4.7 Constructing a Confidence Interval for the Difference Between Two Population Means

Identify an appropriate confidence interval procedure including the parameter for the difference between two population means.

• Based on the sample data, a confidence interval can be calculated to estimate the difference between two population means. The appropriate confidence interval procedure for two independent samples is a twosample t-interval for the difference between population means.
• The parameter for a confidence interval for a two-sample t-interval for the difference between population means should reference the difference in the means, the response variable, and the populations in context.

Justify the appropriateness of constructing a confidence interval for the difference between two population means by verifying conditions.

• A two-sample t-interval for a difference between population means requires that three conditions be met:
  • i. The randomization condition—the data should be collected using two independent random samples or a randomized experiment.
  • ii. The 10% condition—when sampling without replacement, the size of each sample should be less than or equal to 10% of the respective population size: n11≤10%N and n22≤10%N , where N1 is the size of population 1 and N2 is the size of population 2. The sample sizes are represented as n1 and n2. (Note: This condition is unnecessary when the data are from a randomized experiment).
  • iii. The sample data condition—both samples should have a sample size greater than or equal to 30 or it is indicated that both population distributions are approximately normal. If either sample size is less than 30, both sample data distributions should be free from strong skewness and outliers.

Calculate an appropriate confidence interval for the difference between two population means.

• A point estimate for the difference between two population means is the difference in sample means, x12−x .
• For the difference between population means when the population standard deviations are unknown, the confidence interval can be constructed as point estimate ±(margin of error). The confidence interval for the difference between population means is ( ss2 2 xx12- ) ±+t* 1 2 nn1 2 , where t∗ are the critical values for the central C% of a t-distribution with appropriate degrees of freedom that can be found using technology. The degrees of freedom fall between n12+-n 2 and the smaller of n1 −1 and n2 −1.

Calculate the standard error and margin of error for estimating the difference between two population means.

• The standard error for the difference between two sample means is ss2 2 SE 1 2 xx- =+ 12 nn1 2 , where s1 and s2 are the sample standard deviations.
• For the difference between two sample means, the margin of error is the critical value (t∗) times the standard error (SE) of the difference of two sample means, which equals 2 * ss 2 t 1 + 2 nn1 2 . 135 Inference for Quantitative Data: Means UNIT 4