Questions: 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 t-distribution. Therefore, if you incorrectly use a standard normal sampling distribution, the area under the curve at the tails will be different than what it would be for the t-test, meaning the critical value(s) will lie closer to the mean.

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 t-distribution. Therefore, if you incorrectly use a standard normal sampling distribution, the area under the curve at the tails will be different than what it would be for the t-test, meaning the critical value(s) will lie closer to the mean.
Transcript text: 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-taled, or two-taled? 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.
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Solution

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Solution Steps

Step 1: Understanding the Difference Between Normal and t-Distributions

The normal distribution and the t-distribution are both used in hypothesis testing, but they differ in their tail behavior. The t-distribution has thicker tails compared to the normal distribution, especially when the degrees of freedom are small (typically less than 30). This means that the t-distribution assigns more probability to extreme values than the normal distribution.

Step 2: Impact on Critical Values

When you incorrectly use the normal distribution instead of the t-distribution, the critical values for the test will be closer to the mean than they should be. This is because the normal distribution has thinner tails, so the area under the curve at the tails is smaller compared to the t-distribution. As a result, the critical values for the normal distribution will be less extreme (closer to the mean) than those for the t-distribution.

Step 3: Effect on Hypothesis Testing

Using the normal distribution instead of the t-distribution makes it more likely to reject the null hypothesis. This is because the critical values are less extreme, so the test statistic is more likely to fall into the rejection region. This effect is the same regardless of whether the test is left-tailed, right-tailed, or two-tailed, as the critical values are affected symmetrically.

Final Answer

  • The null hypothesis is more likely to be rejected when incorrectly using the normal distribution instead of the t-distribution.
  • For degrees of freedom less than 30, the tails 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.

\\(\boxed{\text{The null hypothesis is more likely to be rejected.}}\\)

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