Lesson Ch. 24

Comparing Two Means

• Comparing two means involves similar concepts as comparing two proportions.

• For independent random variables, variances add together.

• To find the standard deviation for the difference of two sample means:

  • Use the formula:

    1. Sample Variance 1 / n1

    2. Sample Variance 2 / n2

    • Add these variances together.

    • Square root the total to find the standard deviation.

Standard Errors and the t-distribution

• Since population standard deviations for groups are typically unknown, we use standard error instead of standard deviation:

  • Substitute sample variances for population variances in formulas.

• The appropriate model for means is the Student's t-distribution, leading to a:

  • Two-sample t interval for the difference in means.

  • Two-sample hypothesis test.

• Calculators can perform these tests, specifically:

  • Option 4: Two-sample t test

  • Option 0: Two-sample t interval

Calculating the Estimation

• To find the t value:

  • Use the formula:

    • (Difference in sample means - Difference in population means) / Standard error of the difference.

• The model follows a Student's t distribution with degrees of freedom estimated via:

  • Two sample sizes added together minus 2 (n1 + n2 - 2).

  • Exact degrees of freedom may result in a decimal value, provided by the calculator.

Assumptions and Conditions

• Several assumptions and conditions must be met:

  • Independence Assumption: Requires randomization, indicating data can be generated from a simple random sample or randomized experiment.

  • 10% Condition: Generally not checked for difference of means unless sample sizes are quite large relative to population size.

  • Normal Population Assumption: Nearly normal condition must be checked for both groups, confirmed using histograms or normal probability plots.

  • Independence of Groups Assumption: Groups compared must be independent. For dependent groups, a different method is used.

Confidence Intervals for Difference

• The formula to calculate confidence intervals:

  • Difference of sample means ± Margin of error.

  • Margin of error = Critical t value × Standard error of the difference between means.

• Critical value determined by confidence level and degrees of freedom.

Hypothesis Testing for Means

• The hypothesis test checks:

  • Null hypothesis (Ho): The difference in population means = 0 (no difference).

  • Alternative hypothesis (Ha): The difference is not equal to 0 (indicating a difference).

• Conduct the t-test with the t statistic formula:

  • (Difference in sample means - Null hypothesis value) / Standard error.

• Obtain the p-value to decide whether to reject Ho.

Sample Case: Resting Pulse Rates

• Example: Compare mean pulse rates of smokers (mean = 80, sd = 5, n = 26) and nonsmokers (mean = 74, sd = 6, n = 32).

• Hypotheses:

  • Ho: Mean smokers - Mean nonsmokers = 0

  • Ha: Mean smokers - Mean nonsmokers ≠ 0

• Confirm conditions:

  • Random samples, under 10% population, nearly normal data.

• Resulting t value found to be approximately 4.15, with a corresponding p-value indicating strong evidence against the null.

Pooled t-test for Means

• Pooled T-test: Assumes equal variance; often unnecessary unless certain conditions apply.

• Use caution as equal means do not imply equal variances.

• For most tests, it is safer not to pool.

• Assumptions made in randomized comparative experiments can allow for pooling but should be approached with caution.

Conclusion from Case Studies

• Analyze results to determine if one group is significantly different.

• Example analysis of saturated fat content in pizzas:

• Conduct two-sample t-test for fat content:

  • Null: No difference in fat content

  • Alternate: There is a difference

• Results led to conclusions drawn from t-value and p-value calculations, confirming the significant difference in content.

• Visual representation through histograms and normal probability plots assists in corroborating findings.

• Confidence interval results provide a range to quantify the significant differences.