Ch. 23 Lesson

Chapter 23: Inferences About Means

Overview

• Focus on creating confidence intervals and hypothesis tests for means, similar to proportions.

• Use sampling distribution models based on the Central Limit Theorem.

Central Limit Theorem

• Sampling distribution for means follows a normal distribution.

• Mean is the true population mean (μ).

• Standard deviation of the distribution (standard error) = population standard deviation (C3) / √n.

• Requirement: Random sample of quantitative data with known population standard deviation.

  • Problem: Often, we do not have the population standard deviation.

Standard Error

• When C3 is unknown, use sample standard deviation (s) to calculate standard error:

  • Standard Error (SE) = s / √n.

• Extra variation is accounted for, requiring a new sampling model.

Student's t Model

• Developed by William S. Gossett at Guinness Brewery for quality control.

• Uses degrees of freedom (df) for estimating variability in small samples.

• Denoted as t(subscript df): t(model)

• Practical sampling distribution for means follows a Student’s t model for n-1 degrees of freedom.

• Formula for t value:

  • t = (sample mean - population mean) / SE.

Characteristics of Student's t Model

• Unimodal, symmetric, bell-shaped like normal distribution but with fatter tails.

• As degrees of freedom increase, t-model approaches normal distribution.

• t distribution with infinite degrees of freedom equals normal distribution.

Finding t Values

• Use statistical tables or technology to obtain critical t values for varied confidence levels.

• Example:

  • 1 df, two-tail 20% confidence = 3.078; 17 df, two-tail = 1.33.

Calculating Probabilities

• Use TCDF on calculators to determine probabilities for specific t values.

• Example probability calculation:

  • For t > 1.645 with 12 df: Use TCDF to get probability.

• Example completion:

  • Probability for t < 0.53 with 23 df: Result = 0.6994.

t Values for Confidence Intervals

• Identify critical t values for 90% confidence interval:

  • Center = 90% confidence, tails = 5%.

• Use inverse t function to find critical t scores (± for symmetry).

• Example:

  • For 90% CI, 6 df, critical t value ≈ ±1.943.

  • For 95% CI, 35 df, critical t value ≈ ±2.03.

Assumptions and Conditions

• Statistical models rely on assumptions:

  • Independence Assumption: Data are independent from one another.

  • Randomization Condition: Data drawn from random samples.

    • Ideal: Simple Random Samples (SRS).

  • 10% Condition: Sample size should not exceed 10% of the population when sampling without replacement.

  • Normal Population Assumption: Verify using nearly normal condition.

    • Check if data are unimodal and symmetric via histogram or normal probability plot.

    • Sample sizes impact normality assumptions:

      • < 15: Data should be normally distributed.

      • 15-40: Unimodal and symmetric sufficient for t-model.

      • 40: Central Limit Theorem guarantees t-model applicability.

TI Tips for Calculating Confidence Intervals

• Use TI-83/84:

  • Navigate to: STAT > TESTS > T-Interval.

  • Enter either raw data or summary statistics (mean, standard deviation, sample size).

Example Problem

• Context: Nutrition lab tests sodium content in hot dogs.

  • Sample mean = 310 mg, s = 36 mg, n = 40.

  • Calculate 95% CI using t interval.

• Resulting interval: 298.49 mg to 321.51 mg.

• Assumptions made regarding data normality and independent sampling.

Interpretation of Confidence Intervals

• Key: Proper interpretation essential.

• Do not confuse population mean with individual values.

  • Example Incorrect: "90% of all vehicles..."

  • Correct: 90% confidence that the true mean speed of vehicles lies within a specified interval.

Hypothesis Testing

• One-sample t-test for means follows the same conditions as t-interval:

  • Null hypothesis (H0): μ = μ0; Alternative hypothesis (HA): Test directional form (e.g., μ > μ0).

  • t value calculation: (sample mean - μ0) / SE.

Example Testing Scenario

• Context: Men's average marriage age.

  • H0: μ = 23.3 years.

  • Sample: n = 40, sample mean = 24.2 years, s = 5.3 years.

• Run test to find t value (1.074) and p-value (0.1447).

• Conclusion: Fail to reject H0; insufficient evidence for a higher average marriage age.

Additional Example: Maze Navigation by Rats

• Context: Testing maze navigation times with a target average of 60 seconds.

  • Consider conditions & check for outliers before hypothesis test.

• H0: μ = 60 seconds; two-tailed test for deviations from criteria.

• Example findings reveal importance of treatment of outliers in analysis.

Sample Size Calculations

• Formula adjustments based on margin of error and pilot studies.

• Given: Standard deviation and desired margin of error.

  • Example: Margin of error = 3, s = 5.

Conclusion

• Inferences about means utilize similar logic as for proportions, with adjustments to the model.

• Importance of verifying assumptions, confidence intervals, and statistical interpretation remains constant.