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Math 155 Practice Final Exam Questions 1. For a class of 140 students, scores on a recent exam (on a scale from 0 to 100) were as follows: Score 90-100 80-89 70-79 60-69 Relative Frequency .15 .25 .35 .10 What is the frequency of a score between 0 and 59? a. b. c. d. e. .15 .85 0 21 30 2. The statistics π₯Μ = 25, π 2 = 16 are computed from sample data from a population. What is the z-score for an observation: π₯ = 9? a. b. c. d. e. 1 -1 4 -4 None of the above 3. For a sample of 25 observations, the sum of all observations is 455 and the sum of all squared observations is 16,057. The sample standard deviation is equal to which of the following? a. b. c. d. e. 16 18 25.6 256 324 4. If a balanced die is rolled twice, the probability that both rolls are the same is equal to which of the following? a. b. c. d. e. 1/36 2/36 1/6 2/6 None of the above The box plots below are for the reaction time (in seconds) from a study of 100 subjects where 50 were subjected to a non-threatening stimulus (NT) and the other 50 were subjected to a threatening stimulus (T). Boxplot of TIME NT T 2.6 2.4 TIME 2.2 2.0 1.8 1.6 1.4 1.2 Panel variable: STIMULUS Questions 5 and 6 below refer to the box-plots above. 5. What percentage of the subjects who received a threatening stimulus had reaction times exceeding 1.8 seconds? a. b. c. d. e. 0% 25% 50% 75% 100% 6. Which of the following statements is incorrect? a. The inter-quartile range for the group who received a non-threatening stimulus was .3 seconds. b. The twenty-fifth percentile of reaction times for those who received a nonthreatening stimulus was greater than the seventy-fifth percentile for those who received a threatening stimulus. c. The difference in median reaction times for the two groups is .3 seconds. d. In all of the data there was only one outlier. e. The reaction times for the 50 subjects who received the threatening stimulus were from a low of 1.3 seconds to a high of 2.1 seconds. Questions 7-9 refer to the stem and leaf diagram below. The stem and leaf diagram below is for EPA gas mileage (MPG) data for a sample of 80 Honda CRVβs (model year 2008) that were tested for fuel efficiency. In this figure, a mileage of 30.0 would have a stem of 30 and a leaf of 0 (representing the fractional part of a mile per gallon). You are also given for this data the following summary statistics: β80 π=1 π₯π = 2,968.3 Stem-and-leaf of MPG Leaf Unit = 0.10 1 2 5 9 13 21 38 (18) 24 16 11 6 3 2 1 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 N and 2 β80 π=1 π₯π = 109,794 = 80 0 8 579 1269 2489 13566789 00123344556677889 001112233445667789 22345678 00347 02557 002 1 2 9 7. Determine the median mileage for this sample. a. b. c. d. e. 36.95 37.00 37.05 37.10 37.15 8. Determine the mean mileage to the nearest .01 mile per gallon for this sample. a. b. c. d. e. 29.68 36.95 37.00 37.05 37.10 9. Determine the percentage of cars in the sample whose MPG rating met or exceeded 39 miles per gallon. a. b. c. d. e. 14% 16% 18% 20% 22% 10. In a gambling game, two different numbers are selected at random from the whole numbers 1, 2, 3, β¦ , 10 to form the βwinningβ combination. You play the combination 1 and 3. The probability that you will win is: a. b. c. d. e. 1/10 4/10 1/90 1/45 1/100 Use the following information for questions 11-13. The outcome for rolling a pair of balanced dice is the sum of the dots on the upward faces. The probability model for the different outcomes is: Sum 2 3 4 5 6 7 8 9 10 11 12 Probability 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36 Events: π΄ = {7 β€ sum β€ 12} , π΅ = {6 β€ sum β€ 8} 11. Determine the probability that event A will not occur. a. b. c. d. e. 5/12 5/11 6/11 6/12 7/12 12. Determine the probability that event A occurs, given that event B occurs. a. b. c. d. e. 11/16 11/21 11/36 16/36 21/36 13. Determine the probability that event A occurs, or event B occurs, or both events occur. a. b. c. d. e. 11/36 15/36 21/36 26/36 37/36 14. An experiment has 3 possible outcomes denoted by E1 , E2 , E3 . Which of the following is the probability missing in the table below? a. b. c. d. e. Outcome E1 E2 Probability .24 .33 E3 .33 33 .43 43 It cannot be determined without additional information. 15. The probability that event E occurs, or event F occurs, or that both E and F occur is .60. The probability that event E occurs is .48. The probability that event F occurs is .57. Determine the probability that both events E and F occur. a. b. c. d. e. .03 .09 .12 .45 1.05 16. π is a discrete random variable with p ο¨ 1ο© ο½ .6 , p ο¨ 2 ο© ο½.3, p ο¨ 3ο© ο½.1 . The mean value of π is _____. a. b. c. d. e. 1.0 1.5 2.0 2.7 3.0 17. π is a discrete random variable with p ο¨ 1ο© ο½ .6 , p ο¨ 2 ο© ο½.3, p ο¨ 3ο© ο½.1 . The standard deviation of π is _____. a. b. c. d. e. 0.450 0.671 1.000 1.643 2.700 18. A fair coin is flipped 5 times. Determine the probability that at least 4 heads occur in the 5 flips. a. b. c. d. e. 1/32 4/5 6/32 26/32 31/32 19. For an adult population, the average weight is 188 pounds and the standard deviation is 12 pounds. The weights for this population are normally distributed. You randomly select 4 of these adults and compute the average value: πΜ . Determine the probability that πΜ exceeds 194. a. b. c. d. e. .1587 .1915 .3085 .3413 Cannot be determined. 20. A balanced die is rolled 20 times. On average, the number of βsixesβ that you roll is equal to _____. a. b. c. d. e. 3 10/3 6 10 Cannot be determined since the number of sixes rolled is random. 21. An unbalanced die is manufactured so that there is a 20% chance of rolling a βsix.β The die is rolled twenty times. The probability of rolling at least 4 βsixesβ is equal to ______. a. b. c. d. e. .370 .411 .589 .630 .800 22. If X is normally distributed with mean 84 and variance 9, then π(π < 75) = _____. a. b. c. d. e. .0001 .0013 .1587 .3413 .9999 23. The statistics π₯Μ = 25, π 2 = 16 are computed from sample data from a population. What observation π₯ corresponds to a z-score -.5? a. b. c. d. e. 17 19 21 23 27 24. Thirty subjects are selected at random from a population of adult males. In this group, 20% of the population is at least 40% over their βdesirableβ weight. π is the number of males in this sample who are at least 40% over their βdesirableβ weight. Which of the following describes the probability distribution of the random variable π? a. b. c. d. e. πΏ is approximately normal with mean 6. πΏ is approximately normal with mean 12. πΏ is binomial with mean 6. πΏ is binomial with mean 12. None of the above statements is true. 25. Two hundred and fifty-six subjects are randomly selected from a population of adult males. The average weight for this population is 192 pounds and the standard deviation is 16 pounds. πΜ is the average weight for this sample. Which of the following statements is correct? a. The random variable normal. b. The random variable normal. c. The random variable normal. d. The random variable normal. e. The random variable normal. Μ β πππ πΏ ππ has a distribution that is approximately standard Μ β πππ πΏ π has a distribution that is approximately standard Μ β πππ πΏ π has a distribution that is approximately standard Μ β πππ πΏ π/π has a distribution that is approximately standard Μ β πππ πΏ π/ππ has a distribution that is approximately standard 26. The random variable X has mean 72 and variance 36. For a sample of size n ο½ 81 from this population, calculate the z-score of a sample mean equal to 73. a. b. c. d. e. .17 .67 1.00 1.33 1.50 27. Which statement regarding the sampling distribution of the sample mean πΜ is incorrect? a. On average, the sample mean is equal to the population mean. b. The standard deviation of the sample mean is equal to the standard deviation of the population divided by the sample size. c. The distribution of the sample mean is approximately normal if the sample size is sufficiently large. d. The t-distribution is not appropriate for approximating the distribution of the sample mean. e. The mean and variance of the sample mean can be determined from the mean and variance of the population together with the sample size. 28. For a random variable π, the standard deviation is ππ = 10. How large a random sample is required to make the standard error of the mean equal to .1? a. b. c. d. e. 10 100 1,000 10,000 100,000 29. Which of the following statements regarding the large-sample (i.e. π β₯ 30), 95% confidence interval for a population mean is incorrect? a. If the sample size is quadrupled, then the width of the interval is halved. b. The width of the interval is 1.96 times the standard error of the mean. c. The midpoint of the interval is the sample mean, a point estimator of the population mean. d. In repeated use of this procedure, there is a 95% chance that the true mean will be contained in the confidence interval e. The value of πΆ is .05. 30. A small sample t-test is conducted to test the null hypothesis π = 30 against the alternative that the population mean exceeds 30. The population is normally distributed. For a sample of size 16 the value of the test statistic is 2.125. Which of the following statements is correct? a. The null hypothesis is rejected at a 20% level of significance, but it is not rejected at a 10% level of significance. b. The null hypothesis is rejected at a 10% level of significance, but it is not rejected at a 5% level of significance. c. The null hypothesis is rejected at a 5% level of significance, but it is not rejected at a 2.5% level of significance. d. The null hypothesis is rejected at a 2.5% level of significance, but it is not rejected at a 1% level of significance. e. The null hypothesis is rejected at a 1 % level of significance, but it is not rejected at a .5% level of significance. 31. For a sample of size 16 randomly selected from a normal population, the sample standard deviation is 8. Determine the sampling error with a confidence level of 98%. a. b. c. d. e. 1.301 2.583 2.602 5.166 5.204 32. For a large sample, 2-sided, Z-test about a population mean, the p-value associated with the statistic value π = 1.54 is _________. a. b. c. d. e. .0618 .0813 .1236 .4382 .8764 33. For a large sample, 1-sided, Z-test about a population mean where the alternative hypothesis is: π»π : ππ < π0 , the p-value associated with the statistic value π = β2.26 is ______. a. b. c. d. e. -.0119 .0119 .0122 -.4881 .4881 34. From sample data for a sample of size 50, a 97% confidence interval for a population mean is determined to be equal to (54.2 , 57.6). Which of the following statements is false? a. The probability that the interval (54.2 , 57.6) contains the true population mean is .97. b. The value of the sample mean for this set of data is 55.9. c. The sampling error is 1.7. d. We are highly confident that the population mean is at most 57.6. e. The ππΆ/π value used in computing this interval is 2.170. 35. For any hypothesis test, which of the following are true? I. II. III. a. b. c. d. e. The level of significance is equal to the probability of a type I error. The level of significance is equal to the probability that the null hypothesis is accepted when in fact the alternative hypothesis is true. The level of significance is equal to the probability that the test statistic falls in the rejection region. Only statement I is true. Only statement II is true. Only statement III is true. Only statements I and II are true. Only statements I and III are true. 36. Which of the following statements concerning the probability that a random variable π falls within the interval ππ ± 1.50ππ is correct? a. The probability is equal to π·(βπ. ππ β€ π β€ π. ππ) where π is a standardnormal, random variable. b. The probability is midway between .68 and .95. c. The probability is between 4/9 and 5/9. d. The probability is midway between 5/9 and 3/4. e. The probability is at least 5/9. 37. A 95% confidence interval for the proportion of voters favoring a certain ballot measure is computed as (. 48402 , .54598) for a sample of 1,000 voters. Determine the number of voters in the sample who favored this ballot proposition. a. b. c. d. e. 510 515 520 525 530 38. A 95% confidence interval for the proportion of voters favoring a certain ballot measure is computed as (. 48904 , .55096) from a sample of 1,000 voters. Which of the following statements is correct? a. It has been proven that more than 50% of voters are in favor of this ballot measure. b. With 95% confidence, we know that at least 50% of the voters are in favor of this ballot measure. c. There is a 5% chance that fewer than 50% of the voters favor this ballot measure d. There is insufficient evidence to be 95% confident that more than 50% of the voters favor this ballot measure. e. None of the above is correct. 39. Which of the following is the rejection region for a 5% level, t- test of the following type relying on a sample of size 28 from a normal population? π»0 : π = 26 , π»π : π < 26 a. b. c. d. e. |π| < 2.052 π < 2.052 π < β2.052 |π| > 1.703 π < β1.703 40. Which of the following is the p-value for a test of the following type relying on a sample of size 28 from a normal population where the sample data results in πΜ β26 π /β28 = β2.473? π»0 : π = 26 , π»π : π < 26 a. b. c. d. e. .005 .010 .020 .025 .050 41. For a one-sided test concerning a population proportion, the p-value associated with the test statistic computed from the data is .0260. Which of the following statements is false? a. b. c. d. We would fail to reject the null hypothesis at a level of significance of 2%. For a 5% level of significance, the test statistic would be in the rejection region. For a 2-sided test, the same data would have resulted in a p-value of .0520. For a 2-sided test, the same data would have resulted in rejecting the null hypothesis at a 5% level of significance. 42. Which of the following statements regarding the 100(1 β πΌ)% confidence interval π₯Μ ± π§πΌ/2 ππ /βπ for the unknown mean of a population is false? a. The sampling error increases if the level of confidence is increased while the sample size remains the same. b. The sampling error decreases if the level of confidence remains the same while the sample size is increased. c. The midpoint of the interval is always equal to the population mean. Μ . d. The ratio ππΏ /βπ is equal to the standard deviation of the point estimator π e. If the confidence level is increased, then ππΆ/π is also increased. 43. Which of the following statements is false? a. Statistics are used as point estimators of population parameters. b. A 2-sided hypothesis test could be used to detect whether the mean weight of 60 year old males in the US population today has shifted from what it was 2 decades ago. c. Properties of the sampling distribution of a statistic are not needed in creating confidence interval formulas. d. When a large sample z-test results in the null hypothesis π = ππ. π being rejected, it has not been proven that π β ππ. π. e. A matched pairs statistical study results in a 95% confidence interval for ππ β ππ given by (βπ. π , π. π). There is no conclusive evidence that the means are not equal. 44. Which of the following statements is true? Μ β ππ , π Μ + ππ). a. For any sample, 75% of the data lies in the interval (π b. For any mounded and symmetric data set, 95% of the data lies in the interval (π Μ β ππ , π Μ + ππ). Μ β ππ , π Μ + ππ). c. For any sample, 88.9% of the data lies in the interval (π d. For any mounded and symmetric data set, 99.7% of the data lies in the interval (π Μ β ππ , π Μ + ππ). e. The conclusions of Chebychevβs rule are always valid, no matter the shape of the population being sampled. 45. Which of the following statements is false? a. For a given 2-sample data set with independent sampling, the conclusion regarding the equality of the two distributions reached in applying the Wilcoxson rank sum test is the same as the conclusion reached using the Mann-Whitney Utest. b. The advantage of a matched-pairs experimental design is to reduce variance in the test statistic in contrast with the independent sample design. c. Smaller variance in a test statistic lends greater precision to the conclusions of the test. d. Other than the difference in sample size, the validity requirements for a small sample t-test (π < 30) about a population mean are the same as those for a large sample z-test (π β₯ ππ) about a population mean. e. If two population means are truly different, then it becomes easier to detect as the sample size increases. Use the following information for questions 46-49. At the 59,600 student campus of ESU (Enormous State University) a random sample of 500 students was selected to study academic probation by gender. The results were: Probation No Probation Male 15 205 Female 20 260 46. Determine the value of the point estimator for the proportion of all students on probation. a. b. c. d. e. .03 .04 .07 .08 .35 47. Determine the value of the point estimator for the proportion of all female students on probation. a. b. c. d. e. .020 .040 .071 .077 .142 Use the following information for questions 46-49. At the 59,600 student campus of ESU (Enormous State University) a random sample of 500 students was selected to study academic probation by gender. The results were: Probation No Probation Male 15 205 Female 20 260 48. A student is selected at random from this sample. Given that the student is on probation, calculate the probability that the student is a female. a. b. c. d. e. .07 .43 .50 .57 .64 49. Determine the value of the point estimator for the proportion of all students who are male. a. b. c. d. e. .11 .22 .33 .44 .55 Use the following information for questions 50-51. For a discrete random variable π, you are given the following information: π₯ π(π₯) 0 .01 1 .04 2 .15 3 .40 7 4 .18 5 .12 6 .07 7 .03 7 β π₯ π(π₯) = 3.49 , β π₯ 2 π(π₯) = 14.11 π₯=0 π₯=0 50. Determine the standard deviation for this population. a. b. c. d. e. 1.14 1.39 1.93 3.76 14.11 51. Determine the probability that the random variable π is within one standard deviation of its mean. a. b. c. d. e. .40 .55 .58 .68 .73 ------------------------------------------------------------------------------------------------------------------------------52. The probability that the first ball drawn from an urn is red is equal to .44. Given that the first ball drawn is red, the probability that the next ball drawn is blue is equal to .25. You play a game where you win if the first ball you draw is red and the second ball that you draw is blue. Determine the probability that you will not win the game. a. b. c. d. e. .110 .190 .345 .667 .890