Hypothesis Testing — practice test 8С

Apr 18, 2017

Problems with solutions in Elementary Statistics — Basics of Hypothesis Testing; Testing a Claim about a Proportion; Testing a Claim about a Mean: 1) σ known, 2) σ not known; Testing Claim about Variation.

Directions:
Give your answers to the short-answer items. Choose the correct answer for multiple-choice items.

Provide an appropriate response.
- 1. Under what conditions do you reject H₀? Discuss both the traditional and the P-value approach.
  
  Possible answer
  For the traditional method, the test-statistic is the critical region. For the P-value method, the P-values is less than or equal to the significance level α and the test statistic is on the appropriate side in a one-tailed test.
Solve the problem.
- 2. Write the claim that is suggested by the given statement, then write a conclusion about the claim. Do not use symbolic expressions or formal procedures; use common sense.
  
  A person claims to have extra sensory powers. A card is drawn at random from a deck of cards and without looking at the card, the person is asked to identify the suit of the card. He correctly identifies the suit 28 times out of 100.
  
  Possible answer
  The claim is that the person is using his extra sensory powers to determine the suit of the card, and that he correctly determines the suit more often than he would if he were guessing randomly. Even if he were just guessing randomly, he would have a reasonable chance of being correct 28 times out of hundred; this is not improbable, since there are four suits. Therefore, identifying the suit correctly 28 times out 100 does not constitute strong evidence in favor of his claim.
Express the null hypothesis H₀ and the alternative hypothesis H₁ in symbolic form. Use the correct symbol (μ, p, σ) for the indicated parameter.
- 3. The principal of a middle school claims that test scores of the seventh-graders at her school vary less than the test scores of seventh-graders at a neighboring school, which have variation described by σ = 14.7.
  1. H₀: σ < 14.7, H₁: σ > 14.7
  2. H₀: σ = 14.7, H₁: σ > 14.7
  3. H₀: σ > 14.7, H₁: σ ≤ 14.7
  4. H₀: σ = 14.7, H₁: σ < 14.7
Assume that the data has a normal distribution and the number of observations is greater than fifty. Find the critical z value used to test a null hypothesis.
- 4. α = 0.1 for a two-tailed test.
  1. ±2.052
  2. ±1.645
  3. ±1.4805
  4. ±2.33
Find the value of the test statistic z using z = (p̂ – p) / √(p*q/n).
- 5. The claim is that the proportion of accidental deaths of the elderly attributable to residential falls is more than 0.10, and the sample statistics include n = 800 deaths of the elderly with 15% of them attributable to residential falls.
  1. –4.71
  2. 3.96
  3. –3.96
  4. 4.71
Use the given information to find the P-value.
- 6. The test statistic in a two-tailed test is z = 1.95.
  1. 0.0512
  2. 0.3415
  3. 0.0244
  4. 0.4423
Formulate the indicated conclusion in nontechnical terms. Be sure to address the original claim.
- 7. A cereal company claims that the mean weight of the cereal in its packets is at least 14 oz. Assume that a hypothesis test of the claim has been conducted and that the conclusion is to reject the null hypothesis, state the conclusion in nontechnical terms.
  1. There is not sufficient evidence to warrant rejection of the claim that the mean weight is less than 14 oz.
  2. There is sufficient evidence to warrant rejection of the claim that the mean weight is at least 14 oz.
  3. There is not sufficient evidence to warrant rejection of the claim that the mean weight is at least 14 oz.
  4. There is sufficient evidence to warrant rejection of the claim that the mean weight is less that 14 oz.
Assume that a hypothesis test of the given claim will be conducted. Identify the type I or type II error for the test.
- 8. A researcher claims that the amounts of acetaminophen in a certain brand of cold tablets have a standard deviation different from the σ = 3.3 mg claimed by the manufacturer. Identify the type II error for the test.
  1. The error of rejecting the claim that the standard deviation is more than 3.3 mg when it really is more than 3.3 mg.
  2. The error of failing to reject the claim that the standard deviation is 3.3 mg when it is actually different from 3.3 mg.
  3. The error of rejecting the claim that the standard deviation is 3.3 mg when it really is 3.3 mg.
Solve the problem.
- 9. In a hypothesis test, which of the following will cause a decrease in ß, the provability of making a type II error?
  A: Increasing α while keeping the sample size n, fixed.
  B: Increasing the sample size n, while keeping α fixed.
  C: Decreasing α while keeping the sample size n, fixed.
  D: Decreasing the sample size n, while keeping α fixed.
  1. A and D
  2. A and B
  3. C and D
  4. B and C
Identify the null hypothesis, alternative hypothesis, test statistic, P-value, conclusion about the null hypothesis, and final conclusion that addresses the original claim.
- 10. In a clinical study of an allergy drug, 108 of the 202 subjects reported experiencing significant relief from their symptoms. At the 0.01significance level, test the claim that more than half of all those using the drug experience relief.
  
  Possible answer
  H₀: p = 0.5
  H₁: p > 0.5
  Test statistic: z = 0.99
  P-value: p = 0.1611
  Critical value: z = 2.33
  Fail to reject null hypothesis. There is not sufficient evidence to support the claim that more than half of all those using the drug experience relief.
Find the P-value for the indicated hypothesis test.
- 11. An airline claims that the no-show rate for passengers booked on its flights is less than 6%. Of 380 randomly selected reservations, 18 were no-shows. Find the P-value for a test of the airline’s claim.
  1. 0.1492
  2. 0.3508
  3. 0.1230
  4. 0.0746
Determine whether the given conditions justify testing a claim about a population mean μ.
- 12. The sample size is n = 44, σ = 13.4, and the original population is not normally distributed.
  1. No
  2. Yes
Identify the null hypothesis, alternative hypothesis, test statistic, P-value, conclusion about the null hypothesis, and final conclusion that addresses the original claim.
- 13. The health of employees is monitored by periodically weighing them in. A sample employees has a mean weight of 183.9 lb. Assuming that σ is known to be 121.2 lb, use a 0.10 significance level to test the claim that the population mean of all such employees weights is less than 200 lb.
  
  Possible answer
  H₀: µ = 200
  H₁: µ < 200
  Test statistic: z = –0.98
  P-value: p = 0.1635
  Fail to reject H₀_. There is not sufficient evidence to support the claim that the mean is less than 200 pounds.
Determine whether the hypothesis test involves a sampling distribution of means that is a normal distribution, Student t distribution, or neither.
- 14. Claim: µ = 973. Sample data: n = 20, x̅ = 958, s = 29. The sample data appear to come from a normally distributed population with σ = 28.
  1. Normal
  2. Neither
  3. Student t
Assume that a simple random sample has been selected from a normally distributed population. Find the test statistic, P-value, critical value(s), and state the final conclusion.
- 15. Test the claim that for the adult population of one town, the mean annual salary is given by µ = $30,000. Sample data are summarized as n = 17, x̅ = $22,298, and s = $14,200. Use a significance level of α = 0.05.
  
  Possible answer
  α = 0.05
  Test statistic: t = –2.236
  P-value: 0.02 < P-value < 0.05.
  Critical value: t = ±2.120
  Because TS t < –2.120, we reject the null hypothesis. There is sufficient evidence to warrant rejection of the claim that µ = $30,000.
Test the given claim using the traditional method of hypothesis testing. Assume that the sample has been randomly selected from a population with a normal distribution.
- 16. A public company official claims that the mean waiting time for bus number 14 during peak hours is less than 10 minutes. Karen took bus number 14 during peak hours on 18 different occasions. Her mean waiting time was 7.2 minutes with a standard deviation of 1.6 minutes. At the 0.01 significance level, test the claim that the mean is less than 10 minutes.
  
  Possible answer
  Test statistic: t = –7.425
  Critical value: t = –2.567
  Reject H₀. There is sufficient evidence to support the claim that the mean is less than 10 minutes.
- 17. A light-bulb manufacturer advertises that the average life for its light bulbs 900 hours. A random sample of 15 of its light bulbs resulted in the following lives in hours.
  995 590 510 539 739 917 571 555
  916 728 664 693 708 887 849
  At the 10% significance level, do the data provide evidence that the mean life for the company’s light bulbs differs from the advertised mean?
  
  Possible answer
  Test statistic: t = –4.342
  Critical value: t = ±1.761
  Reject H₀: µ = 900 hours. There is sufficient evidence to support the claim that the true mean life differs from the advertised mean.
Find the critical value or values of x² based on the given information.
- 18. H₁: σ < 0.14, n = 23, α = 0.10.
  1. –30.813
  2. 30.813
  3. 14.848
  4. 14.042
Use the traditional method to test the given hypothesis. Assume that the population is normally distributed and that the sample has been randomly selected.
- 19. A manufacturer uses a new production method to produce steel rods. A random sample of 17 steel rods resulted in lengths with a standard deviation of 2.1 cm. At the 0.10 significance level, test the claim that the new production method has lengths with a standard deviation different from 3.5 cm, which was the standard deviation for the old method.
  
  Possible answer
  Test statistic: X² = 5.760
  Critical values: X² = 7.962, 26.296.
  Reject H₀. There is sufficient evidence to support the claim that the standard deviation is different from 3.5.
- 20. With individual lines at the checkouts, a store manager finds that the standard deviation for the waiting times on Monday mornings is 5.7 minutes. After switching to a single waiting line, he finds that for a random sample of 29 customers, the waiting times have a standard deviation of 4.9 minutes. Use a 0.025 significance level to test the claim that with a single line, waiting times vary less that with individual lines.
  
  Possible answer
  Test statistic: X² = 20.692
  Critical value: X² = 15.308
  Fail to reject H₀. There is not sufficient evidence to support the claim that with a single line waiting times have a smaller standard deviation.

tags: Elementary Statistics • Basics of Hypothesis Testing; Testing a Claim about a Proportion; Testing a Claim about a Mean: 1) σ known • 2) σ not known; Testing Claim about Variation.

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