Sampling Distributions for Sample Means - AP Statistics

What is the role of the Central Limit Theorem in sampling distributions?

Allows approximation of sample mean distribution as normal. Enables normal approximation for inference procedures.
Identify the effect of increasing sample size on variability of sample mean.

Variability decreases as sample size increases. Standard error decreases proportional to $\frac{1}{\sqrt{n}}$ .
What is the effect of sample size on the accuracy of a sample mean as an estimator?

Larger sample size increases accuracy. Larger samples reduce sampling error.
What is the shape of the sampling distribution of the sample mean for a small sample size?

Depends on population shape, not always normal. Small samples retain original population distribution shape.
What happens to the spread of the sampling distribution as sample size increases?

The spread decreases as sample size increases. Variability decreases as sample size increases.
What does an unbiased estimator mean?

Estimator equals the population parameter on average. Expected value equals the true population parameter.
Identify the symbol for sample standard deviation.

$s$ . Standard notation for sample standard deviation.
What is a sampling distribution?

Distribution of a statistic over many samples. Shows how a statistic varies across repeated sampling.
What does the term 'finite population correction' mean?

Adjustment to standard error for small population. Modifies standard error when $n$ is large relative to population.
What is the general impact of outliers on the sampling distribution of the mean?

Outliers can skew the sampling distribution. Extreme values affect the distribution's shape.
How does sample size affect the reliability of sample mean?

Larger samples increase reliability. Lower variability means more precise estimates.
What is the effect of larger sample sizes on the sampling distribution?

Results in a narrower, more precise sampling distribution. Smaller standard error means more concentrated distribution.
What condition is required for a sample mean to be an unbiased estimator?

Sample must be random and representative. Random sampling eliminates systematic bias.
Identify the shape of the sampling distribution if the population is normal.

The sampling distribution is normal. Normal population always produces normal sampling distribution.
What does the term 'sampling variability' mean?

Variability of a statistic across different samples. Natural variation in statistics from sample to sample.
What is the sampling distribution of the sample mean for a skewed population?

Approximately normal for large $n$ due to Central Limit Theorem. CLT applies regardless of original population shape.
What is a biased estimator?

An estimator that does not equal the population parameter on average. Expected value differs systematically from true parameter.
Why is the Central Limit Theorem important?

It justifies using normal distribution for sample means. Foundation for statistical inference and hypothesis testing.
What is the importance of random sampling in sampling distributions?

Ensures each sample has an equal chance of selection. Prevents bias and ensures representative sampling.
Identify a condition for independence in sampling.

Sample size $n$ is less than 10% of the population. 10% rule prevents finite population effects.
What happens to the standard error as sample size increases?

Standard error decreases as $n$ increases. Due to inverse square root relationship with $n$ .
What is the notation for sample mean?

$\bar{x}$ . Standard symbol for sample mean.
What is the primary purpose of the Central Limit Theorem?

To allow normal approximation for sample means. Enables statistical inference about population means.
State the relationship between sample size and standard error.

As sample size increases, standard error decreases. Standard error is inversely proportional to $\sqrt{n}$ .

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