How to compare two populations from independent samples?

How to compare two populations from independent samples?

We consider each case separately, beginning with independent samples. As with comparing two population proportions, when we compare two population means from independent populations, the interest is in the difference of the two means.

When to use pooled variance for Population 1?

When we have good reason to believe that the variance for population 1 is equal to that of population 2, we can estimate the common variance by pooling information from samples from population 1 and population 2. An informal check for this is to compare the ratio of the two sample standard deviations.

What does 100% confidence interval mean in population comparison?

100(1 − α)% Confidence Interval for the Difference Between Two Population Means: Small, Independent Samples df = n1 + n2 − 2. The samples must be independent, the populations must be normal, and the population standard deviations must be equal. “Small” samples means that either n1 < 30 or n2 < 30.

How to test hypotheses concerning two population means?

Testing hypotheses concerning the difference of two population means using small samples is done precisely as it is done for large samples, using the following standardized test statistic.

When to use standard deviation to compare two populations?

When the sample sizes are nearly equal (admittedly “nearly equal” is somewhat ambiguous, so often if sample sizes are small one requires they be equal), then a good Rule of Thumb to use is to see if the ratio falls from 0.5 to 2. That is, neither sample standard deviation is more than twice the other.

How does sample size affect the distribution of the population?

As the sample size (n) gets larger, the sample means tend to cluster around the true population mean Holds true, regardless of the distribution of the population

When is the sampling distribution is approximately normal?

Using the Central Limit Theorem, if the population is not normal, then with a large sample, the sampling distribution is approximately normal. The theorem presented in this Lesson says that if either of the above are true, then x ¯ 1 − x ¯ 2 is approximately normal with mean μ 1 − μ 2, and standard error σ 1 2 n 1 + σ 2 2 n 2.

What does large sample mean for two population means?

In the context of estimating or testing hypotheses concerning two population means, “large” samples means that both samples are large. A confidence interval for the difference in two population means is computed using a formula in the same fashion as was done for a single population mean.

How can we compare two types of samples?

The two types of samples require a different theory to construct a confidence interval and develop a hypothesis test. We consider each case separately, beginning with independent samples.

What does it mean to have a small sample?

The test statistic has Student’s t-distribution with df = n1 + n2 − 2 degrees of freedom. The samples must be independent, the populations must be normal, and the population standard deviations must be equal. “Small” samples means that either n1 < 30 or n2 < 30.

How to calculate the difference between two population proportions?

Standardized Test Statistic for Hypothesis Tests Concerning the Difference Between Two Population Proportions The test statistic has the standard normal distribution. must lie wholly within the interval [0, 1].

When is an independent sample a dependent sample?

It is important to be able to distinguish between an independent sample or a dependent sample. The samples from two populations are independent if the samples selected from one of the populations has no relationship with the samples selected from the other population.

How to compare two populations by mean and standard deviation?

Each population has a mean and a standard deviation. We arbitrarily label one population as Population 1 and the other as Population 2, and subscript the parameters with the numbers 1 and 2 to tell them apart. We draw a random sample from Population 1 and label the sample statistics it yields with the subscript 1.