observing the sample data or more extreme results, assuming the null hypothesis is true
Significance level (α \alpha α ): The predetermined probability threshold for rejecting the null hypothesis, typically set at 0.05
Type I error: Rejecting the null hypothesis when it is actually true (false positive)
Type II error: Failing to reject the null hypothesis when it is actually false (false negative)
One-tailed test: A hypothesis test where the alternative hypothesis specifies a direction (greater than or less than)
Two-tailed test: A hypothesis test where the alternative hypothesis does not specify a direction (not equal to)
The Hypothesis Testing Process
State the null and alternative hypotheses based on the research question or problem
Choose an appropriate test statistic and significance level (α \alpha α )
Collect sample data and calculate the test statistic
Determine the p-value associated with the test statistic
Compare the p-value to the significance level (α \alpha α )
If p-value ≤ α \alpha α , reject the null hypothesis in favor of the alternative hypothesis
If p-value > α \alpha α , fail to reject the null hypothesis
Interpret the results in the context of the research question or problem
Consider the limitations and potential sources of error in the study
Types of Hypotheses
One-sample hypothesis: Tests whether a population parameter (mean, proportion) differs from a specified value
Example: Testing if the average height of a population differs from 170 cm
Two-sample hypothesis: Compares two population parameters to determine if they are significantly different
Example: Comparing the mean test scores of two different teaching methods
Paired-sample hypothesis: Tests the difference between two related or dependent samples
Example: Measuring blood pressure before and after a treatment for the same group of patients
ANOVA (Analysis of Variance): Tests the difference between three or more population means
Example: Comparing the average yield of four different fertilizer treatments
Chi-square test: Tests the association between two categorical variables
Example: Determining if there is a relationship between gender and political party affiliation
Common Test Statistics
Z-test: Used for testing hypotheses about population means or proportions when the sample size is large or the population standard deviation is known
T-test: Used for testing hypotheses about population means when the sample size is small and the population standard deviation is unknown
One-sample t-test: Tests if a sample mean differs from a hypothesized population mean
Independent samples t-test: Compares the means of two independent groups
Paired samples t-test: Compares the means of two related or dependent groups
Chi-square test: Used for testing the association between two categorical variables
Goodness-of-fit test: Compares observed frequencies to expected frequencies for a single categorical variable
Test of independence: Determines if two categorical variables are independent or associated
F-test (ANOVA): Used for comparing the means of three or more groups or treatments
Interpreting Results
If the p-value is less than or equal to the significance level (α \alpha α ), reject the null hypothesis
Conclude that there is sufficient evidence to support the alternative hypothesis
Example: If p-value ≤ 0.05, conclude that there is a significant difference between the groups
If the p-value is greater than the significance level (α \alpha α ), fail to reject the null hypothesis
Conclude that there is not enough evidence to support the alternative hypothesis
Example: If p-value > 0.05, conclude that there is no significant difference between the groups
Confidence intervals can be used to estimate the range of plausible values for the population parameter
A 95% confidence interval means that if the study were repeated many times, 95% of the intervals would contain the true population parameter
Effect size measures the magnitude of the difference or relationship between variables
Examples: Cohen's d, Pearson's r, eta-squared
Real-World Applications
Medical research: Testing the effectiveness of a new drug compared to a placebo
Psychology: Comparing the mean scores of two therapy techniques on reducing anxiety
Business: Determining if a new marketing campaign significantly increases sales
Education: Testing if a new teaching method improves student performance compared to traditional methods
Environmental science: Comparing the average pollution levels between two cities
Quality control: Testing if the proportion of defective products exceeds a specified threshold
Market research: Determining if there is an association between age and product preference
Potential Pitfalls and Limitations
Sampling bias: When the sample is not representative of the population, leading to inaccurate conclusions
Type I error (false positive): Rejecting the null hypothesis when it is actually true
Can be reduced by decreasing the significance level (α \alpha α ), but this may increase the risk of Type II error
Type II error (false negative): Failing to reject the null hypothesis when it is actually false
Can be reduced by increasing the sample size or using a more powerful test
Violation of assumptions: Most hypothesis tests rely on certain assumptions about the data (normality, homogeneity of variance)
Violations can lead to invalid results and conclusions
Multiple testing: Conducting many hypothesis tests on the same data increases the likelihood of making a Type I error
Bonferroni correction or other methods can be used to adjust the significance level for multiple comparisons
Practical significance vs. statistical significance: A statistically significant result may not be practically meaningful or important
Consider the effect size and real-world implications of the findings