Ch 7: The Sampling Distribution of the Sample Mean

If a variable x is normally distributed with mean μ and standard deviation σ, then for a sample size n, the variable x̄ ....

is also normally distributed and has - mean μ - standard deviation σ/√n.

Give the equation for the standard deviation of the sample mean

sigma_xbar = the standard deviation of the sample mean sigma = population standard deviation n = sample size

What can we say about the distribution of xBar?

- If x is normally distributed, so is xBar, regardless of sample size. - If the sample size is large, xBar is approximately normally distributed, regardless of the distribution of x. (THE SECOND BIT IS THE CENTRAL LIMIT THEOREM)

How do you find the PERCENT of datapoints that lie within a specified range IF the variable is normally distributed?

Find Z-scores. Find the area under the curve between the enpoints of the range.

How do you find the probability that a variable will be within a specified range IF the variable is NORMALLY DISTRIBUTED?

Find Z-scores. Find the area under the curve between the enpoints of the range.

The Central Limit Theorem (CLT)

For a relatively large sample size, the variable xBAR is approximately normally distributed, REGARDLESS of the distribution of the variable in consideration. The approximation becomes better with increasing sample size.

If a problem tells you that the error can be within 5, what do you do?

If the error can be within five, then the "acceptable range" is.... Mean - 5 to Mean + 5 (Usually, you'll need to find z-scores for these endpoints, then find the area between them.)

What is Sampling Error?

Sampling Error is the error resulting from using a sample to estimate a population characteristic.

Define the Sampling Distribution of the Sample Mean... (THIS IS THE MAJOR THING TO LEARN)

The Sampling Distribution of the Sample Mean is the distribution of all possible sample means of a given sample size.

Compare the sampling error from small samples with the sampling error of large samples.

The sampling error of large samples tends to be less than the sampling error for small samples. Larger samples have less sampling error.

What happens to the mean of the sampling distribution of the sample mean as sample size gets bigger? (PROBABLY IGNORE)

When sample size gets bigger... the mean of the sampling distribution of the sample mean... approaches... The population mean.

What happens to the standard deviation of the sampling distribution of the sample mean as sample size gets bigger? (PROBABLY IGNORE)

When sample size gets bigger... the standard deviation of the sampling distribution of the sample mean... approaches... 0

The mean of all possible sample means always equals...

the population mean