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a)

Two requirements have got to be met before the t-distribution is employed. The first is the sampling distribution's normality. Such that the x-bar uses a traditional distribution. It may accomplish in one of two ways. It would be best to grasp that the individual observations follow a standard distribution, or like to own a large sample size (more than 30), so it can depend upon the central limit theorem. Now, the t-distribution after estimating the population variance with the sample variance.

b)

The distributions are smooth in shape, and the t and z distributions are symmetric. Each of them has a zero mean. Both are continuous, bell-shaped distributions with denser tails than the z-distribution.

The t-statistic employs the sample standard deviation, whereas the z-statistic employs the standard deviation. The exact form of t-distribution changes as df increases. As df increases, the t distribution forms a normal z-score distribution. The t-distribution is flatter and more extensive, while the standard z-distribution has a more prominent central peak.

The zdeviation indicateshow fara fact set is from the suggested or specific facts set in trendy deviations. A z-test compares a pattern to a described populace and is typically used to resolve troubles related to large samples (n > 30). Z-testalso canbebeneficial while checking out a hypothesis. They are usually maximum beneficial while the same olddeviation is known.

T-tests, like z-tests, are calculations used to test a hypothesis. Still, they are most helpfulindetermining whether there is a statistically significant difference between twogroups ofindependentsamples. In other words, a t-test determines whether a difference in the means of two groups is likely to have occurred bychance.

T-tests are generally used when dealing with problems with a small sample size (n = 30).