Z Test and T Test

One-Sample z Test

Here we are dealing with the mean of a sample, whereas in looking at z scores we finding the place of a single score in a distribution.

Two-tailed test.
One-tailed test.

Two-tailed versus one-tailed tests:
In a two-tailed test, we are asking is the result unlikely in either direction. So if we set α = .05, we are asking is it in the 2.5% area of either tail.
If we are doing a one-tailed test, we want to know if the result is in the 5% tail on the side that interests us.
So we could ask, "Is the result of using this drug significant at α = .05 in a two-tailed test?"
This means "Did the drug produce results either so good or so bad that we should expect this result less than 5% of the time by chance?
Or we could ask, "Is the result of using this drug significant at α = .05 in a one-tailed test?"
If our tail is the positive one, we are asking, "Should we expect a result this good less than 5% of the time by chance?"
And the opposite for the negative tail!
The difference: in a two-tailed test, we are asking, "Did this drug have any extreme impact, either good or bad?"
In a one-tailed test, we are asking, "Did the drug have a very good (or bad) impact?"

The equation for the standard deviation of the sample means is:

Standard deviation of the sample mean.

What does this signify? We need to know the SD of the population as a whole. Once we know that, we know that the SD of the sample means will come closer and closer to the population mean, in proportion to the square root of the number of items in the sample we take.
The square root bit explains why it is often not economical to increase the sample size beyond a certain amount to achieve a lower &σ;M: while increasing the sample size by four times will likely increase the cost of the sampling by four times, it will only decrease the SD of the sample mean by two times.

The equation for the z test is:

Z test
Reporting the Results of the One-Sample z Test

When the results of a test like this are reported, we will find a statement like:
"There was evidence the HS students who had low GPAs had lower SAT scores (M = 467) than the general population (z = -2.31, p < .05)."

Confidence Interval for z

When we do this test, we are asking, "Given a sample mean, how likely is that sample mean for a given range of population means?"
So we need to calculate σM, as in the earlier section. Then we use that SD and the z table to get our range.

Wikipedia page on confidence intervals

Assumptions of the One-Sample z Test
  1. We have at hand interval or ratio data.
  2. We have a random sample drawn from the population of interest.
  3. The population of interest is normally distributed.
  4. The population standard deviation is known.
Wrap Up
Purpose of Using the One-Sample z Test
Assumptions of the One-Sample z Test
Summary of the One-Sample z Test
The One-Sample t Test
Reporting the Results of the One-Sample t Test
Confidence Interval for t
Central Limit Theorem
Purpose and Limitations of Using the One-Sample t Test
Assumptions of the One-Sample t Test
Extras