PSYC 2317 Assignment 11 Single Sample T-Test 11.1 What Fact

PSYC 2317 Assignment #11 Single Sample t-Test 11.1 What factor determines whether you should use a z-test or a t-test statistic for a hypothesis test

Determine whether to use a z-test or a t-test for a hypothesis test based on the knowledge of population parameters and sample size. Specifically, a z-test is appropriate when the population standard deviation (σ) is known and the sample size is large (typically n ≥ 30). Conversely, a t-test is used when the population standard deviation is unknown and the sample size is small (n

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The decision between employing a z-test or a t-test in hypothesis testing hinges primarily on the knowledge of the population parameters and the size of the sample. When conducting research, understanding the underlying assumptions about the population is essential to selecting the most appropriate statistical test, ensuring the validity of the results.

A z-test is a parametric test used when the population standard deviation (σ) is known, and the sample size is sufficiently large—generally n ≥ 30, aligning with the Central Limit Theorem which asserts that the sampling distribution of the mean approaches normality regardless of the population distribution under these conditions (Field, 2013). The knowledge of σ allows direct calculation of the standard error of the mean, and the test statistic follows a standard normal distribution (z-distribution). This test is advantageous because it provides precise p-values when the population standard deviation is known, facilitating exact hypothesis testing (McDonald, 2014).

In contrast, a t-test is employed when the population standard deviation is unknown—a common scenario in practice—and the sample size is small (n

To illustrate, consider a scenario where a researcher tests whether a new teaching method affects test scores. If the population standard deviation of test scores is known from previous research, and the sample size is large, a z-test is appropriate because it offers an exact measure of significance. Conversely, if the population standard deviation is unspecified, and only a small sample is available, a t-test should be used to incorporate the extra uncertainty associated with estimating σ from s.

Thus, the key determinants for choosing between a z-test and a t-test are whether the population standard deviation is known and the size of the sample. When both conditions favor a z-test—known σ and large n—its use is justified because of the higher precision. When σ is unknown and the sample is small, the t-test's flexibility and adjustment for degrees of freedom make it the appropriate choice. Recognizing these factors ensures rigorous hypothesis testing and reliable research conclusions (Bewick et al., 2004; Howell, 2012).

References

  • Bewick, V., Cheek, L., & Ball, J. (2004). Statistics review 12: Primary data analysis—Part 1: Comparing groups—t-test and chi-squared test. BMJ, 329(7458), 698–702.
  • Cohen, J., Cohen, P., West, S. G., & Aiken, L. S. (2013). Applied multiple regression/correlation analysis for the behavioral sciences (3rd ed.). Routledge.
  • Field, A. (2013). Discovering statistics using IBM SPSS statistics (4th ed.). Sage Publications.
  • Laerd Statistics. (2017). When to use a t-test. Retrieved from https://statistics.laerd.com/statistical-guides/independent-samples-t-test-statistical-guide.php
  • McDonald, J. H. (2014). Handbook of biological statistics (3rd ed.). Sparky House Publishing.
  • Sharma, S. K., Kanchan, T., & Krishan, K. (2018). Descriptive statistics. The Encyclopedia of Archaeological Sciences, 1–8.

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