Writing You are testing a claim and incorrectly use the standa...

Question

Writing You are testing a claim and incorrectly use the standard normal sampling distribution instead of the t-sampling distribution, mistaking the sample standard deviation for the population standard deviation. 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.

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A clinical researcher claims that the average recovery time meal after a new treatment is 5 days. She takes a small sample of N equals 12 patients and computes the sample standard deviation S. By mistake, she uses a Z test standard normal instead of the correct T test. How does this error affect the probability of rejecting H0 and does the effect Depend on whether the test is left tailed, right-tailed, or two-tailed. Awesome. So first off, let us note that H0 denotes our null hypothesis. So when we look at our problem itself, it appears that we're asked to determine or the the final answer we're trying to solve for is we're trying to figure out. That how, how big of an effect does this mistake have on the solution of this particular problem. By mistake, our researcher uses a Z test instead of a T test, and we're asked, how does this error affect the probability. of rejecting the null hypothesis, and does the effect depend on whether the test is left-tailed, right-tailed, or two-tailed. So now that we know what we're ultimately trying to solve for, let's take a moment to read off our multiple choice answers. A is it makes rejecting H0 more likely regardless of tail direction. B is it makes rejecting H0 less likely regardless of tail direction. C is it makes rejecting H0 more likely only for two tail tests, and D is it makes rejecting H0 more likely only for one tailed tests. Awesome. So, with that in mind, We need to recall a note that for a small sample, for this case, N is equal to 12 and the degrees of freedom DF is equal to 11, the critical values from the T distribution are larger in magnitude than the corresponding Z values. For example, T subscript 0.95 11 will be approximately equal to 1.796 versus Z subscript 0.95 is approximately equal to 1.645. And by using Z critical values instead of the higher T critical values, the researcher therefore sets the rejection cutoff closer to zero, thereby increasing the chance that the test statistic exceeds the threshold and that the null hypothesis is rejected. This holds true for both left-tailed, right-tailed, and two-tailed tests. Because in every case, the Z cutoff is less extreme than the T cutoff. That means that without a doubt, our final answer will have to be multiple choice answer letter A, which once again is it makes rejecting H0 more likely regardless of tail direction. And that's it. We've officially solved for this problem. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Key Concepts

Sampling Distributions

Null Hypothesis and Hypothesis Testing

Type I and Type II Errors

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