Hypothesis Testing
Hypothesis Testing Overview
Key Concepts in Hypothesis Testing
Hypotheses Definition
Two types of hypotheses: Null Hypothesis (H₀) and Alternative Hypothesis (H₁ or Hₐ)
H₀: Represents a statement of no effect or no difference.
H₁: Represents a statement of an effect or a difference.
Symbols and Statements
Hypotheses can be expressed using symbols such as:
≤ (less than or equal to)
< (less than)
≥ (greater than or equal to)
> (greater than)
≠ (not equal to)
Example Hypothesis Testing
Mean Attendance Example
Claim: The mean attendance (μ) is greater than or equal to 20,000.
Null Hypothesis: H₀: μ ≥ 20,000 (the claim about the mean is accepted).
Alternative Hypothesis: H₁: μ < 20,000 (the null is rejected).
Starting Point for Hypothesis Testing
Understand the statement and translate it into statistical language.
Define the null hypothesis (H₀) and alternative hypothesis (H₁).
Example from shipping department: "Number of errors < 3" translates to:
H₀: σ ≥ 3 (null hypothesis)
H₁: σ < 3 (alternative hypothesis)
If the statement involves population standard deviation, it uses the symbol σ.
Lung Cancer Example
Claim: Lung cancer accounts for 25% of all diagnoses.
Null Hypothesis: H₀: p = 0.25
Alternative Hypothesis: H₁: p ≠ 0.25 (two-tailed test)
Types of Decisions in Hypothesis Testing
For H₀, there are only two possible outcomes:
Reject H₀: If sample provides sufficient evidence.
Fail to Reject H₀: If sample does not provide enough evidence.
Note: You can never support H₀. The only actions are to reject or fail to reject.
Errors in Hypothesis Testing
There are two types of errors that can occur:
Type I Error (α): Rejecting H₀ when it is actually true. This is a false positive.
Type II Error (β): Failing to reject H₀ when it is false. This is a false negative.
Level of Significance
Alpha (α): The probability threshold set to determine when to reject H₀.
Common values: 0.10, 0.05, 0.01.
A smaller α decreases the chance of Type I errors but increases the chance of Type II errors.
Every hypothesis test is based on these levels of significance.
p-Value and Decision Making
A p-value is the probability of obtaining the observed results, or something more extreme, given that H₀ is true.
For example: If H₀ is true, how extreme can my observed test statistic be?
Decision Rule: If p-value ≤ α, then reject H₀.
This means that the evidence is sufficiently strong against H₀.
Steps for Hypothesis Testing
State the hypotheses (H₀ and H₁) using proper symbols and population parameters.
Choose a level of significance (α).
Collect data and calculate the test statistic (z-score).
Calculate the p-value based on the test statistic.
Compare the p-value to α to make the decision:
If p-value < α, reject H₀.
If p-value ≥ α, fail to reject H₀.
Calculation of z-Scores
The z-score can be calculated using the formula: z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}
Where \bar{x} is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.
Tail Analysis in Hypothesis Testing
One-tailed Tests:
Left-Tail Test: H₁ involves <; the extreme is to the left. Area to the left of z gives the p-value.
Right-Tail Test: H₁ involves >; the extreme is to the right. Area to the right of z gives the p-value.
Two-tailed Tests: H₁ involves ≠; you look at both tails (area to the left and right).
p-value is calculated by doubling the area from one side of the z-score.
Examples of Tail Cases
Example: Claim that mean time is no more than 102 seconds per day, which is:
Null hypothesis (H₀): μ ≤ 102
Alternative hypothesis (H₁): μ > 102 (Right-Tail Test).
Summary
Hypothesis testing is a structured method for making statistical conclusions. Key steps include forming hypotheses, calculating test statistics and p-values, determining a decision per the significance level, and being aware of potential errors. Understanding these concepts ensures accurate interpretations and decisions in statistical analyses.