Statistics Lecture Review

Introduction

  • Overview of statistics application regarding a cereal company.

  • Importance of calculating specific statistics such as mean (XBAR) and standard deviation (s).

Finding Key Statistical Values

  • Concept of mean: XBAR is calculated from sample data.

  • Concept of standard deviation: s assists in understanding variability in data.

    • Example values presented:

    • XBAR = 14.05

    • s = 0.34

Hypothesis Testing Setup

Claims and Hypothesis

  • Claim regarding mean weight: The mean weight (mu) of the pack is at least 14 ounces.

    • Formulation of null hypothesis (H0): ( H_0: \mu = 14 )

    • Formulation of alternate hypothesis (H1): ( H_1: \mu < 14 )

Calculation of t-Score

  • Formula for t-score calculation:
    [ t = \frac{\text{XBAR} - \mu}{s / \sqrt{n}} ]

  • Substituting known values:

    • ( t = \frac{14.05 - 14}{0.346 / \sqrt{8}} )

    • Computation results in ( t = 0.408 )

Statistical Significance

  • Level of significance set at ( \alpha = 0.01 )

  • Identification of test type: Left-tail test (since we're testing if the mean is less than a certain value).

    • Critical value for one-tailed test at 0.01 significance level: -2.998

Conclusion of the Test

  • Because the calculated t-score (0.408) is within the fail-to-reject region relative to -2.998:

    • Decision made: Fail to reject the null hypothesis.

    • Implication: Insufficient evidence to claim the mean is less than 14 ounces.

Understanding Different Hypothesis Tests

Types of Tests

  1. Hypothesis Testing with Known Standard Deviation (σ):

    • Utilization of z-scores.

    • Formula: [ z = \frac{p_{hat} - p}{\sqrt{\frac{pq}{n}}} ]

  2. Hypothesis Testing with Unknown Standard Deviation:

    • Utilizes t-scores as described above.

  3. Hypothesis Testing for Proportions:

    • Important concepts include:

      • p_hat (sample proportion), p (population proportion), and q (1 - p).

Example of Proportion Testing

  • Problem scenario: University of Michigan believes that more than 600 students support online reprogramming.

  • Survey conducted on a sample of 200 students yielding 132 favorable responses.

    • Hypothesis setup:

    • Claim: ( p > 0.60 )

    • Null hypothesis: ( p = 0.60 )

    • Alternate hypothesis: ( p < 0.60 )

Calculation of Test Statistic for Proportions

  • Determination of p and p_hat:

    • p = 0.60 (the belief from the claim).

    • p_hat = ( \frac{132}{200} = 0.66 )

    • q would be ( 1 - 0.60 = 0.40 )

  • Plugging values into the proportion test formula yields:
    [ z = \frac{0.66 - 0.60}{\sqrt{0.60 * 0.40 / 200}} ]

Reporting Test Findings

  • Test statistic calculated as ( 1.732 )

  • Comparing this z-score to table values to determine critical z-value for right-tail test at ( \alpha = 0.05 ): z critical = 1.645.

  • Conclusion: Since 1.732 > 1.645, we reject the null hypothesis, indicating supporting evidence for the claim that more than 60% support online programming.

Summary on Test Result Interpretation

  • Importance of carefully identifying null and alternate hypotheses.

  • Analysis of test statistics to ensure correct conclusion about the population based on sample data.

    • Notes on reviewing calculation steps to avoid errors.

  • Recommendations for further stat tests: past papers, and problem sets.

Homework Assignments and Review

  • Assignment of Chapter Practice Numbers to reinforce concepts discussed.

    • Example: Chapter 11 practice number three, focusing on hypothesis testing fundamentals.