# One-Tailed vs. Two-Tailed & Type I and Type II Error

1. The SEM for a particular intelligence test (mean of 100, standard deviation of 15) is 3 points. Ken Garoo obtains an IQ score of 126, but needs an IQ of 130 to qualify for a Talented and Gifted (TAG) class. Calculate the range of Ken's true score: (a) with 68% probability, and (b) with 95% probability. Should Ken be admitted into the TAG class?

2. Explain the advantages of a one-tailed verses a two-tailed test. If one-tailed tests are more powerful, then why bother with two-tailed tests at all?

3. Explain why there is an inverse relationship between committing a Type I error and committing a Type II error. What is the best way to reduce both kinds of error?

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#### Solution Preview

Please see the attachment for the missing part of the answers which are math-based.

1. a) Now, for the first part it is important to notice that a 68% percent probability is approximately equal to 1 standard deviation on either side of the mean. This means that the range of scores for 68% probability is calculated by: (see attachment for equation and mathematical steps).

1. b) It is also important to notice for this part of the problem that a 95% confidence interval is approximately equal to 2 standard deviations on either side of the mean. This means that the range of scores for 95% probability is ...

#### Solution Summary

This solution explains the different test types and errors in 434 words. The full formatted math-based discussion is provided in the attached Excel file.

Multiple choice questions on testing of hypothesis

1.The power of a test is measured by its capability of

a. rejecting a null hypothesis that is true.

b. not rejecting a null hypothesis that is true.

c. rejecting a null hypothesis that is false.

d. not rejecting a null hypothesis that is false.

2. If an economist wishes to determine whether there is evidence that average family income in a community exceeds $25,000, then

a. either a one-tailed or two-tailed test could be used with equivalent results.

b. a one-tailed test should be utilized.

c. a two-tailed test should be utilized.

d. None of the above.

3. If the Type I error (alpha) for a given test is to be decreased, then for a fixed sample size n:

a. the Type II error ( beta) will also decrease.

b. the Type II error (beta) will also increase.

c. the power of the test will increase.

d. a one-tailed test must be utilized.

4. Given the following information, calculate the degrees of freedom that should be used in the pooled-variance t test.

s12 = 4 s22 = 6

n1 = 16 n2 = 25

a. df = 41

b. df = 39

c. df = 16

d. df = 25

5. When testing H0: µ1 -µ2 greater than or equal to 0 versus H1: µ1 -µ2 < 0, the observed value of the Z-score was found to be -2.13. The p value for this test would be:

a. 0.0166.

b. 0.0332

c. 0.9668

d. 0.9834