College of Science and Mathematics MSU Statistical Consulting Program

USEFUL HINTS ON VARIOUS KINDS OF

Standard Testing

Z-test, T-tests, ANOVA, and right/left/two-tailed tests
This website was specifically designed to help you better understand what statistical tests are all about, under what conditions they can be used, and how they can be performed. Because most of the statistical calculations these days can be done using computer software, this website will help you become more knowledgeable in interpreting results generated by programs like SAS, JMP, and TI-83 Plus graphing calculator.

 *****This site is under construction*****

Z-tests

T-tests

ANOVA

right/left/two-tailed tests

 

Part 1. Briefly about the Z-test

The reason why we begin with a z-test is certainly not because it is the most common form of the statistical inference. Actually, it is rather rarely used because of its initial conditions, which will be described in more detail later on. A z-test always uses the same density curve to generate the test statistics. Because of the rigorous initial assumptions, like population standard deviation and normality of the data set, these tests are often talked about in statistics textbooks, but not necessarily useful in real life situations. A z-test involves unrealistic assumptions that often can not be addressed in real life cases.

Why is a z-test going to be rarely used in real life situations? In order to perform a z-test to test a claim about the population mean or variance, we must first know the standard deviation of the population. In practice, it is close to impossible. The standard deviation of the population cannot be determined unless we select and study all the individuals in the population. Such task is often impossible to accomplish due to financial limitations, time constraints, and large population sizes. Statisticians prefer to perform t-tests in place of z-tests because they are more applicable and appropriate. Let's than discuss t-tests because they are much more frequently used.

Examples of z-tests can be found above when we discuss right-, left- and two-tailed tests.

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Part 2. T-test (Inference about the population mean)

 Let us first discuss the necessary assumptions that must be met in order to perform a t-test.

THE NECESSARY CONDITIONS TABLE

The first assumption is that the data set that we have to study is a simple random sample.     For details on SRS go to Sampling Strategies.

Another assumption is that the individuals in our sample came from a normally distributed population. Practically speaking, the population distribution curve must be symmetric and it must contain a "peak" in the center of all values.

The standard deviation(s) is are) unknown. For the purpose of proceeding with a  t-test we will use the sample(s) standard deviation(s), estimate(s) of the real         standard deviation(s).

 

 

   T-TESTS:

one-sample t-test

matched pair t-test

 two-sample t-test

pooled two-sample t test

 

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Part 3. Analysis of Variance (ANOVA)

The main purpose of ANOVA is to compare several population means at the same time.

THE EQUAL SAMPLE SIZES CASE

Null Hypothesis: All the means are equal.

Alternative Hypothesis: Not all the means are equal.

When the variability between the samples is "large" in relation to the variability within samples, then we have sufficient evidence to reject the null hypothesis and conclude that not all the means are equal.  What is the between- and within- sample variability?

BETWEEN-SAMPLE VARIABILITY is the product of the variance of the samples' means and the samples' size.

WITHIN-SAMPLE VARIABILITY is the average variability of the samples.

We finally take a ratio of "between" to "within" to generate a statistic known as the F value.

 

Properties of the F distribution

  1. a pair of degrees of freedom for each F distribution (k-1,n-k), where k is the number of populations and n is the total of all values in all samples - numerator (k-1) and denominator (n-k)

  2. graph right-skewed, starting at 0, single-peaked

        

For the alternative hypothesis of "Not all the means are equal" we perform a right-tailed F test.  Similarly to t tests, we generate the test statistic (F=variance between samples/variance within samples).  If the F test statistic is larger than the critical value, we reject the null hypothesis.  The critical values can be obtained from the F Distribution Tables.  Tables differ according with the significance level.  Therefore, given a significance level, you find an appropriate F table, and obtain your critical values off the table by using the degrees of freedom of the numerator (on the horizontal) and the denominator (on the vertical).

If you have access to computer software, the process once again simplifies to comparing the P-value with the significance level.  When the first is smaller than the second, we reject the null hypothesis.

   IMPORTANT...   Necessary assumptions and suggestions: 

  1. Populations were normally distributed.

  2. Population variances were equal (also known as homoscedasticity). 

  3. Even if we reject the null hypothesis, we have gain no knowledge as to which mean is larger or smaller than the rest.

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 Example with 3 samples

THE UNEQUAL SAMPLE SIZES CASE

 Example with 4 samples