# Standardized Scores and Probability: T and Z Tests Coursework

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Updated: Jun 9th, 2022

## Summary

The t-test and the z test are very similar. The real difference between these is the circumstances under which they are useful. The Z-test is used to determine if the difference between a sample mean and the population means is large enough to be statistically significant, or if it is more likely to have occurred by chance. The Z-test is usually used with standardized testing to determine if the scores of a certain sample of test-takers are inside or outside of the standard performance of test-takers. In other words, it is used to find if the deviation is within acceptable parameters to be useful to the research. The Z-test can show if the mean of a normally distributed population has a value specified in a null hypothesis. It can also be used to test the null hypothesis that the means of two populations that have similar make-up and distribution of variables are equal.

The t-test can be used with two data sets, each having its own mean, standard deviation and a similar number of data points, to determine if the means are distinct, provided that the underlying distributions are normal. There are different versions of the t-test. One type is used if the two data sets are unpaired, not connected, such as those which are randomly assigned into two groups, and these are measured after being compared with the other group, so that each unit of one sample has a unique relationship with a particular unit of the other sample (i.e., the same people are measured before and after an intervention or otherwise paired entities).

## Data types for z-tests

• data points must be independent of each other
• n is greater than 30
• distributions are normal if n is low unless n>30 when the distribution of the data may be abnormal normal
• the variances of the samples are the same
• all individuals are selected at random
• all individuals have an equal chance of being selected
• sample sizes should be as equal as possible but small differences can be tolerated

## Data types for t-tests

• data sets are independent of each other except for paired-sample t-tests
• all cases where n<30 the t-tests are used
• distributions are normal for the equal and unequal variance t-test
• the variances of the samples are the same as for the equal variance t-test
• all individuals are selected at random
• all individuals have an equal chance of being selected
• sample sizes are as equal as possible but small differences are tolerated

The percentages are as stated in the above chart. More than 30% variance from the mean should use the z-test, while the t-test works well for less than 30%.

C-Engage Learning, 2008, Research Methods. Web.