MATH-4030F.mukhan4.4030-Final

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Muhammad Khan MATH-4030F Assignment 4030-Final due 12/10/2021 at 04:00pm EST 1. (4 points) Below is a sample of share prices (in dollars) for a particular stock, selected at random over several years: 216 171 170 208 186 199 203 180 173 187 199 196 186 170 198 186 213 215 Use Excel (or other form of electronic assistance) to find the mean, median, mode, variance, standard deviation, and coeffi- cient of variation for this sample. Mean = Median = Mode = Variance = Standard Deviation = Coefficient of Variation = Answer(s) submitted: 192 191.5 186 242.352941 15.567689 0.0810817136 (correct) 2. (2 points) A standardized variable always has A. mean 0 and standard deviation 1 B. changing mean and standard deviation 1 C. mean 0 and changing standard deviation D. changing mean and changing standard deviation The z-score corresponding to an observed value of a variable tells you the number of standard deviations that the observation is from the mean A. True B. False A positive z-score indicates that the observation is A. above the mean B. below the mean Answer(s) submitted: A A A (correct) 3. (1 point) The population proportion is a A. statistic B. parameter C. None of the above The sample proportion is a A. statistic B. parameter C. None of the above Answer(s) submitted: B A (correct) 4. (3 points) A bag contains 8 red marbles, 8 white marbles, and 10 blue marbles. You draw 3 marbles out at random, with- out replacement. What is the probability that all the marbles are red? The probability that all the marbles are red is . What is the probability that exactly two of the marbles are red? The probability that exactly two of the marbles are red is . What is the probability that none of the marbles are red? The probability of picking no red marbles is . Answer(s) submitted: (incorrect) 1
Select True or False from each pull-down menu, depending on whether the corresponding statement is true or false. ? 1. Marginal probability is the probability that a given event will occur, with no other events taken into con- sideration. ? 2. Two or more events are said to be independent when the occurrence of one event has no effect on the probability that the other will occur. ? 3. A useful graphical method of constructing the sample space for an experiment is pie chart. ? 4. Probability refers to a number between 0 and 1, which expresses the chance that an event will occur. Answer(s) submitted: T T F T (correct) 6. (4 points) In a survey of 251 people, the following data were obtained relating gender to political orientation: Republican (R) Democrat (D) Libertarian (L) Total Male (M) 100 27 9 136 Femal (F) 50 50 15 115 Total 150 77 24 251 A person is randomly selected. What is the probability that the person is: a) Male? b) Male and a Democrat? c) Male given that the person is a Democrat? d) Republican given that the person is Male? e) Female given that the person is a Libertarian? f) Are the events Male and Republican independent? Enter yes or no . Answer(s) submitted: 136/251 27/251 27/77 100/136 15/24 no (correct) 7. (3 points) The table below summarizes the number of surface flaws found on the paintwork of new cars following their inspection after primer paint was applied by a new method: No. of flaws No. of cars 0 3 1 7 2 12 3 11 4 3 5 2 6 2 Part a) Find the mean number of flaws per car. Please give your answer to two decimal places. The mean number of flaws per car is: Part b) Find the variance of the number of flaws per car. Please give your answer to two decimal places. The variance of the number of flaws per car is: Answer(s) submitted: 2.45 2.16 (correct) 8. (2 points) If E [ X ] = - 5 and Var ( X ) = 3, then E [( 5 + 6 X ) 2 ] = and Var ( 4 + 3 X ) = . Answer(s) submitted: -1067 (incorrect) 9. (1 point) Which of the following distributions is suitable to model the length of time that elapses before the first telephone call is re- ceived by a switchboard? A. Normal B. Uniform C. Exponential D. Poisson Given that X is a normal variable, which of the following statements is true? A. The variable 5 X is also normally distributed B. The variable X + 5 is also normally distributed 2
C. The variable X - 5 is also normally distributed D. All of the above statements is true Answer(s) submitted: C D (correct) 10. (2 points) For a χ 2 -curve with 22 degrees of freedom, find the χ 2 -value that has area 0.01 to its right. A. 40.290 B. 42.796 C. 9.542 D. None of the above For a χ 2 -curve with 22 degrees of freedom, find the χ 2 -value that has area 0.995 to its right. A. 8.643 B. 9.542 C. 42.796 D. None of the above Answer(s) submitted: A A (correct) 11. (3 points) A random sample of n measurements was selected from a population with unknown mean μ and standard deviation σ . Cal- culate a 90% confidence interval for μ for each of the following situations: (a) n = 110 , x = 73 . 5 , s = 3 . 05 μ (b) n = 90 , x = 4 , s = 2 . 54 μ (c) n = 120 , x = 103 . 2 , s = 4 . 58 μ (d) n = 95 , x = 73 . 2 , s = 4 . 05 μ Answer(s) submitted: 73.022 73.978 3.5595685 4.440431 102.512 103.888 72.516 73.884 (score 0.75) 12. (3 points) Determine the sample size required to estimate a population mean to within 8 units given that the population standard devia- tion is 60. A confidence level of 90% is judged to be appropriate. Sample Size = Determine the sample size required to estimate a population mean to within 8 units given that the population standard devia- tion is 105. Use a confidence level of 90%. Sample Size = Determine the sample size required to estimate a population mean to within 8 units given that the population standard devia- tion is 60. Use a confidence level of 95%. Sample Size = Determine the sample size required to estimate a population mean to within 23 units given that the population standard devi- ation is 60. A confidence level of 90% is judged to be appropri- ate. Sample Size = Answer(s) submitted: 153 467 217 19 (correct) 13. (2 points) The following random sample was selected from a normal distribution: 7 7 9 5 10 20 12 1 15 14 (a) Construct a 90% confidence interval for the population mean μ . μ (b) Construct a 95% confidence interval for the population mean μ . μ Answer(s) submitted: 6.825 13.175 6.082 13.918 (correct) 14. (3 points) Suppose you have selected a random sample of n = 10 measurements from a normal distribution. Compare the standard normal z values with the corresponding t values if you were forming the following confidence intervals. (a) 80% confidence interval z = t = (b) 95% confidence interval z = t = (c) 90% confidence interval z = t = 3
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Answer(s) submitted: 1.282 1.383 1.960 2.262 1.645 1.833 (correct) 15. (4 points) Answer true or false to each statement con- cerning a confidence interval for a population mean. a) The length of a confidence interval can be determined if you know only the margin of error. b) The margin of error can be determined if you know only the length of the confidence interval. c) The confidence interval can be obtained if you know only the margin of error. d) The confidence interval can be obtained if you know only the margin of error and the sample mean. e) The margin of error can be determined if you know only the confidence level. f) The confidence level can be determined if you know only the margin of error. g) The margin of error can be determined if you know only the confidence level, population standard deviation, and sample size. h) The confidence level can be determined if you know only the margin of error, population standard deviation, and sample size. Answer(s) submitted: True True False True False False True True (correct) 16. (3 points) Construct the 99% confidence interval estimate of the popu- lation proportion p if the sample size is n = 500 and the number of successes in the sample is x = 84 . < p < Which of the following is the correct interpretation for your answer in part (a)? A. There is a 99% chance that the percentage of suc- cesses in the population lies in the interval B. We can be 99% confident that the percentage of suc- cesses in the sample lies in the interval C. We can be 99% confident that the percentage of suc- cesses in the population lies in the interval D. None of the above Answer(s) submitted: 0.1249 0.2111 C (correct) 17. (2 points) Suppose that we reject a null hypothesis at the 0.05 level of significance. Then for which of the following α - values do we also reject the null hypothesis? A. 0.06 B. 0.02 C. 0.03 D. 0.04 The critical values z α or z α / 2 are the boundary values for the: A. power of the test B. rejection region(s) C. Type II error D. level of significance Answer(s) submitted: A B (correct) 18. (1 point) The smaller the P -value obtained in a hypothe- sis test, the stronger the evidence against the null hypothesis. the stronger the evidence against the null hypothesis, only if the test is two-sided. the stronger the evidence against the null hypothesis, only if the test is one-sided. the weaker the evidence against the null hypothesis. the weaker the evidence against the null hypothesis, only if the test is two-sided. the weaker the evidence against the null hypothesis, only if the test is one-sided. Answer(s) submitted: the stronger ... hypothesis. (correct) 4
19. (1 point) When one changes the significance level of a hypothesis test from 0.10 to 0.05, which of the following will happen? Check all that apply. A. The chance of committing a Type II error changes from 0.10 to 0.05. B. It becomes easier to prove that the null hypothesis is true. C. The chance of committing a Type I error changes from 0.10 to 0.05. D. The test becomes more stringent to reject the null hypothesis (i.e., it becomes harder to reject the null hy- pothesis). E. It becomes harder to prove that the null hypothesis is true. F. The test becomes less stringent to reject the null hy- pothesis (i.e. it becomes easier to reject the null hypoth- esis). G. The chance that the null hypothesis is true changes from 0.10 to 0.05. Answer(s) submitted: ( C, D ) (correct) A type I error A. is the rejection of a false null hypothesis. B. is the rejection of a true null hypothesis. C. arises when the true null hypothesis is not rejected. D. arises when the false null hypothesis is not rejected. Answer(s) submitted: B (correct) A type II error A. arises when the false null hypothesis is not rejected. B. is the rejection of a true null hypothesis. C. is the rejection of a false null hypothesis. D. arises when the true null hypothesis is not rejected. Answer(s) submitted: A (correct) 22. (2 points) For each statement, select the correct null hy- pothesis, H 0 , and alternative hypothesis, H a , in symbolic form. (a) A certain type of hummingbird is known to have an average weight of 4.55 grams. A researcher wonders if hum- mingbirds (of this same type) living in the Grand Canyon differ in weight from the population as a whole. The researcher finds a sample of 30 such hummingbirds from the Grand Canyon and calculates their average weight to be 3.75 grams. A. H 0 : ¯ x = 3 . 75, H a : ¯ x > 3 . 75 B. H 0 : μ = 4 . 55, H a : μ 6 = 4 . 55 C. H 0 : μ = 4 . 55, H a : μ < 4 . 55 D. H 0 : ¯ x = 4 . 55, H a : ¯ x < 4 . 55 E. H 0 : μ < 4 . 55, H a : μ = 4 . 55 F. H 0 : μ = 3 . 75, H a : μ 6 = 3 . 75 (b) According to the Merck Veterinary Manual, the average resting heart rate for a certain type of sheep dog is 115 beats per minute (bpm). A Montana farmer notices his aging sheep dog has been acting more lethargic than usual and wonders if her heart rate is slowing. He measures her heart rate on 15 oc- casions and finds a sample mean heart rate of 118.2 bpm. A. H 0 : μ = 118 . 2, H a : μ < 118 . 2 B. H 0 : ¯ x = 118 . 2, H a : ¯ x 6 = 118 . 2 C. H 0 : ¯ x = 115, H a : ¯ x > 115 D. H 0 : μ = 115, H a : μ > 115 E. H 0 : μ = 115, H a : μ < 115 F. H 0 : μ = 115, H a : μ 6 = 115 (c) The mean height of all adult American males is 69 inches (5 ft 9 in). A researcher wonders if young American males between the ages of 18 and 21 tend to be taller than 69 inches. A random sample of 100 young American males ages 18 to 21 yielded a sample mean of 71 inches. A. H 0 : ¯ x = 71, H a : ¯ x < 71 B. H 0 : μ = 71, H a : μ < 71 C. H 0 : μ = 69, H a : μ 6 = 69 D. H 0 : ¯ x = 69, H a : ¯ x > 69 E. H 0 : μ > 69, H a : μ < 69 F. H 0 : μ = 69 , H a : μ > 69 Answer(s) submitted: B E F (correct) 23. (3 points) Jen thinks a certain potato chip maker is putting fewer chips in their regular bags of chips. From a ran- dom sample of 19 bags of potato chips she calculated a P value of 0.023 for the sample. (a) At a 5% level of significance, is there evidence that Jen is correct? (Type: Yes or No): (b) At a 10% level of significance, is there evidence that she is correct? (Type: Yes or No): (c) In a statistical test of hypotheses, we say that the data are statistically significant at level α if A. the P - value is less than α . B. α = 0 . 05. C. the P - value is larger than α . D. α is small. Answer(s) submitted: Yes 5
Yes A (correct) 24. (4 points) Suppose that we are to conduct the following hypothesis test: H 0 : μ = 960 H 1 : μ > 960 Suppose that you also know that σ = 160, n = 90, ¯ x = 987 . 2, and take α = 0 . 1. Draw the sampling distribution, and use it to determine each of the following: A. The value of the standardized test statistic: Note: For the next part, your answer should use interval no- tation. An answer of the form ( - , a ) is expressed (-infty, a), an answer of the form ( b , ) is expressed (b, infty), and an answer of the form ( - , a ) ( b , ) is expressed (-infty, a)U(b, infty). B. The rejection region for the standardized test statistic: C. The p-value is D. Your decision for the hypothesis test: A. Do Not Reject H 0 . B. Reject H 1 . C. Do Not Reject H 1 . D. Reject H 0 . Answer(s) submitted: 1.6127616 (1.28155,inf) 0.053405 D (correct) Select True or False from each pull-down menu, depending on whether the corresponding statement is true or false. ? 1. In a criminal trial, a Type I error is made when an inno- cent person is convicted. ? 2. The probability of making a Type I error and the level of significance are the same. ? 3. The probability of a Type II error is represented by β and is the probability of failing to reject a false null hy- pothesis. ? 4. Increasing the probability of a Type I error also in- creases the probability of a Type II error Answer(s) submitted: T T T F (correct) Select True or False from each pull-down menu, depending on whether the corresponding statement is true or false. ? 1. The p -value of a test is the probability of observing a test statistic at least as extreme as the one computed given that the null hypothesis is true. ? 2. An alternative or research hypothesis is an assertion that holds if the null hypothesis is false. ? 3. A two-tail test is a test in which a null hypothesis can be rejected by an extreme result occurring in only one direction. ? 4. The p -value is usually 0.05. Answer(s) submitted: T T F F (correct) 27. (2 points) A random sample of 9 size AA batteries for toys yield a mean of 3.64 hours with standard deviation, 0.99 hours. (a) Find the critical value, t*, for a 99% CI. t* = (b) Find the margin of error for a 99% CI. Answer(s) submitted: 3.3554 3.3554*0.99/[sqrt(9)] (correct) 28. (3 points) A random sample of 130 observations is se- lected from a binomial population with unknown probability of success p . The computed value of ˆ p is 0 . 67. (1) Test H 0 : p = 0 . 6 against H a : p > 0 . 6. Use α = 0 . 01. test statistic z = critical z score The final conclusion is A. There is not sufficient evidence to reject the null hy- pothesis that p = 0 . 6. B. We can reject the null hypothesis that p = 0 . 6 and accept that p > 0 . 6. (2) Test H 0 : p = 0 . 5 against H a : p < 0 . 5. Use α = 0 . 05. test statistic z = critical z score The final conclusion is A. We can reject the null hypothesis that p = 0 . 5 and accept that p < 0 . 5. B. There is not sufficient evidence to reject the null hy- pothesis that p = 0 . 5. (3) Test H 0 : p = 0 . 5 against H a : p 6 = 0 . 5. Use α = 0 . 01. 6
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test statistic z = positive critical z score negative critical z score The final conclusion is A. We can reject the null hypothesis that p = 0 . 5 and accept that p 6 = 0 . 5. B. There is not sufficient evidence to reject the null hy- pothesis that p = 0 . 5. Answer(s) submitted: 1.629 2.326 B 3.877 1.960 B 3.877 2.5758 -2.5758 A (score 0.800000011920929) 29. (3 points) Some car tires can develop what is known as ”heel and toe” wear if not rotated after a certain mileage. To assess this issue, a consumer group investigated the tire wear on two brands of tire, A and B, say. Fifteen cars were fitted with new brand A tires and thirteen with brand B tires, the cars assigned to brand at random. (Two cars initially assigned to brand B suffered se- rious tire faults other than heel and toe wear, and were excluded from the study.) The cars were driven in regular driving con- ditions, and the mileage at which heal and toe wear could be observed was recorded on each car. For the cars with brand A tires, the mean mileage observed was 23 . 69 (in 10 3 miles ) and the variance was 6 . 40 (in 10 6 miles 2 ). For the cars with brand B, the corresponding statistics were 23 . 92 (in 10 3 miles) and 5 . 53 (in 10 6 miles 2 ) respectively. The mileage before heal and toe wear is detectable is assumed to be Normally distributed for both brands. Part a) Calculate the pooled variance s 2 to 3 decimal places. During intermediate steps to arrive at the answer, make sure you keep as many decimal places as possible so that you can achieve the precision required in this question. × 10 6 miles 2 Part b) Determine a 95% confidence interval for μ A - μ B , the difference in the mean 10 3 mileages before heal and toe wear for the two brands of tire. Leave your answer to 2 decimal places. ( , ) Part c) Based on the 95% confidence interval constructed in the pre- vious part, which of the following conclusions can be drawn when we test H 0 : μ A = μ B vs. H a : μ A 6 = μ B with α = 0 . 05. A. Do not reject H 0 since 0 is not in the interval found in part (b). B. Reject H 0 since 0 is in the interval found in part (b). C. Do not reject H 0 since - 0 . 23 is within the interval found in part (b). D. Do not reject H 0 since 0 is within the interval found in part (b). E. Reject H 0 since 0 is not within the interval found in part (b). Answer(s) submitted: 5.998 -2.14 D (correct) 30. (2 points) Suppose we were comparing the mean age of buyers of new domestic cars to the mean age of buyers of new imported cars. What is the variable under consideration? A. buyers of imported cars B. age of buyers C. buyers of domestic cars D. None of the above What are the two populations under consideration? A. age of buyers of new domestic cars and age of buyers of new imported cars B. buyers of new domestic cars and buyers of new im- ported cars C. cars and buyers D. None of the above Answer(s) submitted: B B (correct) 31. (3 points) The number of men and women among pro- fessors in Math, Physics, Chemistry, Linguistics, and English departments from a SRS of small colleges were counted, and the results are shown in the table below. Dept. Math Physics Chemistry Linguistics English Men 64 79 37 23 28 Women 6 8 10 15 17 Test the claim that the gender of a professor is independent of the department. Use the significance level α = 0 . 025 (a) The test statistic is χ 2 = (b) The critical value is χ 2 = (c) Is there sufficient evidence to warrant the rejection of the claim that the gender of a professor is independent of the department? A. No B. Yes Answer(s) submitted: 7
(incorrect) 32. (3 points) Consider the data set below. x 9 7 4 6 4 8 y 3 7 9 9 7 2 For a hypothesis test, where H 0 : β 1 = 0 and H 1 : β 1 6 = 0, and using α = 0 . 05, give the following: (a) The test statistic t = (b) The degree of freedom d f = (c) The rejection region | t | > The final conclustion is A. There is not sufficient evidence to reject the null hy- pothesis that β 1 = 0. B. We can reject the null hypothesis that β 1 = 0 and accept that β 1 6 = 0. Answer(s) submitted: 4 B (incorrect) 33. (3 points) Construct both a 95% and a 98% confidence interval for β 1 . ˆ β 1 = 36 , s = 9 , SS xx = 49 , n = 21 95% : β 1 98% : β 1 Answer(s) submitted: 33.340 38.660 32.736 39.264 (correct) 34. (3 points) The table below shows (lifetime) peptic ulcer rates (per 100 population), U , for various family incomes, x , as reported by the 1989 National Health Interview Survey. Income 4000 6000 8000 12000 16000 20000 30000 45000 60000 Ulcer rate 13.2 13.2 13.5 12.7 12.1 11.9 11 9.1 7.7 (a) Find the equation of the regression line. Ulcer rate, U ( x ) = . (b) Estimate the peptic ulcer rate for an income level of x 0 = 21000 according to the linear model in part (a). Ulcer rate, U ( x 0 ) = . Answer(s) submitted: 13.9158-0.0001x 11.8158 (incorrect) 35. (1 point) Before determining a regression line, it is important to do what? A. Plot the data to make sure it does not appear linear. B. Make sure that every x value has exactly one corre- sponding y value C. Plot the data to make sure it appears somewhat lin- ear. D. None of the above Answer(s) submitted: C (correct) 36. (3 points) Consider the linear equation y = b 0 + b 1 x . a. In the equation, b 0 is A. the independent variable B. the dependent variable C. the y -intercept D. the slope b. In the equation, b 1 is A. the dependent variable B. the y -intercept C. the slope D. the independent variable c. Give the geometric interpretation of b 0 . It indicates A. how much the y -value on the straight line changes when the x -value increases by unit B. the x -value where the straight-line graph of the linear equation intersects the x -axis C. the y -value where the straight-line graph of the linear equation intersects the y -axis D. how much the x -value on the straight line changes when the y -value increases by unit 8
d. Give the geometric interpretation of b 1 . It indicates A. how much the x -value on the straight line changes when the y -value increases by unit B. the x -value where the straight-line graph of the linear equation intersects the x -axis C. how much the y -value on the straight line changes when the x -value increases by unit D. the y -value where the straight-line graph of the linear equation intersects the y -axis Answer(s) submitted: C C C C (correct) 37. (1 point) The coefficienct of determination falls between A. -1 and 1 B. negative infinity to positive infinity C. -1 and 0 D. 0 and 1. E. None of the above Answer(s) submitted: D (correct) 38. (2 points) If the coefficient of determination is 0.975, then the slope of the regression line: A. must be positive B. must be negative C. could be either positive or negative D. none of the above answers is correct If the coefficient of correlation is 0.90, the percentage of the variation in the dependent variable y that is explained by the variation in the independent variable x is: A. 0.90% B. 90% C. 0.81% D. 81% Answer(s) submitted: C D (correct) Select True or False from each pull-down menu, depending on whether the corresponding statement is true or false. ? 1. When the standard deviation is expressed as a percent- age of the mean, the result is the coefficient of correla- tion ? 2. Expressed in percentiles, the interquartile range is the difference between the 25th and the 75th percentiles. ? 3. If the coefficient of correlation r = 0, then there is no lin- ear relationship between the dependent variable y and the independent variable x. ? 4. The coefficient of variation allows us to compare the variation in two sets of data based on different mea- surement units. Answer(s) submitted: F T T F (incorrect) 40. (1 point) The correlation between two quantitative vari- ables X and Y is found to be 0. This implies the two variables are not related at all. A. False B. True Answer(s) submitted: A (correct) 41. (1 point) The slope of a regression line and the correla- tion are similar in the sense that... A. both can be used for prediction. B. they both have the same sign. C. they do not depend on the units of measurement of the data. D. they both fall between -1 and 1 inclusive. E. neither of them can be affected by outliers. Answer(s) submitted: B (correct) 42. (1 point) Which of the following is/are incorrect state- ment(s) about the correlation between two quantitative variables X and Y ? I. A correlation of -0.8 indicates a stronger linear association between X and Y than a correlation of 0.5. II. A correlation of 0 implies X and Y are not related at all. III. A correlation of -1 indicates that Y = - X . A. I only B. II only C. I and II only D. II and III only E. I, II and III Answer(s) submitted: D 9
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(correct) 43. (2 points) Match the following sample correlation coef- ficients with the explanation of what that correlation coefficient means. Type the correct letter in each box. 1. r = 0 2. r = . 92 3. r = 1 4. r = - . 97 A. a strong positive relationship between x and y B. no relationship between x and y C. a strong negative relationship between x and y D. a perfect positive relationship between x and y Answer(s) submitted: B A D C (correct) 44. (2 points) For each problem, select the best response. (a) In a scatterplot of the average price of a barrel of oil and the average retail price of a gallon of gasoline, you expect to see A. a negative association. B. a positive association. C. very little association. D. None of the above. (b) If the correlation between two variables is close to 0, you can conclude that a scatterplot would show A. a strong straight-line pattern. B. a cloud of points with no visible pattern. C. no straight-line pattern, but there might be a strong pattern of another form. D. None of the above. (c) What are all the values that a correlation r can possibly take? A. -1 r 1 B. 0 r 1 C. r 0 D. None of the above. Answer(s) submitted: B C A (correct) Generated by c WeBWorK, http://webwork.maa.org, Mathematical Association of America 10