Question

you are testing a claim and incorrectly use the normal sampling distribution instead of the t - sampling distribution. does this make it more or less likely to reject the null hypothesis? is this result the same no matter whether the test is left - tailed, right - tailed, or two - tailed? explain your reasoning. is the null hypothesis more or less likely to be rejected? explain. more likely; for degrees of freedom less than 30, the tail of the curve are thicker for a t - sampling distribution. therefore, if you incorrectly use a standard normal sampling distribution, the area under the curve at the tails will be smaller than what it would be for the t - test, meaning the critical value(s) will lie closer to the mean. is the result the same? the result is different. with a two - tailed case, the tail thickness does not affect the location of the critical values, however, in a left - and right - tailed case, the tail thickness does affect the location of the critical value. the result the same. in each case, the tail thickness does not affect the location of the critical value(s). the result is the same. in each case, the tail thickness affects the location of the critical value(s). the result is different. with a left - and right - tailed case, the tail thickness does not affect the location of the critical value, however, in a two - tailed case, the tail thickness does affect the location of the critical value.

Explanation:

When using the normal sampling distribution instead of the t - sampling distribution (for degrees of freedom less than 30), the tails of the normal distribution are thinner. This means that the critical values (for a given significance level) are closer to the mean compared to the t - distribution. Since the critical region (the region where we reject the null hypothesis) is determined by the critical values, having critical values closer to the mean makes it more likely to reject the null hypothesis.

For the second part, in a left - tailed test, the critical value \(z_{crit}\) (for normal) is less extreme (closer to the mean) than \(t_{crit}\) (for t - distribution). In a right - tailed test, \(z_{crit}\) is less extreme (closer to the mean) than \(t_{crit}\). In a two - tailed test, the combined critical regions (two tails) also have the normal critical values closer to the mean. So, in all cases (left - tailed, right - tailed, two - tailed), using the normal distribution instead of the t - distribution makes it more likely to reject the null hypothesis.

Answer:

The null hypothesis is more likely to be rejected. The result is the same for left - tailed, right - tailed, and two - tailed tests.