Hypothesis Testing and P-Value Approach in Statistics

Chapter 17: Tests of Significance

Introduction to Hypothesis Testing

Hypothesis testing is a fundamental method in inferential statistics used to make decisions or judgments about population parameters based on sample data. It involves formulating two competing hypotheses and using sample evidence to determine which hypothesis is more plausible.

• Hypothesis: A statement that something is true about a population parameter.

• Hypothesis Test: A statistical procedure to evaluate the validity of a hypothesis using sample data.

Types of Hypotheses

• Null Hypothesis (H0): The hypothesis to be tested, usually stating that there is no effect or no difference. It is assumed to be true unless evidence suggests otherwise.

• Alternative Hypothesis (Ha): The hypothesis that is considered as an alternative to the null hypothesis, often representing the researcher's claim.

Definitions

• Null Hypothesis (H0): The statement being tested. It is denoted by H0 and is assumed true unless sample data provide substantial contradictory evidence.

• Alternative Hypothesis (Ha): The statement that contradicts H0. It is denoted by Ha and is accepted if H0 is rejected.

Possible Conclusions of a Hypothesis Test

• Reject H0

• Fail to reject H0

Formulating Hypotheses

The choice of the alternative hypothesis depends on the research question:

• If the concern is whether a parameter is different from a specified value: $H_a: \mu \neq \mu_0$ (Two-tailed test)

• If the concern is whether a parameter is greater than a specified value: $H_a: \mu > \mu_0$ (Right-tailed test)

• If the concern is whether a parameter is less than a specified value: $H_a: \mu < \mu_0$ (Left-tailed test)

Examples of Hypothesis Testing

• Example 1: A snack company produces 454g bags of cookies. The quality assurance department tests whether the mean net weight of all bags is 454g.

  1. Determine the null hypothesis: $H_0: \mu = 454$

  2. Determine the alternative hypothesis: $H_a: \mu \neq 454$

  3. Classify the test: Two-tailed test

• Example 2: In 2005, the mean retail price of history books was $78.01. Test if this year's mean price has increased.

  1. Null hypothesis: $H_0: \mu = 78.01$

  2. Alternative hypothesis: $H_a: \mu > 78.01$

  3. Test type: Right-tailed test

• Example 3: Test if the average adult with income below the poverty level gets less than the RAI of calcium (1000mg).

  1. Null hypothesis: $H_0: \mu = 1000$

  2. Alternative hypothesis: $H_a: \mu < 1000$

  3. Test type: Left-tailed test

P-Value Approach to Hypothesis Testing

The P-value approach is a widely used method for making decisions in hypothesis testing. It quantifies the probability of obtaining a test statistic as extreme as the observed one, assuming the null hypothesis is true.

• P-value: The probability of observing a test statistic at least as extreme as the one calculated, under the assumption that H0 is true.

• Significance Level (α): The threshold probability for rejecting H0. Common values: 0.01, 0.05, 0.10.

• Decision Rule: If P-value ≤ α, reject H0. If P-value > α, do not reject H0.

Example of P-value Decision

• P-value = 0.1142

• At α = 0.01, 0.05, 0.10, would you reject H0?

  • At α = 0.01: Do not reject H0

  • At α = 0.05: Do not reject H0

  • At α = 0.10: Do not reject H0

Assumptions for the Z-Test

• Observations are from a Simple Random Sample (SRS).

• Population is normally distributed.

• Population mean is unknown.

• Population standard deviation is known (rare in practice).

Z-Test Statistic Formula

The Z-test statistic is used to test hypotheses about a population mean when the population standard deviation is known.

• Formula: $Z = \frac{\bar{X} - \mu_0}{\sigma / \sqrt{n}}$

• Where: $\bar{X}$ = sample mean $\mu_0$ = hypothesized population mean $\sigma$ = population standard deviation $n$ = sample size

Determining P-Value for a One-Mean Z-Test

• Null hypothesis: $H_0: \mu = \mu_0$

• Test statistic: $Z = \frac{\bar{X} - \mu_0}{\sigma / \sqrt{n}}$

• P-value is calculated based on the Z value and the type of test (one-tailed or two-tailed).

Example Calculations

• Example 6: Two-tailed test, Z = 1.71

  • At 5% significance level, check if P-value < 0.05 to reject H0.

• Example 7: Left-tailed test, Z = -1.92

  • At 5% significance level, check if P-value < 0.05 to reject H0.

• Example 8: Right-tailed test, Z = 2.85

  • At 5% significance level, check if P-value < 0.05 to reject H0.

Steps in the P-Value Approach to Hypothesis Testing

1. State the null and alternative hypotheses.

2. Decide on the significance level (α).

3. Compute the value of the test statistic.

4. Determine the P-value.

5. Compare the P-value to α and make a decision.

6. Interpret the results of the hypothesis test.

Comprehensive Example

Suppose in 2005, the mean retail price of history books was $78.01. This year's sample mean for 40 randomly selected books is $81.44, with a population standard deviation of $7.61. At the 1% significance level, test if the mean price has increased.

• Null hypothesis: $H_0: \mu = 78.01$

• Alternative hypothesis: $H_a: \mu > 78.01$

• Sample mean: $\bar{X} = 81.44$

• Population standard deviation: $\sigma = 7.61$

• Sample size: $n = 40$

• Significance level: $\alpha = 0.01$

• Calculate Z: $Z = \frac{81.44 - 78.01}{7.61 / \sqrt{40}}$

• Find P-value for calculated Z.

• If P-value < 0.01, reject H0; otherwise, do not reject H0.

Summary Table: Types of Hypothesis Tests

Test Type	Alternative Hypothesis	Decision Rule
Two-tailed	$H_a: \mu \neq \mu_0$	Reject H0 if sample mean is significantly different from hypothesized mean
Right-tailed	$H_a: \mu > \mu_0$	Reject H0 if sample mean is significantly greater than hypothesized mean
Left-tailed	$H_a: \mu < \mu_0$	Reject H0 if sample mean is significantly less than hypothesized mean

Additional info: The notes have been expanded with academic context, including definitions, formulas, and step-by-step procedures for hypothesis testing and the P-value approach, to ensure completeness and clarity for exam preparation.