Describe what is measured by the estimated standard error in the bottom of the independent-measures t statistic

Answer 1

The estimated standard error, also known as the standard error of the difference, is a measure of the variability or dispersion in the sampling distribution of the difference between two independent sample means in a t-test. It represents the standard deviation of the sampling distribution of the mean difference, and it quantifies the average amount of sampling error that is expected in estimating the true population difference between the means.

In the context of an independent-measures t-test, this estimated standard error is calculated by taking into account the sample sizes, variances, and covariance of the two independent samples. It helps us understand how much the observed difference between the sample means might deviate from the true difference between the population means due to random sampling variability alone.

To illustrate with an example, let's consider a study comparing the test scores of two groups of students, Group A and Group B. Group A consists of 50 students, and Group B consists of 60 students. The researcher wants to determine if there is a significant difference in the average test scores between the two groups.

The researcher collects the data and calculates the mean test score for Group A (mean_A) and Group B (mean_B). The researcher also calculates the standard deviation for Group A (SD_A) and Group B (SD_B). Additionally, the researcher computes the covariance (Cov_AB) between the two groups.

Using these values, the researcher can calculate the estimated standard error for the independent-measures t-test. It can be computed using the following formula:

estimated standard error = sqrt[(SD_A^2 / n_A) + (SD_B^2 / n_B) - (2 * Cov_AB / sqrt(n_A * n_B))]

where n_A and n_B represent the sample sizes of Group A and Group B, respectively.

The estimated standard error measures the average amount of variability expected in the difference between the sample means. A smaller estimated standard error indicates less variability and greater precision in estimating the true difference. Conversely, a larger estimated standard error suggests more variability and less precision in estimating the true difference.

In summary, the estimated standard error in the bottom of the independent-measures t statistic measures the expected variability in the difference between two sample means. It helps assess the precision of the estimated difference and provides an indication of how likely the observed difference is due to random sampling variability. A smaller estimated standard error suggests a more reliable estimate of the population difference, while a larger estimated standard error indicates greater uncertainty in the estimate.