AP Stats sampling distributions and central limit theorem review

What is a sampling distribution?

It is the probability distribution of a statistic (like the sample mean) based on all possible samples from a population.

Fill in the blank: The mean of the sampling distribution is equal to the _____.

population mean (μ)

True or False: Larger sample sizes decrease the variability of the sampling distribution.

True. Larger samples lead to smaller standard errors.

If a population has a mean of $\displaystyle 100 and a standard deviation of$20, what is the mean of the sampling distribution?

The mean remains $100.

How does sample size affect the standard error?

Standard error (SE) decreases as sample size (n) increases: $\displaystyle SE = \\frac{\sigma}{\sqrt{n}}$.

Example: Find the standard error for a sample size of 25 with population SD of 10.

Calculate: $\displaystyle SE = \\frac{10}{\sqrt{25}} = 2$.

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

It becomes approximately normal, regardless of the population shape, due to the Central Limit Theorem.

Comparison: Sample mean vs. Population mean.

- Sample mean: Average of sample - Population mean: Average of whole population

How do you calculate the sampling distribution for sample means?

1. Take multiple samples 2. Calculate the mean for each 3. Create a distribution of those means.

If a store sells 200 items with a mean price of $50, what is the mean of the sample distribution?

It is still $50, same as the population mean.

Fill in the blank: The variability of a sampling distribution is measured by the _____.

standard error (SE)

True or False: A smaller sample size leads to a larger standard error.

True. Smaller n increases SE.

Example: Sample size of 36, population SD of $12. Find SE.

Calculate: $\displaystyle SE = \\frac{12}{\sqrt{36}} = 2$.

What is the role of the Law of Large Numbers?

As sample size increases, the sample mean converges to the population mean.

How is the sampling distribution of proportions defined?

It is the distribution of sample proportions from all possible samples of size n.

If you have a population proportion of 0.6 and a sample size of 50, how do you estimate the SE?

$\displaystyle SE = \sqrt{\\frac{p(1-p)}{n}} = \sqrt{\\frac{0.6(0.4)}{50}} \approx 0.109$.

What is the Central Limit Theorem?

The theorem states that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the population's distribution.

True or False: CLT applies only to normal populations.

False. CLT applies to any population as long as the sample size is sufficiently large (usually n ≥ 30).

Fill in the blank: The mean of the sampling distribution is equal to the ____ of the population.

mean

How does sample size affect the standard error?

As sample size increases, the standard error decreases, resulting in a tighter distribution of sample means.

Example: Mean salary is $50,000. Sample size is 40. What's the expected sample mean?

$50,000. The expected sample mean equals the population mean.

What is the formula for standard error (SE)?

The formula is: $\displaystyle SE = \\frac{\sigma}{\sqrt{n}}$, where σ is the population standard deviation and n is the sample size.

True or False: A sample size of 15 is sufficient for CLT.

False. Generally, a sample size of at least 30 is recommended for the CLT to hold.

What happens to the distribution of sample means as n increases?

It becomes increasingly normal, regardless of the shape of the population distribution.

If the population standard deviation is $10, what is SE for n=16?

$\displaystyle SE = \\frac{10}{\sqrt{16}} = 2.5$

Comparison: Normal vs. Non-normal distribution.

- Normal: Sample mean distribution is normal for any n. - Non-normal: Requires larger n (n ≥ 30) for sample mean to be approximately normal.

What does a smaller standard error indicate?

It indicates that the sample means are closer to the population mean, showing more precision.

Example: Population mean is $\displaystyle 200, SD is$50. n=25, what is SE?

$\displaystyle SE = \\frac{50}{\sqrt{25}} = 10$

What is the impact of outliers on the CLT?

Outliers can skew the population distribution, but the sample mean will still approximate normality with a large enough sample size.

True or False: CLT guarantees the population is normal.

False. CLT allows for non-normal populations; it just requires large samples for the sample means to be normal.

How does the Central Limit Theorem apply to polling?

It ensures that the average results from large samples of polls will follow a normal distribution, allowing for better predictions.

What is the relationship between sample size and confidence intervals?

Larger sample sizes yield narrower confidence intervals for the sample mean, increasing precision.

Example: Sample means from a skewed distribution. What happens as n increases?

The sample means will tend to form a normal distribution as n becomes larger.

True or False: The CLT only applies to independent samples.

True. Samples must be independent for CLT to hold.

What is the importance of the Central Limit Theorem?

It provides a foundation for inferential statistics, allowing for hypothesis testing and confidence interval construction.

True or False: Larger samples improve the CLT's accuracy.

True - Larger sample sizes lead to sample means that are closer to the population mean, making the sampling distribution more normal.