4.6 Sampling Distributions for the Difference Between Two Population Means

Calculate the mean and standard deviation of a sampling distribution for the difference between two sample means.

• For two independent populations with population means µ1 and µ2 and population standard deviations σ1 and σ2, when the sampled values are independent, the sampling distribution of the difference in sample means x12−x has a mean μμ()xx μ 12 12 and standard deviation σσ2 2 σ 1 2 ()xx- =+ 12 nn1 2 .

Justify the appropriateness of conditions for the sampling distribution of the difference between two sample means.

• Sampling without replacement requires that two conditions must be met:
  • i. The randomization condition—the data should be collected using two independent random samples.
  • ii. The 10% condition—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.
• If the data come from an experiment, the data only need to meet the randomization condition. The treatments must be randomly assigned to experimental units to meet the randomization condition.
• The sampling distribution for the difference between sample means, x12−x , can be modeled with a normal distribution if the two population distributions can each be modeled by a normal distribution.
• The sampling distribution for the difference between sample means, x12−x , can be modeled approximately by a normal distribution if the two population distributions cannot be modeled by a normal distribution but n1 ≥30 and n2 ≥30.

Interpret the mean, standard deviation, and probabilities for a sampling distribution for the difference between sample means.

• The mean, standard deviation, and probabilities for a sampling distribution for the difference between sample means should be interpreted within the context of specific populations. 133 Inference for Quantitative Data: Means UNIT 4