We considered three hypothesis-testing procedures based on...

Name: Problem 195, Chapter 10: Introductory Statistics
Uploaded: 2021-07-06T15:53:24-08:00
Duration: 3 min 31 s
Channel: Maxime Rossetti
Description: We considered three hypothesis-testing procedures based on independent simple random samples to compare the means of two populations with unknown standard deviations. a. Identify the three procedures by name. b. List the conditions for using each procedure. c. Identify the test statistic for each procedure.

We considered three hypothesis-testing procedures based on independent simple random samples to compare the means of two populations with unknown standard deviations. a. Identify the three procedures by name. b. List the conditions for using each procedure. c. Identify the test statistic for each procedure.

We considered three hypothesis-testing procedures based on independent simple random samples to compare the means of two populations with unknown standard deviations.
a. Identify the three procedures by name.
b. List the conditions for using each procedure.
c. Identify the test statistic for each procedure.

Key Concepts

Pooled Two?Sample t-Test

This test is used when you have two independent simple random samples from normally distributed populations and you can assume that the population variances are equal. Under these conditions, the two sample variances are pooled into one combined estimate of variance. The test statistic is calculated as the difference between the two sample means minus the hypothesized difference (usually zero) divided by the pooled standard error, and it follows a t-distribution with degrees of freedom equal to the sum of the sample sizes minus two.

Welch's Two?Sample t-Test

When the assumption of equal variances does not hold, Welch’s t-test is the proper method for comparing two means. This procedure does not assume that the two population variances are equal and instead uses each sample’s variance to estimate its own standard error. The test statistic is computed as the difference between the sample means minus the hypothesized difference, divided by the square root of the sum of the individual estimated variances divided by their respective sample sizes. The degrees of freedom are approximated by the Welch–Satterthwaite formula.

Large?Sample (Normal Approximation) Test

When both sample sizes are sufficiently large, the Central Limit Theorem guarantees that the sampling distribution of the difference between the means is approximately normal, even if the underlying populations are not perfectly normal. In this situation, even though the population standard deviations are unknown, they can be replaced by the sample standard deviations. The test statistic is computed in a similar fashion to the t-tests, but because of the large sample size the normal distribution (i.e., the z distribution) is used as an approximation.

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