T test of means

Key Assumptions in Hypothesis Testing

  • Alternative Hypothesis ((H_a))

    • Defines whether (\mu) (population mean) is either less than, greater than, or not equal to (\mu_0) (hypothesized mean).

  • Assumption 1: Simple Random Sample (SRS)

    • Must ensure sample is randomly selected.

  • Assumption 2: Population Size

    • Population must be greater than 10 times the sample size (n).

  • Assumption 3: Sample Size Conditions

    • Large Sample Size: If ( n \geq 45 )

      • This supports the Central Limit Theorem.

      • Distribution's skewness or outliers do not impact the results.

    • Medium Sample Size: If ( 15 \leq n \leq 45 )

      • For moderate skewness, a maximum of one to two outliers is permissible.

      • Data can be symmetric or moderately skewed.

    • Small Sample Size: If ( n \leq 15 )

      • Data must be symmetric with no outliers.

Evaluating Skewness in Sample Data

  • To demonstrate medium or small sample conditions:

    • Visual Tools:

      • Box-and-Whisker Plot:

        • Effective for showing outliers.

        • Can indicate skewness visually.

      • Histograms:

        • Useful but not as effective for outlier detection.

  • Judgment in Samples:

    • If sample size is 44 or equal to 45, judgment call on skewness is needed.

    • A larger sample size can accommodate more skewness.

    • Lower sample sizes require stricter adherence to symmetry guidelines.

Choosing Between One-Sample T-Test and Z-Test

  • Z-Test: Used when the population standard deviation is known and sample size is large (( n \geq 45 )).

  • T-Test: Used in cases where population standard deviation is unknown:

    • The formula utilized:

      • ( t = \frac{\bar{x} - \mu}{s / \sqrt{n}} )

      • Involves sample standard deviation (s) instead of the population standard deviation (( \sigma )).

  • Degrees of Freedom:

    • Calculated as ( n - 1 ): represents the number of independent pieces of information.

Practical Application of the T-Test

  • Probability Determination:

    • Finding (t) scores using tables or statistical software to determine critical values or p-values.

  • Example Scenario:

    • If given a ( t ) score of 1 with 10 degrees of freedom, understand how to interpolate between values in a t-table to determine probabilities.

Conclusion and Exploration of New Tools

  • Familiarity with statistical software or calculators can ease calculations.

  • Discussion of visual aids and their importance in assessing data distributions.

  • Suggested exploration of additional learning resources or videos for clarity on concepts discussed.