6.51 In what ways are the distributions of the z-statistic and t-statistic alike? How do they differ?

6.52 Under what circumstances should you use the t-distribution in testing a hypothesis about a population mean?

6.53 For each of the following rejection regions, sketch the sampling distribution of t and indicate the location of the rejection region on your sketch:

1. $t>1.440,\text{where}\text{df}=6$

2. $t<-1.782,\text{where}\text{df}=12$

3. $t<-2.060\text{or}t>2.060,\text{where}\text{df}=25$

6.54 For each of the rejection regions defined in Exercise 6.53 , what is the probability that a Type I error will be made?

6.55 A random sample of n observations is selected from a normal population to test the null hypothesis that $\mu =10.$ Specify the rejection region for each of the following combinations of $H_{\mathrm{a}},\alpha ,$ and n:

1. $H_{\mathrm{a}}:\mu \ne 10;\alpha =.05;n=14$

2. $H_{\mathrm{a}}:\mu >10;\alpha =.01;n=24$

3. $H_{\mathrm{a}}:\mu >10;\alpha =.10;n=9$

4. $H_{\mathrm{a}}:\mu <10;\alpha =.01;n=12$

5. $H_{\mathrm{a}}:\mu \ne 10;\alpha =.10;n=20$

6. $H_{\mathrm{a}}:\mu <10;\alpha =.05;n=4$

6.56 The following sample of six measurements was randomly selected from a normally distributed population: $1,3,-1,5,1,2.$

1. Test the null hypothesis that the mean of the population is 3 against the alternative hypothesis, $\mu <3.$ Use $\alpha =.05.$

2. Test the null hypothesis that the mean of the population is 3 against the alternative hypothesis, $\mu \ne 3.$ Use $\alpha =.05.$

3. Find the observed significance level for each test.

6.57 A sample of five measurements, randomly selected from a normally distributed population, resulted in the following summary statistics: $\bar{x}=4.8,s=\mathrm{1.3.}$

1. Test the null hypothesis that the mean of the population is 6 against the alternative hypothesis, $\mu <6.$ Use $\alpha =.05.$

2. Test the null hypothesis that the mean of the population is 6 against the alternative hypothesis, $\mu \ne 6.$ Use $\alpha =.05.$

3. Find the observed significance level for each test.

6.58 Suppose you conduct a t-test for the null hypothesis $H_0:\mu =1,000$ versus the alternative hypothesis $H_{\mathrm{a}}:\mu >1,000,$ based on a sample of 17 observations. The test results are $t=1.89,p-\mathrm{v}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{e}=.038.$

1. What assumptions are necessary for the validity of this procedure?

2. Interpret the results of the test.

3. Suppose the alternative hypothesis had been the two-tailed $H_{\mathrm{a}}:\mu \ne 1,000.$ If the t-statistic were unchanged, what would the p-value be for this test? Interpret the p-value for the two-tailed test.

TRAPS 6.59 Lobster trap placement. Refer to Consider the Bulletin of Marine Science (Apr. 2010) observational study of lobster trap placement by teams fishing for the red spiny lobster in Baja California Sur, Mexico, Exercise 5.41 (p. 273). Trap spacing measurements (in meters) for a sample of seven teams of red spiny lobster fishermen are reproduced in the accompanying table. Let $\mu$ represent the average of the trap spacing measurements for the population of red spiny lobster fishermen fishing in Baja California Sur, Mexico.The In Exercise 5.41 you computed the mean and standard deviation of the sample measurements to be $\bar{x}=89.9$ meters and $s=11.6$ meters, respectively. Suppose you want to determine if the true value of $\mu$ differs from 95 meters.

Alternate View

93

99

105

94

82

70

86

1. Specify the null and alternative hypotheses for this test.

2. Since $\bar{x}=89.9$ is less than 95, a fisherman wants to reject the null hypothesis. What are the problems with using such a decision rule?

3. Compute the value of the test statistic.

4. Find the approximate p-value of the test.

5. Select a value of $\alpha$, the probability of a Type I error. Interpret this value in the words of the problem.

6. Give the appropriate conclusion, based on the results of parts d and e.

7. What conditions must be satisfied for the test results to be valid?

8. In Exercise 5.41Suppose you found a 95% confidence interval for $\mu$. Does the interval support your conclusion in part f?

PAI 6.60 Music performance anxiety. Refer to Consider the British Journal of Music Education (Mar. 2014) study of performance anxiety by music students, Exercise 5.37 (p. 272). Recall that theThe Performance Anxiety Inventory (PAI) was used to measure music performance anxiety on a scale from 20 to 80 points. The table below gives PAI values for participants in eight different studies. A MINITAB printout of the data analysis is shown.

Alternate View

54

42

51

39

41

43

55

40

1. Set up the null and alternative hypotheses for determining whether the mean PAI value, $\mu$, for all similar studies of music performance anxiety exceeds 40.

2. Find the rejection region for the test, part a, using $\alpha =.05$.

3. Compute the test statistic.

4. State the appropriate conclusion for the test.

5. What conditions are required for the test results to be valid?

6. Locate the p-value for the test on the MINITAB printout and use it to make a conclusion. (Your conclusion should agree with your answer in part d.)

7. How would your conclusion change if you used $\alpha =.01$?

6.61 Dental anxiety study. Refer to Consider the BMC Oral Health (Vol. 9, 2009) study of adults who completed the Dental Anxiety Scale, presented in Exercise 4.85 (p. 207). Recall that scores range from 0 (no anxiety) to 25 (extreme anxiety). Summary statistics for the scores of 15 adults who completed the questionnaire are $\bar{x}=10.7\mathrm{a}\mathrm{n}\mathrm{d}s=\mathrm{3.6.}$ Conduct a test of hypothesis to determine whether the mean Dental Anxiety Scale score for the population of college students differs from $\mu =11.$ Use $\alpha =.05.$

SPIDER 6.62 Crab spiders hiding on flowers. Refer toConsider the Behavioral Ecology (Jan. 2005) experiment on crab spiders’ use of camouflage to hide from predators (e.g., birds) on flowers, presented in Exercise 2.42 (p. 51). Researchers at the French Museum of Natural History collected a sample of 10 adult female crab spiders, each sitting on the yellow central part of a daisy, and measured the chromatic contrast between each spider and the flower. The data (for which higher values indicate a greater contrast, and, presumably, an easier detection by predators) are shown in the accompanying table. The researchers discovered that a contrast of 70 or greater allows birds to see the spider. Of interest is whether the true mean chromatic contrast of crab spiders on daisies is less than 70.

Alternate View

57	75	116	37	96	61	56	2	43	32

1. Define the parameter of interest, $\mu$.

2. Set up the null and alternative hypotheses of interest.

3. Find $\bar{x}$ and s for the sample data, and then use these values to compute the test statistic.

4. Give the rejection region for $\alpha =.10.$

5. State the appropriate conclusion in the words of the problem.

MOLARS 6.63 Cheek teeth of extinct primates. Consider Refer to the American Journal of Physical Anthropology (Vol. 142, 2010) study of the characteristics of cheek teeth (e.g., molars) in an extinct primate species, Exercise 2.38 (p. 50). Recall that theThe researchers recorded the dentary depth of molars (in millimeters) for a sample of 18 cheek teeth extracted from skulls. These depth measurements are listed in the accompanying table. Anthropologists know that the mean dentary depth of molars in an extinct primate species—called Species A—is 15 millimeters. Is there evidence to indicate that the sample of 18 cheek teeth come from some other extinct primate species (i.e., some species other than Species A)? Use the SPSS printout at the bottom of the page to answer the question.

SPSS Output for Exercise 6.63

TOMBS 6.64 Radon exposure in Egyptian tombs. Refer toConsider the Radiation Protection Dosimetry (Dec. 2010) study of radon exposure in Egyptian tombs, Exercise 5.39 (p. 272). The radon levels—measured in becquerels per cubic meter (Bq/m³)—in the inner chambers of a sample of 12 tombs are listed in the table. For the safety of the guards and visitors, the Egypt Tourism Authority (ETA) will temporarily close the tombs if the true mean level of radon exposure in the tombs rises to 6,000 Bq/m³. Consequently, the ETA wants to conduct a test to determine if the true mean level of radon exposure in the tombs is less than 6,000 Bq/m³, using a Type I error probability of .10. A SAS analysis of the data is shown on p. 399. Specify all the elements of the test: H₀, H_a, test statistic, p-value, $\alpha$, and your conclusion.

Alternate View

50	910	180	580	7800	4000
390	12100	3400	1300	11900	1100

SAS Output for Exercise 6.64

6.65 Yield strength of steel connecting bars. To protect against earthquake damage, steel beams are typically fitted and connected with plastic hinges. However, these plastic hinges are prone to deformations and are difficult to inspect and repair. An alternative method of connecting steel beams—one that uses high-strength steel bars with clamps—was investigated in Engineering Structures (July 2013). Mathematical models for predicting the performance of these steel connecting bars assume the bars have a mean yield strength of 300 megapascals (MPa). To verify this assumption, the researchers conducted material property tests on the steel connecting bars. In a sample of three tests, the yield strengths were 354, 370, and 359 MPa. Do the data indicate that the true mean yield strength of the steel bars exceeds 300 MPa? Test using $\alpha =.01$.

6.66 Pitch memory of amusiacs. Amusia is a congenital disorder that adversely impacts one’s perception of music. Refer toConsider the Advances in Cognitive Psychology (Vol. 6, 2010) study of the pitch memory of individuals diagnosed with amusia, Exercise 5.45 (p. 273). Recall that eachEach in a sample of 17 amusiacs listened to a series of tone pairs and was asked to determine if the tones were the same or different. In the first trial, the tones were separated by 1 second; in a second trial, the tones were separated by 5 seconds. The difference in accuracy scores for the two trials was determined for each amusiac (where the difference is the score on the first trial minus the score on the second trial). The mean score difference was .11 with a standard deviation of .19.

1. In theory, the longer the delay between tones, the less likely one is to detect a difference between the tones. Consequently, the true mean score difference should exceed 0. Set up the null and alternative hypotheses for testing the theory.

2. Carry out the test, part a, using $\alpha =.05$. Is there evidence to support the theory?

RECALL 6.67 Free recall memory strategy. Psychologists who study memory often use a measure of “free recall” (e.g., the number of correctly recalled items in a list of to-be-­remembered items). The strategy used to memorize the list—for example, category clustering—is often just as important. Researchers at Central Michigan University developed an algorithm for computing measures of category clustering in Advances in Cognitive Psychology (Oct. 2012). One measure, called ratio of repetition, was recorded for a sample of 8 participants in a memory study. These ratios are listed in the table. Test the theory that the average ratio of repetition for all participants in a similar memory study differs from .5. Select an appropriate Type I error rate for your test.

6.68 Increasing hardness of polyester composites. Polyester resins reinforced with fiberglass are used to fabricate wall panels of restaurants. It is theorized that adding cement kiln dust (CKD) to the polyester composite will increase wall panel hardness. In a study published in Advances in Applied Physics (Vol. 2, 2014), hardness (joules per squared centimeters) was determined for three polyester composite mixtures that used a 40% CKD weight ratio. The hardness values were reported as 83, 84, and 79 j/cm². Research has shown that the mean hardness value of polyester composite mixtures that use a 20% CKD weight ratio is $\mu =76{\text{j/cm}}^2$. In your opinion, does using a 40% CKD weight ratio increase the mean hardness value of polyester composite mixtures? Support your answer statistically.

SKID 6.69 Minimizing tractor skidding distance. Refer toConsider the Journal of Forest Engineering (July 1999) study of minimizing tractor skidding distances along a new road in a European forest, presented in Exercise 5.48 (p. 274). The skidding distances (in meters) were measured at 20 randomly selected road sites. The data are repeated in the accompanying table. Recall that aA logger working on the road claims that the mean skidding distance is at least 425 meters. Is there sufficient evidence to refute this claim? Use $\alpha =.10.$

LAKES 6.70 Dissolved organic compound in lakes. The level of dissolved oxygen in the surface water of a lake is vital to maintaining the lake’s ecosystem. Environmentalists from the University of Wisconsin monitored the dissolved oxygen levels over time for a sample of 25 lakes in the state (Aquatic Biology, May 2010). To ensure a representative sample, the environmentalists focused on several lake characteristics, including dissolved organic compound (DOC). The DOC data (measured in grams per cubic meters) for the 25 lakes are listed in the table on p. 400. The population of Wisconsin lakes has a mean DOC value of $15{\text{grams/m}}^3.$ Use a hypothesis test (at $\alpha =.10$) to make an inference about whether the sample is representative of all Wisconsin lakes for the characteristic dissolved organic compound.

CIRCLES 6.71 Walking straight into circles. When people get lost in unfamiliar terrain, do they really walk in circles, as is commonly believed? To answer this question, researchers conducted a field experiment and reported the results in Current Biology (Sept. 29, 2009). Fifteen volunteers were blindfolded and asked to walk as straight as possible in a certain direction in a large field. Walking trajectories were monitored every second for 50 minutes using GPS and the average directional bias (degrees per second) recorded for each walker. The data are shown in the table below. A strong tendency to veer consistently in the same direction will cause walking in circles. A mean directional bias of 0 indicates that walking trajectories were random. Consequently, the researchers tested whether the true mean bias differed significantly from 0. A MINITAB printout of the analysis is shown below.

1. Interpret the results of the hypothesis test for the researchers. Use $\alpha =.10$.

2. Although most volunteers showed little overall bias, the researchers produced maps of the walking paths showing that each occasionally made several small circles during the walk. Ultimately, the researchers supported the “walking in circles” theory. Explain why the data in the table are insufficient for testing whether an individual walks in circles.

MINITAB Output for Exercise 6.71

SHARK 6.72 Lengths of great white sharks. One of the most feared predators in the ocean is the great white shark. It is known that the white shark grows to a mean length of 21 feet; however, one marine biologist believes that great white sharks off the Bermuda coast grow much longer owing to unusual feeding habits. To test this claim, some full-grown great white sharks were captured off the Bermuda coast, measured, and then set free. However, because the capture of sharks is difficult, costly, and very dangerous, only three specimens were sampled. Their lengths were 24, 20, and 22 feet. Do these data support the marine biologist’s claim at $\alpha =.10$?