Sampling Distribution

Definition

The sampling distribution is a probability distribution that describes the possible values of a statistic, such as the sample mean or sample proportion, obtained from all possible samples of the same size drawn from a population. It represents the distribution of a statistic across all possible samples, rather than the distribution of the population itself.

5 Must Know Facts For Your Next Test

1. The sampling distribution is the foundation for many statistical inference techniques, including confidence intervals and hypothesis testing.
2. The shape of the sampling distribution of the sample mean depends on the shape of the population distribution and the sample size, according to the Central Limit Theorem.
3. The standard error of the sample mean is inversely proportional to the square root of the sample size, indicating that larger samples lead to more precise estimates of the population mean.
4. The sampling distribution of the sample proportion also follows the Central Limit Theorem, and its standard error is inversely proportional to the square root of the product of the sample size and the population proportion.
5. The finite population correction factor is used to adjust the standard error of the sample mean when the sample size is a significant proportion of the population size.

Review Questions

• Explain how the Central Limit Theorem relates to the sampling distribution of the sample mean.
  • The Central Limit Theorem states that as the sample size increases, the sampling distribution of the sample mean approaches a normal distribution, regardless of the shape of the population distribution. This means that for large enough sample sizes, the sample mean will be approximately normally distributed, with a mean equal to the population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size. This property of the sampling distribution of the sample mean is crucial for making inferences about the population mean using confidence intervals and hypothesis tests.
• Describe how the finite population correction factor is used to adjust the standard error of the sample mean when the sample size is a significant proportion of the population size.
  • When the sample size is a significant proportion of the population size, the standard error of the sample mean needs to be adjusted to account for the fact that the population is finite. The finite population correction factor is used to make this adjustment. The formula for the finite population correction factor is $\sqrt{(N-n)/(N-1)}$, where $N$ is the population size and $n$ is the sample size. This factor is multiplied by the standard error of the sample mean calculated under the assumption of an infinite population to obtain the adjusted standard error. Using the finite population correction factor is important when the sample size is a large proportion of the population size, as it can significantly affect the precision of inferences made about the population mean.
• Analyze how the sampling distribution of the sample proportion is related to the Central Limit Theorem and the standard error of the sample proportion.
  • Similar to the sampling distribution of the sample mean, the sampling distribution of the sample proportion also follows the Central Limit Theorem. As the sample size increases, the sampling distribution of the sample proportion approaches a normal distribution, regardless of the underlying population proportion. The standard error of the sample proportion is inversely proportional to the square root of the product of the sample size and the population proportion. This means that as the sample size increases, the standard error of the sample proportion decreases, leading to more precise estimates of the population proportion. The relationship between the sampling distribution of the sample proportion, the Central Limit Theorem, and the standard error of the sample proportion is crucial for making inferences about population proportions using confidence intervals and hypothesis tests.

Related terms

Central Limit Theorem:

The Central Limit Theorem states that as the sample size increases, the sampling distribution of the sample mean approaches a normal distribution, regardless of the shape of the population distribution.

Standard Error:

The standard error is the standard deviation of the sampling distribution of a statistic, such as the sample mean or sample proportion. It measures the variability of the statistic across all possible samples.

Hypothesis Testing:

Hypothesis testing involves making inferences about a population parameter based on a sample statistic and the sampling distribution of that statistic.