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What is a sampling distribution? What does the knowledge of σ contribute to a researcher’s understanding of the theoretical sampling distribution (regarding its characteristics and shape)? Contrast a t distribution from that of the standard normal distribution. In what ways might N affect the CI?
A sampling distribution is a probability distribution of a statistic (e.g., the mean or standard deviation) that is calculated from a large number of random samples drawn from a single population. It's a theoretical concept that shows the distribution of all possible values of that statistic for a given sample size.
σ
)
The knowledge of the population standard deviation (
σ
) is crucial to a researcher's understanding of the theoretical sampling distribution.
Shape: If the original population is normally distributed, the sampling distribution of the sample mean will also be normal, regardless of sample size. However, even if the population is not normal, the Central Limit Theorem states that the sampling distribution of the mean will approach a normal distribution as the sample size (n) increases. The knowledge of
helps confirm the normality assumption and the applicability of the Central Limit Theorem.
Characteristics (Spread): The most direct impact of knowing
is on the calculation of the standard error (
), which is the standard deviation of the sampling distribution. The formula for the standard error of the mean is:
σA smaller
leads to a smaller standard error, meaning the sample means are more tightly clustered around the population mean. Conversely, a larger results in a wider, more spread-out sampling distribution, indicating more variability in the sample means.
The t-distribution and the standard normal distribution (z-distribution) are both bell-shaped and symmetric, with a mean of 0. However, they are used in different scenarios and have key differences in shape:
Known vs. Unknown Standard Deviation: The z-distribution is used when the population standard deviation (
) is known. The t-distribution is used when
is unknown and must be estimated using the sample standard deviation (s).
Degrees of Freedom: The t-distribution is defined by its degrees of freedom (df), which is typically calculated as the sample size minus 1