Inferences from Two Samples — practice test 9A

May 9, 2017

Problems with solutions in Elementary Statistics — Inferences About Two Proportions; Inferences About Two Means: Independent Samples; Inferences from Matched Pairs; Comparing Variation in Two Samples.

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

Provide an appropriate response.
- 1. What is the effect on the P-value when a test is changed from a two-tailed hypothesis with = and ≠ to a one-tailed hypothesis such as ≥ and
  Possible answer
  The P-value is cut in half.
Find the number of successes x suggested by the given statement.
- 2. Among 1410 randomly selected car drivers in one city, 7.52% said that they had been involved in an accident during the past year.
  1. 105
  2. 106
  3. 104
  4. 107
From the sample statistics, find the value of p-bar used to test the hypothesis that the population proportions are equal.
- 3. n₁ = 100, x₁ = 34; n₂ = 100, x₂ = 30
  1. 0.288
  2. 0.224
  3. 0.320
  4. 0.352
Compute the test statistic used to test the null hypothesis that p1 = p2.
- 4. A report on the nightly news broadcast stated that 15 out of 140 households with pet dogs were burglarized and 20 out of 180 without pet dogs were burglarized.
  1. 0.000
  2. –0.192
  3. –0.045
  4. –0.113
Find the appropriate p-value to test the null hypothesis, H₀: p₁ = p₂, using a significance level of 0.05.
- 5. n₁ = 50, x₁ = 20; n₂ = 75, x₂ = 15
  1. .0032
  2. .1201
  3. .0146
  4. .0001
Use the traditional method to test the given hypothesis. Assume that the samples are independent and that they have been randomly selected.
- 6. Use the given sample data to test the claim p₁ > p₂. Use a significance level of 0.01.
  Sample 1: n₁ = 85, x₁ = 38
  Sample 2: n₂ = 90, x₂ = 23
  
  Possible answer
  H₀: p₁ = p₂, H₁: p₁ > p₂.
  Test statistic: z = 2.66.
  Critical value: z = 2.33.
  Reject the null hypothesis. There is sufficient evidence to support the claim that p₁ > p₂.
- 7. A researcher finds that of 1,000 people who said that they attend a religious service at least once a week, 31 stopped to help a person with car trouble. Of 1,200 people interviewed who had not attended a religious service at least once a month, 22 stopped to help a person with car trouble. At the 0.05 significance level, test the claim that the two proportions are equal.
  
  Possible answer
  H₀: p₁ = p₂, H₁: p₁ ≠ p₂.
  Test statistic: z = 1.93.
  Critical values: z = ±1.96.
  Fail to reject the null hypothesis. There is not sufficient evidence to warrant rejection of the claim that the two proportions are equal.
Construct the indicated confidence interval for the difference between population proportions p₁— p₂. Assume that the samples are independent and that they have been randomly selected.
- 8. x1 = 10, n1 = 42 and x2 = 27, n2 = 47. Construct a 90% confidence interval for the difference between population proportions p₁ — p₂_.
  1. 0.076 < p₁ – p₂ < 0.400
  2. 0.047 < p₁ – p₂ < 0.429
  3. –0.497 < p₁ – p₂ < –0.176
  4. 0.429 < p₁ – p₂ < 0.045
Determine whether the samples are independent or consist of matched pair.
- 9. The accuracy of verbal responses is tested in an experiment in which individuals report their heights and then are measured. The data consist of the reported height and measured height for each individual.
  1. Independent samples
  2. Matched pairs
Test the indicated claim about the means of two populations. Assume that the two samples are independent and that they have been randomly selected.
- 10. Two types of flares are tested for their burning times (in minutes) and sample results are given below:
  Brand X: n = 35, x̅ = 19.4, s = 1.4;
  Brand Y: n = 40, x̅ = 15.1, s = 0.8.
  
  Refer to the sample date to test the claim that the two populations have equal means. Use a 0.05 significance level.
  
  Possible answer
  H₀: µ₁ = µ₂, H₁: µ₁ ≠ µ₂.
  Test statistic t = 16.025.
  Critical values: t = ± 2.032.
  Reject the null hypothesis. There is sufficient evidence to warrant rejection of the claim that the two populations have equal means.
Construct the indicated confidence interval for the difference between the two population means. Assume that the two samples are independent and that they have been randomly selected.
- 11. A researcher wishes to determine whether people with high blood pressure can reduce their blood pressure by following a particular diet. Use the sample data below to construct a 99% confidence interval for µ₁ – µ₂ where µ₁ and µ₂ represent the mean for the treatment group and the control group respectively.
  
  Treatment Group: n₁ = 85, x̅₁ = 189.1, s₁ = 38.7
  Control Group: n₂ = 75, x̅₂ = 203.7, s₂ = 39.2
  1. –29.0 < µ₁ – µ₂< –0.2
  2. –26.7 < µ₁ – µ₂< –2.5
  3. –30.9 < µ₁ – µ₂< 1.7
  4. –1.3 < µ₁ – µ₂< 30.5
Find s_d.
- 12. Consider the set of differences between two dependent sets: 84, 85, 83, 63, 61, 100, 98.
  Round to the nearest tenth.
  1. 16.2
  2. 15.3
  3. 13.1
  4. 15.7
The two data sets are dependent. Find the mean value of the differences d for the paired sample data (d-bar) to the nearest tenth.
- 13. ***
  A: 60, 61, 64, 63, 51
  B: 25, 23, 29, 25, 22
  1. 43.8
  2. 45.5
  3. 21.0
  4. 35.0
Determine whether the hypothesis test involves a sampling distribution of means that is a normal distribution, Student t distribution, or neither.
- 14. When testing for a difference between the means of a treatment group and a placebo group, the computer display below is obtained. Using a 0.04 significance level, is there sufficient evidence to support the claim that the treatment group (variable 1) comes from a population with a mean that is different from the mean for the placebo population? Explain.
  
  Possible answer
  Yes, the P-value for a two-tailed test is 0.0316, which is smaller than the significance level of 0.04. There is sufficient evidence to support the claim that the two population means are different.
Assume that you want to test the claim that the paired sample data come from a population for which the mean difference is µ_d = 0. Compute the value of the t test statistic.
- 15.
  1. t = –1.480
  2. t = 0.690
  3. t = –0.185
  4. t = –0.523
Determine the decision criterion for rejecting the null hypothesis in the given hypothesis test; i.e., describe the values of the test statistic that would result in rejection of the null hypothesis.
- 16. A farmer has decided to use a new additive to grow his crops. He divided his farm into 10 plots and kept records of the corn yield (in bushels) before and after using the additive. The results are shown below.
  
  You wish the test the following hypothesis at the 5 percent level of significance.
  H₀: µ_D = 0 against H₁: µ_D > 0.
  What decision rule would you use?
  1. Reject H0 if test statistic is less than 1.833.
  2. Reject H0 if test statistic is greater than –1.833 or less than 1.833.
  3. Reject H0 if test statistic is greater than 1.833.
  4. Reject H0 if test statistic is greater than –1.833.
Use the traditional method of hypothesis testing to test the given claim about the means of two populations. Assume that two dependent samples have been randomly selected from normally distributed populations.
- 17. A test of abstract reasoning is given to a random sample of students before and after they completed a formal logic course. The results are given below. At the 0.05 significance level, test the claim that the mean score is not affected by the course.
  
  Possible answer
  Test statistic t = 2.366.
  Critical values: t = ± 2.262.
  Reject the null hypothesis. There is sufficient evidence to warrant rejection of the claim that the mean is not affected by the course.
Construct a confidence interval for µ_d, the mean of the differences d for the population of paired data. Assume that the population of paired differences is normally distributed.
- 18. If d-bar = 3.125, s_d = 2.911 and n = 8, determine a 99 percent confidence interval for µ_d.
  1. 1.851 < µ_d < 4.399
  2. 1.851 < µ_d < 6.726
  3. –0.476 < µ_d < 6.726
  4. –0.360 < µ_d < 4.399
Test the indicated claim about the variances or standard deviations of two populations. Assume that the populations are normally distributed. Assume that the two samples are independent and that they have been randomly selected.
- 19. Two types of flares are tested for their burning times (in minutes) and sample results are given below. Use a 0.05 significance level to test the claim that the two brands have equal variances.
  
  Brand X: n = 35, x̅ = 19.4, s = 1.4
  Brand Y: n = 40, x̅ = 15.1, s = 0.8
  
  Possible answer
  H₀: σ₁² = σ₂², H₁: σ₁² ≠ σ₂².
  Test statistic: F = 3.0625.
  Critical value: 1.8752 < F > 2.0739.
  Reject the null hypothesis. There is sufficient evidence to warrant rejection of the claim that the two brands gave equal variances.
Solve the problem.
- 20. A test for homogeneity of variance is conducted at the 5% level of significance. Sample sizes are n1 = 31 and n2 = 31. The test statistic is F = 6.4071. What do you know about the variance of the populations from which the samples were taken?
  
  Possible answer
  The critical value F-value for a two-tailed test of homogeneity of variance is F = 2.0739, df = 30, 30 and with 0.025 in the right tail. Since the TS F is in the critical region, the null hypothesis of equal variances is rejected. We conclude that the populations have significantly different variances.

tags: Elementary Statistics • Inferences About Two Proportions; Inferences About Two Means: Independent Samples; Inferences from Matched Pairs; Comparing Variation in Two Samples

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