Statistics Exam Preparation Notes

Statistics Exam Preparation Notes

Important Dates and Information

  • Office Hours:

    • Tuesday: 5:15 PM - 7:15 PM (Tony in Milbank 415Q)

    • Wednesday: 11:30 AM - 1:30 PM (Me in Milbank 415D)

    • Thursday: 7 PM - 9 PM (Tony on Zoom)

  • Current Assignments:

    • Assignment 4 is due next Monday.

Course Overview

  • Hypothesis Testing: Comparison between groups and implications of sample and population data.

  • Understanding Statistics: Importance of such concepts to business and research.

Hypothesis Testing Basics

  • Types of Errors:

    • Type I Error: Rejecting the null hypothesis when true (false positive).

    • Type II Error: Failing to reject the null hypothesis when false (false negative).

  • Four Steps in Hypothesis Testing:

    1. State the Hypothesis.

    2. Set decision criteria.

    3. Collect Data and Compute Statistics.

    4. Make a decision.

  • Mean Difference: Difference between the means of two groups must be evaluated for significance.

Testing Types

  • One-sample z-test: To evaluate whether sample mean differs from the known population mean when population's SD is known.

    • Formula:
      [ z = \frac{M - \mu}{\sigma/\sqrt{n}} ]

    • Results inform if there's a significant difference from the population mean.

    • Use critical z-values (e.g., ±1.96 for 95% CI).

  • Independent Measures t-Test: Comparing two sample means to see if they differ significantly.

    • Formula: [ t = \frac{(M1 - M2)}{s_p} ]

      • Where
        [ sp = \sqrt{\frac{SS1 + SS2}{df1 + df_2}} ]

    • Hypotheses:

      • Null: ( H0: \mu1 - \mu_2 = 0 )

      • Alternative: ( H1: \mu1 - \mu_2 \neq 0 )

  • Repeated Measures t-Test: Two conditions measured on the same participants to control for individual differences.

    • Difference Scores: ( D = X2 - X1 )

    • Calculated similarly to the independent measures t-test but depends on difference scores.

    • Hypotheses:

      • Null: ( H0: \muD = 0 )

      • Alternative: ( H1: \muD \neq 0 )

Effect Size and Statistical Power

  • Effect Size: Quantifies how large the difference is.

    • Cohen's d:
      [ d = \frac{M1 - M2}{s_p} ]

    • Significance does not equal practical importance.

    • Effects sizes are useful tools in evaluating the magnitude of the findings.

  • Statistical Power: The probability of correctly rejecting a false null hypothesis.

    • Factors Affecting Power: Sample size, effect size, alpha level.

    • Increasing sample size increases power but may also increase complexity in interpretation.

Confidence Intervals (CIs)

  • CI Basics: Provides a range in which we expect the population mean to lie with a certain level of confidence.

    • Formula:
      [ CI = M \pm (t_{critical} * SE) ]

    • Mean Interval Interpretation: For example, 95% CI means that if the same study were repeated, 95 out of 100 times the population mean would fall within this range.

Key Terms and Concepts

  • Standard Error (SE): The estimated standard deviation of the sampling distribution of a statistic.

  • Statistical Significance: Often determined by p-values and thresholds like 0.05 or 0.01.

  • True Population Parameters: Formulated through hypothesis testing and estimation techniques.

Practice Questions

  • Be sure to review assignments and past quizzes for practical application and understanding.

  • Example Questions:

    • What are the implications of a Type I error in a clinical trial?

    • When would you choose an independent measures design over repeated measures?

    • How does increasing sample size impact hypothesis testing?

Remember to review your notes frequently and utilize office hours to clarify any questions! Good luck on your exam!