Statistics Hypothesis Testing Review

Hypothesis Testing Formulas

• Three main formulas for hypothesis testing have been introduced.

• Focus on Z-score calculation, as p-values will not feature prominently in the test.

Z Statistic Formula

• The formula for the Z statistic is given by:

  • Z = \frac{\hat{p} - p}{\sqrt{pq/n}}

  • Where:

    • \hat{p} = sample proportion

    • p = population proportion

    • q = 1 - p (complement of p)

    • n = sample size

• Emphasis on understanding and using the Z statistic correctly for hypothesis testing.

Concepts of Hypothesis Testing

• Students should be able to leave the elementary statistics class with a fundamental understanding of p-values, even if not tested directly.

• Transitioning from theory to includes application in practice problems.

Example Problem Review

• Working through example problems during class to practice application of concepts.

• Focus on interpreting claims and hypotheses in various contexts.

Example Problem #8

• Scenario involves a nationwide study of marathon estimates:

  • Claim states that the estimate is too low;

  • Therefore, hypothesis setup is:

  • Alternative Hypothesis ( H_a ): \mu > 0.64 (claiming the true mean is higher than 64%)

  • Null Hypothesis ( H_0 ): \mu \leq 0.64

• Information from the survey:

  • Survey result: 490 out of 331

  • Calculate \hat{p} as a fraction first.

Step-by-Step Calculation

• Calculate Z statistic step-by-step:

  • Keep intermediate results in the calculator for accuracy.

• Key calculation to convert proportions:

  • \hat{p} = \frac{331}{490}

  • Calculate the difference from 0.64.

• Calculate the square root portion using:

  • \sqrt{p \cdot q / n} where p = 0.64 and q = 0.36 ; the divisor is 490.

Testing the Hypothesis

• Find Z from calculated values, comparing against critical value Z_{0.05}

  • Z-table critical values indicate if the null hypothesis should be rejected.

• Affordable statistic value leads to decision:

  • If value falls within the reject area, reject the null; if below, fail to reject.

Introduction to Chapter 13: Two Means Hypothesis Testing

• Chapter focuses on hypothesis testing for the difference between two means, requiring different inputs and equations:

  • Basic formula:

  • Z = \frac{\bar{X}1 - \bar{X}2}{\sqrt{(\frac{\sigma1^2}{n1}) + (\frac{\sigma2^2}{n2})}}

  • Where:

    • \bar{X}1, \bar{X}2 = sample means

    • \sigma1, \sigma2 = standard deviations of populations

    • n1, n2 = sample sizes

• Zero hypothesized difference initially until more complex questions arise.

Example Problem: Shopping Duration

• Claim: Women spend more time shopping.

  • Based on sampled data:

  • Women (Group 1):

    • n_1 = 50

    • \bar{X}_1 = 25.36

    • \sigma_1 = 7.24

  • Men (Group 2):

    • n_2 = 45

    • \bar{X}_2 = 23.45

    • \sigma_2 = 3.75

Hypotheses Setup

• Null Hypothesis: H0: \mu1 - \mu_2 \leq 0

• Alternative Hypothesis: Ha: \mu1 - \mu_2 > 0 $$ (right tail test)

Calculating the Z Statistic

• Leading to formula utilization, entering values into the Z statistic formula step-by-step:

  • Compute differences and ensure proper order of operations.

  • Provide a final conclusion based on Z score and compare to critical justification.

Test Structure

• Focus primarily on fill-in-the-blank structure as opposed to multiple-choice.

• Emphasis on calculations and justifying decisions based on hypothesis.

Conclusion on Hypothesis Results

• Importance of failing to reject the null hypothesis in real-world applications.

• Ensure clarity in terms of one-tailed and two-tailed tests based on hypotheses setup.

  • Clarifying conditions under which hypothesis tests necessitate different interpretations according to tail direction.