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.
for degrees of freedom less than 30, the tail of the curve are thicker for a distribution. therefore, if you incorrectly use a standard normal sampling distribution, the area under the curve at the tails will be what it would be for the t - test, meaning the critical value(s) will lie the mean.

Explanation:

For degrees of freedom less than 30, the tails of the curve are thicker for a \(t -\)distribution. When we use the normal sampling distribution instead of the \(t -\)sampling distribution, the area under the curve at the tails is less than what it would be for the \(t -\)test. In hypothesis testing, the critical values for a normal distribution (when \(t -\)distribution should be used) are closer to the mean compared to the \(t -\)distribution critical values. A smaller area in the tails (for the normal distribution used inappropriately) means that for a given test statistic, it is more likely to fall in the rejection region (since the critical values are closer to the mean). This result is the same for left - tailed, right - tailed, or two - tailed tests because the relationship between the normal and \(t -\)distribution (with \(df<30\)) regarding the thickness of tails and position of critical values relative to the mean holds for all types of one - sided and two - sided tests.

Answer:

More likely. For degrees of freedom less than 30, the tail of the curve are thicker for a \(t -\)distribution. Therefore, if you incorrectly use a standard normal sampling distribution, the area under the curve at the tails will be less than what it would be for the \(t -\)test, meaning the critical value(s) will lie closer to the mean. This result is the same for left - tailed, right - tailed, or two - tailed tests.