Explain the rejection regions for small samples and large samples.

In hypothesis testing, a rejection region is a set of values that, if a test statistic falls within it, will lead to the rejection of the null hypothesis. The size of the rejection region depends on the level of significance chosen for the test.

The rejection region can be different for small samples and large samples. The distinction between small and large samples is based on the sample size in relation to the population size.

For small samples, where the sample size is less than or equal to 30, the distribution of the test statistic is usually non-normal and skewed. In such cases, the rejection region is determined using a t-distribution. The t-distribution has fatter tails than the normal distribution, which accounts for the increased variability in small samples. The rejection region is determined by finding the t-value for the desired level of significance (alpha) and degrees of freedom (n-1), where n is the sample size. This t-value represents the boundary beyond which the null hypothesis is rejected. The rejection region is the area in the tails of the t-distribution beyond this boundary.

For large samples, where the sample size is greater than 30 or the population size is infinite, the distribution of the test statistic is usually normal. In such cases, the rejection region is determined using a standard normal distribution. The rejection region is determined by finding the z-value for the desired level of significance (alpha). This z-value represents the boundary beyond which the null hypothesis is rejected. The rejection region is the area in the tails of the standard normal distribution beyond this boundary.

In summary, the rejection region for small samples is determined using a t-distribution, while the rejection region for large samples is determined using a standard normal distribution. The distinction between small and large samples is based on the sample size in relation to the population size.

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