MATH102 11SP Midterm ch7-10
A [ answers in web view ] Total points: 70

• Please show all your work! No partial credit will be given for incorrect answers with no work shown.
• Please draw a box around your final answer.
• You are only permitted to use your own calculator and writing implements. Cell phones should be muted and left in your pocket or bag.
• All relevant tables are attached to the back. You may detach them for your reference.
• Assume α = 0.05 everywhere unless indicated otherwise.
• For t-tests on two groups, if the df is not given, you may use the conservative estimate of df = min(n₁, n₂) - 1.

1. Does HBE concentration in men increase after they exercise?
   Data from a study of 6 men are below.

   							Mean	SD
   Before:	42	58	38	50	49	48	47.5	6.921
   After:	47	57	44	53	49	53	50.5	4.722

   1. State the null and alternative hypotheses, both in words and in notation. [3]
      H₀: μ_d ≤ 0 (presuming d = after - before; you could also subtract in the other order), HBE concentration does not increase after exercise.
      H_A: μ_d > 0, HBE concentration does increase after exercise.
   2. What statistical test is be appropriate to test the hypothesis?
      Should it be 1-tailed or 2-tailed? [2]
      Dependent t-test on pairwise differences. 1-tailed.
   3. Run the test (either p-value or classical approach) and draw a conclusion. [4]
      mean diff = 3.00, SD of diffs = 2.898.
      SE_d = 1.183, t = 2.535, df = 5, one-tailed.
      P-value approach: ⇒ 0.025 < p < 0.05 (real p = 0.026).
      Classical approach: t(0.05) = 2.02
   4. Interpret your conclusion in the context of the original research question.
      Please use complete English sentences. [2]
      Reject H₀, HBE concentration rises after exercise.
   5. What assumptions did you rely upon in conducting the test? [2]
      Random sampling of men; change in HBE concentration is normally distributed.
2. A factory needs to ensure that the widgets it produces have variance no more than 2.5mm². An inspector from corporate headquarters randomly selects 41 widgets from the factory, to check if the factory is within specifications. Those 41 widgets have a variance of 3.18mm² in length.
   1. State the null and alternative hypotheses, both in words and in notation. [3]
      H₀: variance σ² ≤ 2.5
      H_A: variance σ² > 2.5
   2. What statistical test is be appropriate to test the hypothesis?
      Should it be 1-tailed or 2-tailed? [2]
      χ² (chi-squared), 1-tailed.
   3. Run the test (either p-value or classical approach) and draw a conclusion. [4]
      χ² = 50.88, df = 40.
      P-value: p > 0.10 (actual p = 0.116)
      Classical: χ²* = 55.8.
   4. Interpret your conclusion in the context of the original research question.
      Please use complete English sentences. [2]
      We fail to reject the null hypothesis: there is insufficient evidence to show that the factory is out of spec.
   5. What assumptions did you rely upon in conducting the test? [2]
      Random sampling; length of widgets is normally distributed.
3. Drug company "Faizer" claims their antidepressant is preferred by 30 out of 40 physicians. In response, drug company "Vonartis" claims their antidepressant is preferred by 60 out of 70 physicians. Is Vonartis' drug significantly more preferred than Faizer's?
   1. State the null and alternative hypotheses, both in words and in notation. [3]
      H₀: binomial proportion p_V ≤ p_F
      H_A: binomial proportion p_V > p_F
   2. What statistical test is be appropriate to test the hypothesis?
      Should it be 1-tailed or 2-tailed? [2]
      Comparing two independent proportions, 1-tailed.
   3. Run the test (either p-value or classical approach) and draw a conclusion. [4]
      SE_V = sqrt(pq/n) = sqrt(0.857*0.143/70) = 0.04182
      SE_F = sqrt(pq/n) = sqrt(0.75*0.25/40) = 0.068465
      SE_p1-p2 = sqrt(0.857*0.143/70 + 0.75*0.25/40) = 0.08023
      z = ( (p'_V - p'_F) - 0 ) / SE = (0.857 - 0.75) / 0.08023 = 1.335
      P-value: 0.05 < p < 0.10 (actual p = 0.09)
      Classical: z* = 1.65
   4. Interpret your conclusion in the context of the two drug companies.
      Please use complete English sentences. [2]
      Fail to reject H₀, insufficient evidence to show that more physicians prefer Vonartis' drug than Faizer's.
   5. What assumptions did you rely upon in conducting the test? Are the assumptions met? Why? [2]
      Random sampling of physicians (this is a big assumption here!), np = 30 > 5, nq = 10 > 5, np = 60 > 5, nq = 10 > 5.
4. "Faizer" and "Vonartis" also produce competing blood glucose monitors. An independent lab obtains one glucose monitoring device from each company. A single sample of blood is tested 15 times in each company's glucose monitoring device. Faizer's device yields a standard deviation of 0.35 mmol/L; Vonartis' device yields a standard deviation of 0.28 mmol/L. Is there a difference in the precision of the two companies' devices?
   1. State the null and alternative hypotheses, both in words and in notation. [3]
      H₀: σ_F = σ_V, or σ_F² / σ_V² = 1
      H_A: σ_F ≠ σ_V, or σ_F² / σ_V² ≠ 1
   2. What statistical test is be appropriate to test the hypothesis?
      Should it be 1-tailed or 2-tailed? [2]
      F-test comparing two variances, 2-tailed.
   3. Run the test (either p-value or classical approach) and draw a conclusion. [5]
      F = 0.35² / 0.28² = 1.5625, df = (14, 14)
      2-tailed p-value: p > 0.10 (actual p = 0.414)
      Classical: F*(12,14,0.025) = 3.05, F*(15,14,0.025) = 2.95
   4. Interpret your conclusion in the context of the two drug companies.
      Please use complete English sentences. [2]
      Fail to reject H0: no significant difference in precision between the two companies' blood glucose monitors.
   5. What assumptions did you rely upon in conducting the test? [2]
      Random sampling of blood, normal distribution of blood glucose levels.
5. Human beta-endorphin (HBE) is a hormone secreted by the pituitary gland under conditions of stress (like exams!). Suppose we wish to determine whether blood concentration of HBE (pg/mL) is different for men who exercise regularly as compared with men who do not exercise regularly.
   1. State the null and alternative hypotheses, both in words and in notation. [3]
      [H₀: μ_ex = μ_no: HBE levels are the same for both groups.
      H_A: μ_ex ≠ μ_no: HBE levels are different for the two groups.
   2. What statistical test is be appropriate to test the hypothesis?
      Should it be 1-tailed or 2-tailed? [2]
      t-test on independent groups, 2-tailed.
   3. Data for this experiment are given below. Sketch boxplots for the data, on a common axis (number line). [4]

      								Mean:	SD:
      Exercisers:	60	58	62	49	51	58	54	56	4.7958
      Non-exercisers:	41	37	51	60	28	35		42	11.6276
   4. Run the test (either p-value or classical approach) and draw a conclusion. [4]
      SE_E = 1.8127, SE_N = 4.7469, SE = 5.0812.
      mean diff = 14, so t = 2.755.
      df = min(n₁, n₂) - 1 = 5 (real df = 6.45).
      P-value: Two-tailed: 0.02 < p < 0.05 (real p = 0.0401)
      Classical: t* = 2.57
   5. Interpret your conclusion in the context of the original research question.
      Please use complete English sentences. [2]
      Reject H₀, HBE levels are different for exercisers as compared to non-exercisers.
   6. What assumptions did you rely upon in conducting the test? [2]
      Random sampling on both groups; HBE levels are normally distributed in both groups; variance of HBE levels is similar in both groups.