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8. Hypothesis Tests Concerning the Mean: T-Test vs. Z-Test

g. identify the appropriate test statistic and interpret the results for a hypothesis test concerning the population mean of both large and small samples when the population is normally or approximately distributed and the variance is 1) known or 2) unknown;

What is the difference between a t-statistics and a z-statistic? - A t-statistic follows a t-distribution - A z-statistic follows a z-distribution (a standard normal distribution)

When should a t-test be used? The t-test should be used if the population variance is unknown and either of the following conditions hold: - The sample size is large (n >=30) - The sample size is small (n < 30) but the population is normally distributed or approximately normally distributed

What is the t-test formula? \(t_{n-1} = \frac{\bar{X} - \mu_0}{\frac{s}{\sqrt{n}}}\) - t_{n-1} = t-statistic with n-1 degrees of freedom - \bar{X} = the sample mean - n = the sample size - \mu_0 = the null hypothesis population mean - s = the sample standard deviation - s/sqrt{n} = the standard error of the sample mean

How is a t-table used? A t-table is similar to a z-table except the rows use “degrees of freedom” instead of “standard deviations”, and the columns use the “level of significance” instead of standard deviation.

How do you find the “degrees of freedom” that is used in a t-table? The degrees of freedom is n-1 where n = the sample size

If you are solving a two-tailed test and are only given a one-tailed t-table, how do you convert? Divide the level of significance by 2 and find the appropriate column. E.g. if your level of significance is 0.05, you would find the column for 0.025. Then use the t-score given for both the upper and lower rejection points.

What is the difference for the test statistic if the population mean is known vs unknown? - If it is unknown you use the t-statistic and the sample population mean. If known you use the z-statistic and the overall population mean - Unknown: \(t_{n-1} = \frac{\bar{X} - \mu_0}{\frac{s}{\sqrt{n}}}\) - Unknown: \(t_{n-1} = \frac{\bar{X} - \mu_0}{\frac{\sigma}{\sqrt{n}}}\)

When hypothesis testing a population mean and you don’t know the population mean, should you use the z-statistic or the t-statistic? You should use the t-statistic if you don’t know the population mean unless you have a very big population size (when the t-score begins to look a lot like the z-score).

The Jones Fund example on this page is an excellent walk through of using a Hypothesis with t-test: https://analystnotes.com/cfa-study-notes-hypothesis-tests-concerning-the-mean-t-test-versus-z-test.html

In practice, is the t-test or z-test used more often? Why? The t-test is used more often than the z-test, because the z-test requires that the population variance be known, but in the real world we usually won’t know the population variance.

When is it OK to use a z-test (instead of a t-test) even if you don’t know the population variance? If you have a very large population sample it’s ok to use a z-test because the central limit theorem implies that that distribution of a sample mean will be approximately normally distributed as the sample size increases.

When does the t-graph begin to resemble a z-graph? Degrees of freedom for the t-distribution are represented as n - 1, so as n increases so do the degrees of freedom, and the t-graph starts to look like the z-graph. At “infinity” degrees of freedom the two graphs are identical.