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Chapter 9: Problem 48

For each of the following examples of tests of hypothesis about \(\mu\), show the rejection and nonrejection regions on the \(t\) distribution curve. a. A two-tailed test with \(\alpha=.02\) and \(n=20\) b. A left-tailed test with \(\alpha=.01\) and \(n=16\) c. A right-tailed test with \(\alpha=.05\) and \(n=18\)

Short Answer

Expert verified
a. For the two-tailed test, the rejection regions are \(t < -2.539\) and \(t > 2.539\). b. For the left-tailed test, the rejection region is \(t < -2.602\). c. For the right-tailed test, the rejection region is \(t > 1.74\).

Step by step solution

01

Understand Tails of Tests

The two-tailed test allows for the possibility that \(\mu\) is either significantly higher or lower than hypothesized. The left-tailed test considers that \(\mu\) could be significantly lower than hypothesized, and the right-tailed test, conversely, considers that \(\mu\) could be significantly higher.
02

Calculate T-Values for Two-tailed Test

A two-tailed test with \(\alpha=.02\) and \(n=20\) will have a degree of freedom of \(20-1=19\). We need to find t-values which cut off the highest 1% and lowest 1% areas of the t-distribution curve (0.02/2). If we look these up in a t-table or use a statistics software package, we find the values are approximately \(\pm 2.539\).
03

Visualize the Two-tailed test

Draw a t-distribution curve, mark and shade these rejection areas (t < -2.539, t > 2.539), and label the nonrejection area (-2.539 < t < 2.539).
04

Calculate T-Value for Left-tailed Test

For a one-tailed test with \(\alpha=.01\) and \(n=16\), the degree of freedom will be \(16-1=15\). We need to find the t-value which cuts off the lowest 1% area. If we look this up, it is approximately -2.602.
05

Visualize the Left-tailed test

Draw a t-distribution curve, mark and shade the rejection area (t < -2.602) and label the nonrejection area (t > -2.602).
06

Calculate T-Value for Right-tailed Test

For a right-tailed test with \(\alpha=.05\) and \(n=18\), the degree of freedom will be \(18-1=17\). We need to find the t-value that cuts off the highest 5% area. If we look this up, it is approximately 1.74.
07

Visualize the Right-tailed test

Draw a t-distribution curve, mark and shade the rejection area (t > 1.74), and label the nonrejection area (t < 1.74).

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Most popular questions from this chapter

According to the Kaiser Family Foundation, U.S. workers who had employer- provided health insur. ince paid an average premium of $$\$ 4129$$ for family health insurance coverage during 2011 (USA TODAY. October 10,2011 ). Suppose a recent random sample of 25 workers with employer-provided health insur. ance selected from a city paid an average premium of $$\$ 4517$$ for family health insurance coverage with a standard deviation of $$\$ 580 .$$ Assume that such premiums paid by all such workers in this city are normally distributed. Does the sample information support the alternative hypothesis that the average premium for such coverage paid by all such workers in this city is different from $$\$ 4129 ?$$ Use a $$5 \%$$ significance level IIse both the \(p\) -value approach and the critical-value approach

Brooklyn Corporation manufactures DVDs. The machine that is used to make these DVDs is known. to produce not more than \(5 \%\) defective DVDs. The quality control inspector selects a sample of \(200 \mathrm{DVDs}\) each week and inspects them for being good or defective. Using the sample proportion, the quality con trol inspector tests the null hypothesis \(p \leq .05\) against the alternative hypothesis \(p>.05\), where \(p\) is th proportion of DVDs that are defective. She always uses a \(2.5 \%\) significance level. If the null hypothesi: is rejected, the production process is stopped to make any necessary adjustments. A recent sample of 200 DVDs contained 17 defective DVD: Using a \(2.5 \%\) significance level, would you conclude that the prod" should be stoppe to make necessary adjustments b. Perform the test of part a using a \(1 \%\) significance level. Is your decision different from the one in.par Comment on the results of parts a and \(b\)

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What is the difference between the critical value of \(z\) and the observed value of \(z\) ?
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