9.1 Introduction to Statistical Tests

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Directions: Exercises 1 -3: Write the claim as a mathematical sentence. State the null and alternative hypotheses and identify which represents the claim.

Exercise 1: A car dealership announces that the mean time for an oil change is less than 15 minutes.

Exercise 2: A company advertises that the mean life of its furnaces is more than 18 years

Exercise 3: A company that manufactures ball bearings claims the average diameter is 6 mm.

Types of Errors • A type I error occurs if the null hypothesis is rejected when it is true. • A type II error occurs if the null hypothesis is not rejected when it is false.

Exercise 4: The USDA limit for salmonella contamination for chicken is 20%. A meat inspector reports that the chicken produced by a company exceeds the USDA limit. You perform a hypothesis test to determine whether the meat inspector’s claim is true. When will a type I or type II error occur? Which is more serious?

Type I Error

H0 (TRUE)

H1 (Decision)

With a type I error, ______

Type II Error

H0 (FALSE) (Decision)

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With a type II error, ______

Level of Significance The level of significance is the probability we are willing to risk rejecting H0 when it is true; (i.e. a Type I Error). It is typically 1% or 5%.

Type I Error

H0 (TRUE)

H1 (Decision)

P-Value The P-value is the probability, if the null hypothesis is true, of obtaining a sample statistic with a value as extreme or more extreme than the one determined from the sample data.

Directions: Exercises 5-7: For each claim, state H0 and H1. Then determine whether the hypothesis test is a left-tailed, right-tailed, or two-tailed test. Sketch a normal sampling distribution and shade the area for the P-value.

Exercise 5 A school publicizes that the proportion of its students who are involved in at least one extracurricular activity is 61%.

H1 Test Type

Exercise 6 A car dealership announces that the mean time for an oil change is less than 15 minutes.

H1 Test Type

Exercise 7 A car dealership announces that the mean time for an oil change is less than 15 minutes.

H1 Test Type

Making Decision

Condition Decision Rule Reject Fail to Reject

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Exercise 8 You perform a hypothesis test for the following claim. How should you interpret your decision if you reject H0? If you fail to reject H0?

H0 (Claim): A school publicizes that the proportion of its students who are involved in at least one extracurricular activity is 61%.

Case: Reject H0

H0 (Claim)

Conclusion:

Case: Fail to Reject H0

H0 (Claim)

Conclusion:

Exercise 9 You perform a hypothesis test for the following claim. How should you interpret your decision if you reject H0? If you fail to reject H0?

H0 (Claim): A car dealership announces that the mean time for an oil change is less than 15 minutes.

Case: Reject H0

H1 (Claim)

Conclusion:

Case: Fail to Reject H0

H1 (Claim)

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Conclusion:

Exercise 10 Find the P-value for a left-tailed hypothesis test with a test statistic of z = –2.23. Decide whether to reject H0 if the level of significance is α = 0.01.

Test Type

Standardized Test Statistic α = 0.01 Pval = Conclusion

Exercise 11 Find the P-value for a two-tailed hypothesis test with a test statistic of z = 2.14. Decide whether to reject H0 if the level of significance is α = 0.05.

Test Type

Standardized Test Statistic α = Pval =

Conclusion

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Exercise 12 Let x be a random variable that represents the heart rate in beats per minute of Rosie, and old sheep dog. From past experience the vet knows that x is normally distributed with a mean of 115 bpm and standard deviation of 12 bpm. Over the past several weeks Rosie’s heart rate (beats / min) was measured at

93 109 110 89 112 117 The sample mean is 105.0. The vet is concerned that Rosie’s heart rate may be slowing. At a 5% level of significance, do the data indicate that this is the case?

Solution: Let equal the population mean (beats/minute) heart rate for Rosie. According to the Merc Veterinary Manual, for dogs of this breed.

H1 (claim)

Conclusion:

Exercise 13 The environmental Protection Agency has been studying Miller Creek regarding ammonia nitrogen concentration. For many years, the concentration has been 2.3 mg/l. However, a new golf course and housing developments re raising concern that the concentration may have changed because of lawn fertilizer. Any change (either an increase or a decrease) in the ammonia nitrogen concentration can affect plant and animal life in and around the creek. Let x be a random variable representing ammonia nitrogen concentration (in mg/l).

Based on recent studies of Miller Creek, we may assume that x has a normal distribution with population standard deviation 0.3. Recently, a random sample of eight water tests from the creek gave the following x values.

2.1 2.5 2.2 2.8 3.0 2.2 2.4 2.9

The sample mean is about 2.51. Construct a statistical test to examine the claim that the concentration of ammonia nitrogen has changed from 2.3 ml/g. Use 0.01 as our level of significance. State the null hypothesis and alternate hypothesis. What is your conclusion?

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2.3 (claim)

Population Normally distributed. Case: σ is known Test Type Two-Tailed Test Test Statistic Standardized Test Statistic

α = 0.01 Pval =

Conclusion Interpretation At the 1% level of significance,

9.2 Testing the mean µ: Case σ is known

Exercise 1 In auto racing, a pit crew claims that its mean pit stop time (for 4 new tires and fuel) is less than 13 seconds. A random selection of 32 pit stop times has a sample mean of 12.9 seconds. Assume that the population standard deviation is 0.19 second. Is there enough evidence to support the claim at α = 0.01? Use a P-value.

H1 Population Distribution unknown. Case: σ is known , Test Type Left-tailed test Test Statistic Standardized Test Statistic

α = 0.01 Pval = Conclusion Interpretation At the 1% level of significance,

Exercise 2 The National Institute of Diabetes and Digestive and Kidney Diseases reports that the average cost of bariatric (weight loss) surgery is $22,500. You think this information is incorrect. You randomly select 30 bariatric surgery patients and find that 6 NAME:______BLOCK:______DATE:______

the average cost for their surgeries is $21,545. Assume that 𝜎=$3015. Is there enough

evidence to support your claim at α = 0.05? Use a P-value. (Adapted from National Institute of Diabetes and Digestive and Kidney Diseases)

H1 Population Test Type Test Statistic Standardized Test Statistic α = 0.05 Pval = Conclusion Interpretation At the 5% level of significance,

9.2 Testing the mean µ: Case σ is Unknown

Exercise 1 Find the critical value for a left-tailed test given and

Test Type

Exercise 2 Find the critical values and for a two-tailed test given given and

Test Type

Exercise 3 The drug 6-mP (6-mercoptopurine) is used to treat leukemia. The following data represent the remission times (in weeks) for a random sample of 21 patients using 6- mP.

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10 7 32 23 22 6 16 34 32 25 11 20 19 6 17 35 6 13 9 6 10

The sample mean is 17.1 weeks with a sample standard deviation of 10.0 weeks. Let x be a random variable representing the remission times (in weeks) for all patients. Assume the x-distribution is mound-shaped and symmetric. A previous drug treatment had a remission time of 12.5 weeks. At a 1% level of significance do the data indicate the mean remission time for 6-mP is different (either way)?

H1 Population Test Type Two-tailed test Test Statistic

10.0 Standardized Test Statistic α = Pval = Conclusion

Interpretation At the 1% level of significance,

NOTE: If using Table 6, Appendix II, note that the sample statistic falls between 2.086 and 2.528. The P-value for the sample t falls between the corresponding two-tail areas 0.050 and 0.020. This means . The entire range is greater than α. This means the specific P-value is greater than α, so we cannot reject H0.

α =0.01 0.020 0.050

Exercise 4: A random sample of 46 adult coyotes in a region of northern Minnesota showed the average age to be 2.05 years with sample standard deviation of 0.82 years. However, it is thought that the overall population mean age of coyotes is 1.75 years. Does the sample data indicate that coyotes in this region of northern Minnesota tend to live longer than the average of 1.75 years? Use α =0.01.

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H1 Population Test Type Test Statistic Standardized Test Statistic α = Pval = Conclusion Interpretation

Exercise 5: A used car dealer says that the mean price of a 2008 Honda CR-V is at least $20,500. You suspect this claim is incorrect and find that a random sample of 14 similar vehicles has a mean price of $19,850 and a standard deviation of $1084. Is there enough evidence to reject the dealer’s claim at α = 0.05? Assume the population is normally distributed. (Adapted from Kelley Blue Book)

H1 Population Test Type Test Statistic

Standardized Test Statistic α = Pval = Conclusion Interpretation At the 5% level of significance,

9.2 Testing the mean µ: Critical Region Method

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Exercise 1: Find the critical value and sketch rejection region for a left-tailed z-test with α = 0.05 and with α = 0.01.

Exercise 2: Find the critical value and sketch rejection region for a right-tailed z- test with α = 0.05 and with α = 0.01.

Exercise 3: Find the critical values and sketch rejection region for a two-tailed z-test with α = 0.05 and with α = 0.01.

Exercise 4: In auto racing, a pit crew claims that its mean pit stop time (for 4 new tires and fuel) is less than 13 seconds. A random selection of 32 pit stop times has a sample mean of 12.9 seconds. Assume that the population standard deviation is 0.19 second. Is there enough evidence to support the claim at α = 0.01? Use the critical region method.

Standardized Test Statistic

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Exercise 5: A used car dealer says that the mean price of a 2008 Honda CR-V is at least $20,500. You suspect this claim is incorrect and find that a random sample of 14 similar vehicles has a mean price of $19,850 and a standard deviation of $1084. Is there enough evidence to reject the dealer’s claim at α = 0.05? Assume the population is normally distributed. Use the critical region method.

α = 0.05

Exercise 6: An industrial company claims that the mean pH level of the water in a nearby river is 6.8. You randomly select 19 water samples and measure the pH of each. The sample mean and standard deviation are 6.7 and 0.24, respectively. Is there enough

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evidence to reject the company’s claim at α = 0.05? Assume the population is normally distributed. Use the critical region method.

Standardized Test Statistic α = 0.05

Conclusion

Interpretation At the 5% level of significance,

Exercise 7: Employees at a construction and mining company claim that the mean salary of the company’s mechanical engineers is less than that of the one of its competitors, which is $68,000. A random sample of 30 of the company’s mechanical engineers has a mean salary of $66,900 with a standard deviation of $5500. At α = 0.05, test the employees’ claim. Use the critical region method. H0

α = 0.05

Conclusion Interpretation At the 5 % level of significance,

9.3 Testing a Proportion

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Exercise 1: A team of eye surgeons has developed a new technique for a risky eye operation to restore the sight of people blinded from a certain disease. Under the old method, it is known that only 30% of the patients who undergo this operation recover their eyesight. Suppose that surgeons in various hospitals have performed a total of 225 operations using the new method and that 88 have been successful (i.e., the patients fully recovered their sight). Can we justify the claim that the new method is better than the old one? (Use a 1% level of significance.)

H1 Population Binomial Distribution

Test Type Test Statistic

α = 0.01

Exercise 2: A botanist has produced a new variety of hybrid wheat that is better able to withstand drought than other varieties. The botanist knows that for the parent plants, the proportion of seeds germinating is 80%. The proportion of seeds germinating for the hybrid variety is unknown, but the botanist claims that it is 80%. To test this claim, 400 seeds from the hybrid plant are tested, and it is found that 312 germinate. Use a 5% level of significance to test the claim that the proportion germinating for the hybrid is 80%.

Test Type

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Test Statistic

α = 0.05

Exercise 3: Repeat Exercise 2 using the critical region method.

Test Type Test Statistic

Standardized Test Statistic α = 0.05 Conclusion Interpretation At the 5% level of significance,

TI-83/84 STAT / TESTS menu Section Description 1: Z-Test 9.2 Testing the mean when is known. Be able to do these problems without using the Z- Test function. That is, sketch the distribution and compute the p-value using the normalcdf function. 2: T-Test 9.2, 9.4 Testing the mean when is not known, or 14 NAME:______BLOCK:______DATE:______

testing dependent paired data . 5: 1-PropZTest 9.3 Testing a proportion p. 7: ZInterval 8.1 Estimating when is known. Be able to do these problems without using the ZInterval function. That is, sketch the distribution and compute the interval using the invNorm function. 8: TInterval 8.2 Estimating when is not known. 9: 2-SampZInt 8.5 Estimating when and are known. A: 1-PropZInt 8.3 Estimating when the Binomial Distribution. B: 2-PropZInt 8.5 Estimating