independent sample

(noun)

Two samples are independent as they are drawn from two different populations, and the samples have no effect on each other.

Related Terms

Examples of independent sample in the following topics:

  • t-Test for Two Samples: Independent and Overlapping

    • Two-sample t-tests for a difference in mean involve independent samples, paired samples, and overlapping samples.
    • The two sample t-test is used to compare the means of two independent samples.
    • Two-sample t-tests for a difference in mean involve independent samples, paired samples and overlapping samples.
    • The independent samples t-test is used when two separate sets of independent and identically distributed samples are obtained, one from each of the two populations being compared.
    • In this case, we have two independent samples and would use the unpaired form of the t-test .

  • Comparing Two Sample Averages

    • Student's t-test is used in order to compare two independent sample means.
    • In the t-test comparing the means of two independent samples, the following assumptions should be met:
    • Two-sample t-tests for a difference in mean involve independent samples, paired samples and overlapping samples.
    • The independent samples t-test is used when two separate sets of independent and identically distributed samples are obtained, one from each of the two populations being compared.
    • In this case, we have two independent samples and would use the unpaired form of the t-test.

  • Using Two Samples

    • The groups are classified either as independent or matched pairs.
    • Independent groups mean that the two samples taken are independent, that is, sample values selected from one population are not related in any way to sample values selected from the other population.
    • Matched pairs consist of two samples that are dependent.
    • In this section, we explore hypothesis testing of two independent population means (and proportions) and also tests for paired samples of population means.
    • Distinguish between independent and matched pairs in terms of hypothesis tests comparing two groups.

  • Summary of Types of Hypothesis Tests

    • Populations are independent and population standard deviations are known (not likely).

  • Point estimates and standard errors for differences of means

    • The two samples are independent of one-another, so the data are not paired.
    • Because we are examining two simple random samples from less than 10% of the population, each sample contains at least 30 observations, and neither distribution is strongly skewed, we can safely conclude the sampling distribution of each sample mean is nearly normal.
    • Finally, because each sample is independent of the other (e.g. the data are not paired), we can conclude that the difference in sample means can be modeled using a normal distribution.
    • If the sample means, $\bar{x}_1$ and $\bar{x}_2$, each meet the criteria for having nearly normal sampling distributions and the observations in the two samples are independent, then the difference in sample means, $\bar{x}_1-\bar{x}_2$, will have a sampling distribution that is nearly normal.
    • The sample difference of two means, $SE_{\bar{x}_1-\bar{x}_2}=\sqrt{\frac{s^2_1}{n_1}+\frac{s^2_2}{n_2}}$, is nearly normal with mean µ1 −2 and estimated standard error: $SE_{\bar{x}_1-\bar{x}_2}=\sqrt{\frac{s^2_1}{n_1}+\frac{s^2_2}{n_2}}$when each sample mean is nearly normal and all observations are independent.

  • The t distribution as a solution to the standard error problem

    • This is true for both small and large samples.
    • Use the t distribution for inference of the sample mean when observations are independent and nearly normal.
    • Independence of observations.
    • We collect a simple random sample from less than 10% of the population, or if it was an experiment or random process, we carefully check to the best of our abilities that the observations were independent.
    • When examining a sample mean and estimated standard error from a sample of n independent and nearly normal observations, we use a t distribution with n − 1 degrees of freedom (df).

  • Identifying when the sample proportion is nearly normal

    • A sample proportion can be described as a sample mean.
    • Conditions for the sampling distribution of ˆ p being nearly normal
    • The sampling distribution for , taken from a sample of size n from a population with a true proportion p, is nearly normal when
    • If data come from a simple random sample and consist of less than 10% of the population, then the independence assumption is reasonable.
    • Alternatively, if the data come from a random process, we must evaluate the independence condition more carefully.

  • Standard error of the mean

    • (a) Would you rather use a small sample or a large sample when estimating a parameter?
    • Computing SE for the sample mean Given n independent observations from a population with standard deviation σ, the standard error of the sample mean is equal to:
    • A reliable method to ensure sample observations are independent is to conduct a simple random sample consisting of less than 10% of the population.
    • Because the sample is simple random and consists of less than 10% of the population, the observations are independent.
    • A histogram of 1000 sample means for run time, where the samples are of size n = 100.

  • Examining the Central Limit Theorem

    • The normal model for the sample mean tends to be very good when the sample consists of at least 30 independent observations and the population data are not strongly skewed.
    • To make the assumption of independence we should perform careful checks on such data.
    • While the supporting analysis is not shown, no evidence was found to indicate the observations are not independent.
    • Since we should be skeptical of the independence of observations and the very extreme upper outlier poses a challenge, we should not use the normal model for the sample mean of these 50 observations.
    • Be especially cautious about independence assumptions regarding such data sets.

  • Sampling Distributions and the Central Limit Theorem

    • The central limit theorem for sample means states that as larger samples are drawn, the sample means form their own normal distribution.
    • The central limit theorem states that, given certain conditions, the mean of a sufficiently large number of independent random variables, each with a well-defined mean and well-defined variance, will be (approximately) normally distributed.
    • The central limit theorem for sample means specifically says that if you keep drawing larger and larger samples (like rolling 1, 2, 5, and, finally, 10 dice) and calculating their means the sample means form their own normal distribution (the sampling distribution).
    • Consider a sequence of independent and identically distributed random variables drawn from distributions of expected values given by $\mu$ and finite variances given by $\sigma^2$.
    • Illustrate that as the sample size gets larger, the sampling distribution approaches normality