test3-fa22
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Apr 3, 2024
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MATH 3342-022 Test 3 Fall 2022 Test 3 Key Name (Last, First): INSTRUCTIONS e This exam consists of two parts. Part I is comprised of 10 multiple-choice questions (no partial credit) and is worth 50 points. Part II consists of partial-credit-type questions and is worth 50 points. e Write the answer of your choice, A-E, to the Part I questions in the Answer: space provided. Use CAPITALS, e.g. B instead of b. e Give complete answers to the Part II questions in order to get maximum credit. Even if you do not know the answer try to write something down; remember that blank space equals zero points. e The following aids are allowed on this test: — A table of standard normal curve areas (Table A.3) and a table of critical values for the t-distribution (Table A.5). — A formula-sheet consisting of no more than 1 sheet (2 pages) of 8.5 x 11 paper. — A standard scientific calculator. Usage of a smartphone is permitted provided it is ONLY used as a scientific calculator (you may NOT use it as a communication device, to access the internet, stored text, etc.).
MATH 3342-022 Test 3 Fall 2022 PART 1 1. Which of the following statements is correct or appropriate (select one)? (a) The sample mean regarded as a random variable, is denoted by X, while the calculated value of the sample mean from data is denoted by . (b) The distribution of the sample mean X is sometimes called its sampling distribution. d ) (c) A statistic is any quantity that can be calculated from a sample of data. (d) All of the above are correct/appropriate *** ) (e) None of the above is correct/appropriate Answer: 2. Let X1,...,Xe4 be a random sample (IID) from a population with mean of p = 72 and a standard deviation of ¢ = 16. Then, the distribution of the sample mean X has a standard deviation of: (a) (b) (c) 8 (d) 1 ) (e) None of the above Answer: 3. In addition to predicting the standard deviation of X in Question 2, the Central Limit Theorem also predicts that: X is approximately normal with mean 72 *** X is exactly normal with mean 72 and variance 162 X is approximately gamma with o = 72/5 and 8 =5 he Central Limit Theorem does not apply here None of the above is accurate Answer: 4. A random sample (IID) of size 36 is drawn from a population with mean of 5 and standard deviation of 10. Let T be the sample total. Then, E(T) is: None of the above Answer: None of the above Answer:
MATH 3342-022 Test 3 Fall 2022 6. For quality control purposes, a manufacturer of bike helmets wants to determine the true proportion of defectives, p, in their manufacturing process. To this end, a random sample of 50 bike helmets is selected, and it is found that 11 of them are flawed. Based on these data, the best estimate of p would be: None of the above Answer: 7. Construct a 95% confidence interval for p in Question 6. (a) (0.105, 0.335) *** (b) (0.023, 0.197) (c) (0.361, 0.639) (d) (0.124, 0.316) ) (e) None of the above Answer: 8. A 99% confidence interval for the mean u of a normal population when the standard deviation o is known is found to be (98.6, 118.4). If the confidence level is reduced to 95%, the confidence interval for pu: (a) Becomes wider (b) Becomes narrower *** (c (d (e) None of the above Answer: ) ) ) Remains unchanged ) Can be constructed from the given information ) 9. In Question 8, what is the value of the sample mean x? Cannot be determined from the given information None of the above Answer: 10. In Question 8, what is the minimum sample size required for the 99% confidence interval for u to have a width of w = 10, if o = 127 None of the above Answer:
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MATH 3342-022 Test 3 Fall 2022 PART II 1. (15 points) The sediment density (g/cm) of randomly selected specimens from a certain region are normally distributed with mean u = 65 and variance 02 = 20. If a random sample (IID) of 3 such specimens is selected, X7, X3, X3, let T'= X1 + X3 + X3 denote the sample total. (a) What is the probability that X3 yields a value of at least 70 g/cm? P(X >70) = P(Z>M) V20 — P (Z > 1.1180) 1 — 0.8682 = 0.1318 (b) Compute the mean and variance of T. The mean is: E(T) = 3(65) = 195 The variance of the sum is the sum of the variances: V(T) = 3(20) = 60 (¢) Compute P(T < 210). Since the sum of independent normals is again normal, T' ~ N(195, 60), so that: P(T <210) = P(ZS M) V60 = P(Z<194) 0.9736
MATH 3342-022 Test 3 Fall 2022 2. (20 points) The flexural strength (Mpa) of a random sample of 14 concrete beams yielded the following values: 6.9, 7.3, 7.8, 8.0, 8.2, 8.3, 8.4, 8.7, 8.8, 9.1, 9.2, 9.7, 10.7, 10.7 The sample mean and standard deviation of this sample are, respectively, * = 8.7 and s = 1.124. (a) Calculate a point estimate of the strength value that separates the weakest 50% of all such beams from the strongest 50%, and state which estimator you used. The median would be a good estimate of the 50th percentile: 4+ 8. 5;:8+537:8,55 (b) Calculate a 95% confidence interval (two-sided) for the true mean flexural strength of all such beams. Using ta/2,n—1 = 1.025,13 = 2.16, the CI is: S 1.124 Tt 1— = 87x£216—— a/2,n—1 \/fi \/fi = 8.7+0.6489 — (8.05, 9.35) (¢) What the manufacturer really wants is to be 95% confident that the true mean flexural strength of all such beams is at least L. Calculate an appropriate value for L. Using tan—1 = t.05,13 = 1.771, the 95% lower confidence bound L is: 8 1.124 T—ton-1——= = 87—-17711—— a,n 1 \/fi \/fi = 8.7—0.5320 = &.168 (d) What distributional assumption(s) should be checked in order to ensure the validity of the inferences made in (b) and (c)? The population of strength values must be normally distributed.
MATH 3342-022 Test 3 Fall 2022 3. (25 points) An article gave the following summary information for fracture strengths (MPa) of n = 165 ceramic bars fired in a particular kiln: T = 88.26 and s = 3.88. (a) Find a 95% confidence interval (two-sided) for the mean fracture strength of all such ceramic bars. Since this is a large sample we use a normal-based CI: T+t Za/QL /n which gives: 88.26:|:z,025fl = 88.26:|:1.96fi = (87.67, 88.85) V165 V165 (b) Suppose the investigators had believed a priori that the population standard deviation was o = 3 MPa. Based on this, how large a sample would be required to estimate the mean to within 0.5 MPa? (IL.e., what is the minimum 7 if the 95% CI is to have a width of w = 1.) The minimum n = 139 because: 9 2 n> (2za/2%) - (2(1.96)%) — 138.3 4. (5 points) In a random sample of 507 adult Americans, only 142 correctly described the Bill of Rights as the first ten amendments to the US Constitution. Calculate a (two-sided) confidence interval using a 99% confidence level for the proportion of all US adults that can give a correct description of the Bill of Rights. Using 2,005 = 2.576 and p = 142/507 = 0.28, the CI is: . p(1 — p) 0.28(0.72) + = 0.28 2. —= D 242 - 0.28 576 507 = 0.28£0.05 (0.23, 0.33)
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