Distribution of Sample Means & The Central Limit

Law of Large numbers

As the sample size (n) gets larger, the sample mean ( ¯x) will be close to μ. If the sample size is large, then the sample mean will typically be close to the population mean, u. This happens because the standard deviation σ/√n‾ will get smaller.

When is x-bar normally distributed?

If the data was from a normal population, or if the sample is sufficiently large.

The distribution of sample means is normal when

When ¯x is normally distributed or when, thanks to Central Limit Theorem (CLT), our sample size (n) is large.

For the mean of draws from a random variable with mean μ and standard deviation σ, the following are true:

The mean of the random variable X¯ is μ. The standard deviation of the random variable X¯ is σ/√n.

The distribution of sample mean

is a distribution of all possible sample means (x¯) for a particular sample size. It has a mean of u and a standard deviation of σ/√n‾.

What is the shape of the sampling distribution of the sample mean when the data are not normally distributed, but the sample size is large?

Approximately normal

How does the shape of the sampling distribution of the sample mean change when the sample size is increased?

As the sample size increase it gets closer to normal distribution. Standard deviation decreases, mean is unchanged.

What is the difference between the Central Limit Theorem and the Law of Large Numbers?

Central Limit Theorem - As the sample size gets larger it will get closer to normal. The shape will be approximately more distributed. If the sample size is large, the sample mean will be approximately normally distributed. Law of large numbers - The standard deviation of the sample mean will get smaller (closer to the true mean). The mean will get more accurate. The Law of Large Numbers states that as the sample size increases, the standard deviation of the sample mean will get smaller.

What are the two situations when the distribution ¯x is guaranteed to be normal (or as near as makes no difference).

If the parent population is normal, the distribution of the sample ¯x will be normal, for every sample size n. Even if the parent population is not normal, the Central Limit Theorem guarantees that the distribution of the sample mean x¯ will be approximately normal if the sample size n is large enough. (For this course, if n >_ 30, we will say the distribution of the sample means will be approximately normal, even if the parent population is not normal).

Central Limit Theorem

If the sample size is large that ¯x will be approximately normal.

The Central Limit Theorem

If the sample size is large, the random variable x will be approximately normally distributed.

How do you compute the z-score for any value of this random variable

z = value−mean / standard deviation= x¯−μ σ/√n Notice that we just replaced the "value" with the normal random variable, the "mean" with its mean and "standard deviation" with its standard deviation.

What is the shape of the sampling distribution of the sample mean when the data are normally distributed?

Normal

When is the sample mean normally distributed (or approximately normally distributed)?

The parent population is normally distributed, so the sample mean is automatically normally distributed. The sample size is large, and the Central Limit Theorem implies that the sample mean is normally distributed.

What is the P-value?

The probability of getting a test statistic at least as extreme as the one you got, assuming H0 is true. A P-value is calculated by finding the area under the normal distribution curve that is more extreme (farther away from the mean) than the z-score.

What is a sample distribution?

The sampling distribution is the set of all possible sample means from a population. Although we will only see one sample mean in our study, there are many sample means that could have possibly been observed.

How does the shape of the sampling distribution of the sample mean change when the sample size is increased?

The sampling distribution more closely approaches a normal distribution and the standard deviation decreases. The mean is unchanged.

How do we calculate the z-score?

When the distribution of sample means is normally distributed, we can use z-scores and the probability applet to calculate proportions and probabilities. A z-score is calculated as: z = value−mean standard deviation divided by the square root of the sample size n

Two ways that x can be (approximately) normally distributed.

if the data were drawn from a normal population if the sample size is sufficiently large