Statistics for Business and Economics - Chapter 7 Inferences Based on a Single Sample: Tests of Hypothesis

Statistics for Business and Economics: Chapter 7 Overview

  • This chapter focuses on inferences based on a single sample, specifically tests of hypothesis.

1. The Elements of a Test of Hypothesis

  • Null Hypothesis ($H_0$): Represents the hypothesis that will be accepted unless convincing evidence suggests it is false. Typically represents the status quo or claim about the population parameter.

  • Alternative Hypothesis ($H_a$): Represents the hypothesis that will be accepted only with sufficient evidence from data. It contradicts the null hypothesis and supports the researcher's assumptions.

  • Test Statistic: A sample statistic calculated from the sample data, used to make a decision between the null and alternative hypotheses.

  • Rejection Region: The set of values of the test statistic for which the null hypothesis is rejected. The size of this region is determined by the significance level ($α$).

  • Significance Level ($α$): The probability of committing a Type I error, typically set at small values like 0.01, 0.05, or 0.10.

  • Type I Error: Rejecting the null hypothesis when it is true. Denoted as $α$.

  • Type II Error: Failing to reject the null hypothesis when it is false, denoted as $β$.

2. Formulating Hypotheses and Setting Up the Rejection Region

  • To formulate hypotheses, follow these steps:

    1. Select the alternative hypothesis that the experimental design aims to verify.

    2. Set the null hypothesis as the status quo, which will be presumed true unless enough evidence is found to support the alternative.

Types of Hypotheses

One-Tailed Test:

  • Upper-Tailed: Tests if a parameter is greater than a certain value. Notation: $H_a: heta > k$.

  • Lower-Tailed: Tests if a parameter is less than a certain value. Notation: $H_a: heta < k$.

Two-Tailed Test:

  • Tests if a parameter is different from a certain value, indicated by $H_a: heta
    eq k$.

3. Observed Significance Levels: p-Values

  • p-Value: The probability, under the null hypothesis, of observing a test statistic as extreme or more extreme than the one observed. Used to evaluate the evidence against the null hypothesis.

    • If p-value $< α$, reject $H_0$.

    • If p-value $≥ α$, do not reject $H_0$.

Calculation of p-Values

  1. Calculate the test statistic $z$ from the sample data.

  2. For one-tailed tests:

    • If $H_a: heta > k$, find the area to the right of $z$.

    • If $H_a: heta < k$, find the area to the left of $z$.

  3. For two-tailed tests, the p-value is twice the tail area beyond the observed value.

4. Test of Hypothesis about a Population Mean: Normal (z) Statistic

Conditions for Valid Hypothesis Testing:

  1. A random sample is obtained from the population.

  2. The sample size must be large ($n ≥ 30$) to apply the Central Limit Theorem.

One-Tailed and Two-Tailed Tests:

  • Define rejection regions based on the significance level and calculate test statistics to determine appropriate actions.

5. Test of Hypothesis about a Population Mean: Student’s t-Statistic

  • Applicable for small samples where the population standard deviation is unknown. Conditions:

  1. Sample drawn from approximately normally distributed population.

  2. Use t-distribution for the test statistics with $n - 1$ degrees of freedom.

6. Test of Hypothesis about a Population Proportion

  • Larger samples for proportion tests follow similar rules:

  1. Define hypotheses:

    • $H0: p = p0$

    • $Ha: p < p0$ (or $p > p0$, or $p eq p0$ in two-tailed tests).

  2. Determine test statistic and rejection region accordingly based on sample size and significance levels.

7. Test of Hypothesis about a Population Variance

  • Rationale for testing variances.

  • Establish hypotheses regarding variances and calculate test statistics under specified conditions.

8. Type II Error Probabilities (β)

  • Understanding the Type II error in statistical testing, its implications, and how to calculate it using specified rejection regions and z-values.

9. Power of Test

  • Defined as the probability of rejecting the null hypothesis when it is false, with the formula: Power = $1 - β$. Factors influencing test power include true parameter values, significance levels, standard deviations, and sample sizes.

10. Example Calculations and Applications

  • Practical examples of calculating test statistics, rejection regions, interpreting results in context, including hypotheses on means and proportions.

  • Key calculations on Type I and Type II error implications and their effect on hypothesis testing outcomes.

11. Key Recap and Formulae

  • Key definitions of parameters, hypotheses, test statistics, errors, and conclusions in hypothesis testing.

  • Formulas and processes to determine p-values, rejection regions, and power of tests based on varying statistics across different tests of hypotheses.