THE z - TEST n Purpose: Compare a sample mean to a hypothesized population mean n Design: Any design where a sample mean is found.

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There are two statistical tests for mean: 1) z test – Used for large samples (n ≥ 30) 1) t test – Used for small samples (n < 30)

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Presentation transcript:

THE z - TEST n Purpose: Compare a sample mean to a hypothesized population mean n Design: Any design where a sample mean is found.

Assumptions 1. Independent observations 1. Independent observations 2. Normal population (or large N) 2. Normal population (or large N) 3. Population  is known. 3. Population  is known. 4. Interval or ratio level data. 4. Interval or ratio level data.

How it Works n Where does your sample mean fall in the sampling distribution? n The sampling distribution is made up of the sample means you would get if the Ho is true.

How it Works n If your sample mean is fairly typical (in the middle) for a population where the Ho is true, then fail to reject Ho. n If your sample mean is very unusual (on the tail of the distribution) for a population where the Ho is true, then reject Ho.

unusual typical

One-Tailed Test n Direction of difference is predicted. n Set a critical value on one tail of the sampling distribution. n If the observed statistic meets or beats the critical value, the test is significant and Ho is rejected.

one-tailed z-crit upper 5%

Two-Tailed Test n Direction of difference is not predicted. n Set two critical values, one on each tail of the sampling distribution. n If the observed statistic meets or beats either critical value, the test is significant and Ho is rejected.

two-tailed z-crit upper 2.5% z-crit lower 2.5%

Comparing One- and Two-Tailed n One-tailed is more powerful. n Two-tailed can be significant in either direction. n If you hypothesize in the wrong direction one-tailed, it can’t be significant no matter how big the difference.

Computation of the z-Test

Computing Standard Error

Example A standardized achievement test has a mean of 50 and a population standard deviation of 14. My class of 49 people got a mean of 56 on the test. Is this sample mean significantly different from the population mean? A standardized achievement test has a mean of 50 and a population standard deviation of 14. My class of 49 people got a mean of 56 on the test. Is this sample mean significantly different from the population mean?

STEP 1: Calculate the standard error of the mean.

STEP 2: Calculate the z. STEP 2: Calculate the z.

STEP 3: Find the critical value of z. For one-tailed,  =.05, z-crit = 1.65 For two-tailed,  =.05, z-crit = 1.96

STEP 4: Compare z to z-critical. If z is equal to or greater than z-crit, it is significant. (For 2-tailed tests, ignore the sign). STEP 4: Compare z to z-critical. If z is equal to or greater than z-crit, it is significant. (For 2-tailed tests, ignore the sign). z = 3.00, z-crit (2 tailed) = 1.96 Reject Ho; significant Reject Ho; significant

APA Format Sentence A z-test showed that the mean of the class was significantly different from the mean of the population, z = 3.00, p <.05. A z-test showed that the mean of the class was significantly different from the mean of the population, z = 3.00, p <.05.