Note that when we generate estimates for a population parameter in a single sample (e.g., the mean ) or population proportion ) the resulting confidence interval provides a range of likely values for that parameter. The standard error of the difference is 0.641, and the margin of error is 1.26 units. Our best estimate of the difference, the point estimate, is 1.7 units. Interpretation: With 95% confidence the difference in mean systolic blood pressures between men and women is between 0.44 and 2.96 units. Therefore, the confidence interval is (0.44, 2.96) Next we substitute the Z score for 95% confidence, Sp=19, the sample means, and the sample sizes into the equation for the confidence interval. Notice that for this example Sp, the pooled estimate of the common standard deviation, is 19, and this falls in between the standard deviations in the comparison groups (i.e., 17.5 and 20.1). The ratio of the sample variances is 17.5 2/20.1 2 = 0.76, which falls between 0.5 and 2, suggesting that the assumption of equality of population variances is reasonable.įirst, we need to compute Sp, the pooled estimate of the common standard deviation. Next, we will check the assumption of equality of population variances. If n 1 30 for both men and women), so we can use the confidence interval formula with Z. Use Z table for standard normal distribution If n 1 > 30 and n 2 > 30, we can use the z-table:.If either sample size is less than 30, then the t-table is used. If the sample sizes are larger, that is both n 1 and n 2 are greater than 30, then one uses the z-table. Therefore, the standard error (SE) of the difference in sample means is the pooled estimate of the common standard deviation (Sp) (assuming that the variances in the populations are similar) computed as the weighted average of the standard deviations in the samples, i.e.:Īnd the pooled estimate of the common standard deviation isĬomputing the Confidence Interval for a Difference Between Two Means If we assume equal variances between groups, we can pool the information on variability (sample variances) to generate an estimate of the population variability. The standard error of the point estimate will incorporate the variability in the outcome of interest in each of the comparison groups. The use of Z or t again depends on whether the sample sizes are large (n 1 > 30 and n 2 > 30) or small. The confidence interval will be computed using either the Z or t distribution for the selected confidence level and the standard error of the point estimate. The point estimate for the difference in population means is the difference in sample means: In the two independent samples application with a continuous outcome, the parameter of interest is the difference in population means, μ 1 - μ 2. We could begin by computing the sample sizes (n 1 and n 2), means ( and ), and standard deviations (s 1 and s 2) in each sample. Both of these situations involve comparisons between two independent groups, meaning that there are different people in the groups being compared. For example, we might be interested in comparing mean systolic blood pressure in men and women, or perhaps compare body mass index (BMI) in smokers and non-smokers. There are many situations where it is of interest to compare two groups with respect to their mean scores on a continuous outcome. Confidence Interval for Two Independent Samples, Continuous Outcome
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