Sampling Distributions and Central Limit Theorem in Statistics

What does the Central Limit Theorem state regarding the sampling distribution of the sample mean?

When the sample size n is large, the sampling distribution of the sample mean is approximately Normal.

What is a point estimate?

A single value used to estimate a population parameter.

What is a point estimator?

A statistic that provides an estimate of a population parameter.

What is the primary reason that repeated samples of the same size result in different sample means?

Sampling variability.

How does increasing the sample size affect the mean of the sampling distribution of the sample mean?

The mean of the sampling distribution remains approximately equal to the population mean.

What happens to the standard deviation of the sampling distribution as the sample size increases?

It decreases.

How does the shape of the sampling distribution change as the sample size increases?

It becomes more Normal.

What does the Law of Large Numbers state?

As the sample size increases, the sample mean tends to get closer to the population mean.

What is the mean of the sampling distribution of the sample mean (μx̄)?

It is equal to the population mean (μ).

What is the formula for the standard deviation of the sampling distribution of the sample mean (σx̄)?

σ / √n

If the population distribution is Normal, what is the shape of the sampling distribution of the sample mean?

Exactly Normal, regardless of the sample size.

If the population distribution is non-normal, what condition must be met for the sampling distribution to be approximately Normal?

The sample size n must be at least 30.

What is the status of the sampling distribution if the population is non-normal and the sample size is less than 30?

The sampling distribution is unknown.

What is an unbiased estimator?

A statistic whose sampling distribution mean is equal to the true value of the parameter being estimated.

What does 'bias' refer to in the context of a sampling distribution?

The center of the sampling distribution.

What determines the variability of a statistic?

The sampling design and the sample size n.

What are the two key properties of unbiased estimators?

Low bias and low variability.

What is the standard deviation of the sampling distribution for a population with σ = 1.2 and a sample size of 9?

0.4 (calculated as 1.2 / √9)

Why is it important to check the sample size when the population distribution is not Normal?

To determine if the Central Limit Theorem applies (n ≥ 30) so the sampling distribution can be assumed approximately Normal.

What does the notation x̄ ~ N(μ, σ/√n) represent?

The sampling distribution of the sample mean follows a Normal distribution with mean μ and standard deviation σ/√n.

How is the variability of a statistic described?

By the spread of its sampling distribution.

What is the most common way to describe quantitative data?

Using sample means.