Hypothesis Testing: Concepts, Errors, and P-Value Approach

Hypothesis Testing

Introduction to Hypothesis Testing

Hypothesis testing is a fundamental statistical method used to make inferences about population parameters based on sample data. It involves evaluating claims or statements about a population using sample evidence.

• Hypothesis: A claim or statement about a property of a population (e.g., mean, proportion, standard deviation).

• Examples:

  • The coin being used is fair.

  • The mean body temperature of healthy adults is less than 98.6˚ F.

  • The standard deviation of IQ scores of actors is more than 15.

Null and Alternative Hypotheses

Every hypothesis test involves two competing hypotheses:

• Null Hypothesis (H0): A statement that the value of a population parameter is equal to some claimed value. It is tested directly, assuming it is true until evidence suggests otherwise.

• Alternative Hypothesis (H1): A statement that the parameter differs from the null hypothesis. It uses inequality symbols: <, >, or ≠.

Examples:

• H0: $p = 0.5$     H1: $p \ne 0.5$

• H0: $\mu = 98.6$     H1: $\mu < 98.6$

• H0: $\sigma = 15$     H1: $\sigma > 15$

Identifying Null and Alternative Hypotheses

To correctly identify hypotheses:

1. State the original claim in symbolic form.

2. State the opposite of that claim.

3. The claim with strict inequality (<, >, or ≠) is the alternative hypothesis (H1).

4. The null hypothesis (H0) is the condition of equality (=).

Examples of Hypothesis Identification

Claim	H0	H1
The percentage of viewers tuned to 60 Minutes is 24%.	$p = 0.24$	$p \ne 0.24$
The mean fuel consumption for Chevrolet Corvettes is less than 17 miles/gallon.	$\mu = 17$	$\mu < 17$
It takes no more than 3 days on average to process an auto accident claim.	$\mu = 3$	$\mu > 3$
The mean IQ of statistics students is at least 110.	$\mu = 110$	$\mu < 110$
More than half of the drivers in Bismarck fail to use their turn signals consistently.	$p = 0.5$	$p > 0.5$
A defendant on trial is “innocent until proven guilty.”	The defendant is innocent.	The defendant is guilty.

Note: The original claim may become either the null or alternative hypothesis, especially if it involves ≤ or ≥.

Types of Errors in Hypothesis Testing

Two types of errors can occur in hypothesis testing:

• Type I Error: Rejecting the null hypothesis when it is actually true. The probability of a Type I error is denoted by $\alpha$ (alpha).

• Type II Error: Failing to reject the null hypothesis when it is actually false. The probability of a Type II error is denoted by $\beta$ (beta).

Examples:

• Type I error: Reject $p = 0.5$ when in reality $p = 0.5$.

• Type II error: Fail to reject $p = 0.5$ when in reality $p > 0.5$.

• Type I error: An innocent person goes to jail.

• Type II error: A guilty person goes free.

P-Value Approach to Hypothesis Testing

The P-value is a key concept in modern hypothesis testing. It represents the probability of obtaining a sample statistic as extreme as the observed one, assuming the null hypothesis is true.

• P-Value: The probability of getting a value of the sample test statistic at least as extreme as the one found from the sample data, assuming H0 is true.

• The null hypothesis is rejected if the P-value is very small (typically less than $\alpha$).

Steps in the P-Value Approach

1. State H0 and H1.

2. Calculate the test statistic.

3. Compute the P-value.

4. State the conclusion:

   • Reject H0 if P-value $\leq \alpha$.

   • Fail to reject H0 if P-value $> \alpha$.

Stating Conclusions

Conclusions should be stated in non-technical terms, referencing the original claim:

• There is enough evidence to reject H0.

• There is enough evidence to support H1.

• There is not enough evidence to reject H0.

• There is not enough evidence to support H1.

Note: We always reject H0 or support H1; we do not "accept" or "support" H0.

Summary Table: Errors in Hypothesis Testing

Decision	Reality	Type of Error
Reject H0	H0 is true	Type I error ($\alpha$)
Fail to reject H0	H0 is false	Type II error ($\beta$)

Example: In a criminal trial, H0 is "the defendant is innocent." A Type I error means convicting an innocent person; a Type II error means acquitting a guilty person.

Additional info: The P-value approach is widely used because it provides a direct measure of evidence against the null hypothesis. Typical values for $\alpha$ are 0.05 or 0.01, representing the maximum probability of a Type I error that is considered acceptable.