ARKANSAS COUNCIL OF TEACHERS OF MATHEMATICS
ARKANSAS COUNCIL OF TEACHERS OF MATHEMATICS2017 STATISTICS REGIONAL EXAMNAME:-DirectionsThis exam includes 25 multiple-choice questions and three open-response questions that mightbe used as tie breakers. For questions 1 through 25 (the multiple-choice questions), mark youranswer choice in the appropriate location on the sheet provided. After completing questions 1through 25, answer each tie breaker question in sequential order (i.e. complete Question #1 first,then Question #2, and then Question #3 last). Be sure that your name is printed on each of the tiebreak questions. When time is called, you will be asked to turn in your multiple-choice questionanswer sheet and your written responses to the tie breaker questions.1. All of the following increase the width of a confidence interval except:a. Increased confidence levelb. Increased variabilityc. Increased sample sized. Decreased sample size2. The statement that βπ(π΄ π΅) π(π΅ π΄) whenever π΄ and π΅ are independent eventsβ is:Please select the best answer of those provided below.a. Always Trueb. Never Truec. Not Enough Information; we would need to know if π΄ and π΅ are disjoint eventsd. Not Enough Information; we would need to know if the events are equally likely3. The p-value in hypothesis testing represents which of the following:Please select the best answer of those provided below.a. The probability of failing to reject the null hypothesis, given the observed resultsb. The probability that the null hypothesis is true, given the observed resultsc. The probability that the observed results are statistically significant, given that thenull hypothesis is trued. The probability of observing results as extreme or more extreme than currentlyobserved, given that the null hypothesis is true-1- PageACTM-Statistics
ARKANSAS COUNCIL OF TEACHERS OF MATHEMATICS2017 STATISTICS REGIONAL EXAMNAME:4. Assume that the difference between the observed, paired sample values is defined in thesame manner and that the specified significance level is the same for both hypothesis tests.Using the same data, the statement that βa paired/dependent two sample t-test is equivalent toa one sample t-test on the paired differences, resulting in the same test statistic, same p-value,and same conclusionβ is:Please select the best answer of those provided below.a. Always Trueb. Never Truec. Sometimes Trued. Not Enough Informationπ(π΄)5. Note for this question that the odds in favor of an event π΄ are defined as follows: 1 π(π΄)1For fraternal twins, the odds in favor of having children that are twins are 16. Based upon thisinformation, what is the probability of a fraternal twin not having children that are twins?a.b.c.d.116151611716176. Refer to the back-to-back stemplot at the right.Which of the following are true statements regarding the datasummarized?I.II.III.IV.The distributions have the same meanThe distributions have the same rangeThe distributions have the same varianceThe distributions have the same coefficient of variationa. II onlyb. II and IIIc. I, III, and IVd. II, III, and IV-2- PageACTM-Statistics
ARKANSAS COUNCIL OF TEACHERS OF MATHEMATICS2017 STATISTICS REGIONAL EXAMNAME:For Questions 7β10, refer to the table, which relates to the possible epilepsy-depression link.Depressive DisorderDiagnosed EpilepsyPresent (Yes)Absent (No)Epilepsy3751No Epilepsy24787. What is the probability of one randomly selected individual presenting with a depressivedisorder given the individual has diagnosed epilepsy? Round to 3 decimal places.a. 0.421b. 0.420c. 0.195d. 0.6078. Assume simple random sampling for the data summarized in the table above.Let ππΈ represent the proportion of individuals with diagnosed epilepsy (βEpilepsyβ) thatpresent with a depressive disorder. Let πππΈ represent the proportion of individuals withoutdiagnosed epilepsy (βNo Epilepsyβ) that present with a depressive disorder.What is the 95% confidence interval to estimate ππΈ πππΈ , the difference between thepopulation proportions of individuals presenting with a depressive disorder among those withdiagnosed epilepsy and among those without diagnosed epilepsy? Round to 3 decimal places.a. (0.040, 0.386)b. (0.258, 0.577)c. (-0.005, 0.142)d. (0.053, 0.317)9. A researcher believes that the proportion of individuals with diagnosed epilepsy that presentwith a depressive disorder, ππΈ , is higher than the proportion of individuals without diagnosedepilepsy that present with a depressive disorder, πππΈ . Testing this claim, what would theresulting p-value be? Round to 3 decimal places.a. 0.006b. 0.069c. 0.003d. 0.035-3- PageACTM-Statistics
ARKANSAS COUNCIL OF TEACHERS OF MATHEMATICS2017 STATISTICS REGIONAL EXAMNAME:10. Refer to Question 9. Using a 0.10 significance level, which of the following is the mostappropriate conclusion given the results?a. Reject the null hypothesis; there is sufficient evidence to support the researcherβsclaim.b. Fail to reject the null hypothesis; there is sufficient evidence to support theresearcherβs claim.c. Accept the null hypothesis; there is not sufficient evidence to support the researcherβsclaim.d. Accept the null hypothesis; there is sufficient evidence to support the researcherβsclaim.11. A sociologist focusing on popular culture and media believes that the average number ofhours per week (hrs/week) spent using social media is greater for women than for men.Examining two independent simple random samples of 100 individuals each, the researchercalculates sample standard deviations of 2.3 hrs/week and 2.5 hrs/week for women and menrespectively. If the average number of hrs/week spent using social media for the sample ofwomen is 1 hour greater than that for the sample of men, what conclusion can be made froma hypothesis test where:{π»0 : ππ ππ 0π»1 : ππ ππ 0a. The observed difference in average number of hrs/week spent using social media isnot significantb. The observed difference in average number of hrs/week spent using social media issignificantc. A conclusion is not possible without knowing the average number of hrs/week spentusing social media in each sampled. A conclusion is not possible without knowing the population sizes-4- PageACTM-Statistics
ARKANSAS COUNCIL OF TEACHERS OF MATHEMATICS2017 STATISTICS REGIONAL EXAMNAME:12. A 99% t-based confidence interval for the mean price for a gallon of gasoline (dollars) iscalculated using a simple random sample of gallon gasoline prices for 50 gas stations. Giventhat the 99% confidence interval is 3.32 π 3.98, what is the sample mean price for agallon of gasoline (dollars)?Please select the best answer of those provided below.a. 0.33b. 3.65c. Not Enough Information; we would need to know the variation in the sample ofgallon gasoline pricesd. Not Enough Information; we would need to know the variation in the population ofgallon gasoline prices13. A quiz consists of 9 True/False questions. Assume that the questions are independent. Inaddition, assume that (T) and (F) are equally likely outcomes when guessing on any one ofthe questions. What is the probability of guessing on each of the 9 quiz questions and gettingmore than one of the True/False questions wrong? Round to 3 decimal places.a. 0.998b. 0.018c. 0.020d. 0.98014. Five students take AP Calculus AB one year and AP Calculus BC the next year. Their overallcourse grades (%) are listed below for both courses. Which of the following statisticalprocedures would be most appropriate to test the claim that student overall course grades arethe same in both courses? Assume that any necessary normality requirements hold.StudentAP Cal ABAP Cal 9%74.8%a. Two-tailed two-sample paired/dependent t-test of meansb. Two-tailed two-sample independent t-test of meansc. Two-tailed two-sample independent z-test of meansd. One-tailed two-sample z-test of proportions-5- PageACTM-Statistics
ARKANSAS COUNCIL OF TEACHERS OF MATHEMATICS2017 STATISTICS REGIONAL EXAMNAME:15. Referring to the setting and data provided in Question 14 above, what is the test statistic fortesting the claim that student overall course grades are the same in both courses? Round to 3decimal places.a. -0.516b. -0.157c. 4.306d. Not Enough Information; we would need to know the variation in the population16. The histogram to the right represents the hospitallength of stay (in days) for patients at a nearby medicalfacility. How many patients are included in thehistogram?a. 5b. 21c. 17d. 917. Using the histogram to the right that represents thehospital lengths of stay (in days) for patients at anearby medical facility, determine the relationshipbetween the mean and the median.a. Mean Medianb. Mean Medianc. Mean Mediand. Mean Median18. Refer to the discrete probability distribution provided in the table below.X xP(X x)00.04010.11020.45030.2304?Find the probability that x is equal to 0 or 4. Round to 3 decimal places.a. 0.040b. 0.210c. 0.007d. 1.000-6- PageACTM-Statistics
ARKANSAS COUNCIL OF TEACHERS OF MATHEMATICS2017 STATISTICS REGIONAL EXAMNAME:19. Green sea turtles have normally distributed weights, measured in kilograms, with a mean of134.5 and a variance of 49.0. A particular green sea turtleβs weight has a z-score of -2.4.What is the weight of this green sea turtle? Round to the nearest whole number.a. 17 kgb. 151 kgc. 118 kgd. 252 kg20. What percentage of measurements in a dataset fall above the median?a. 49%b. 50%c. 51%d. Cannot Be Determined21. Which of the following exam scores is better relative to other students enrolled in the course? A psychology exam grade of 85; the mean grade for the psychology exam is 92 with astandard deviation of 3.5 An economics exam grade of 67; the mean grade for the economics exam is 79 with astandard deviation of 8 A chemistry exam grade of 62; the mean grade for the chemistry exam is 62 with astandard deviation of 5a. The psychology exam score is relatively betterb. The economics exam score is relatively betterc. The chemistry exam score is relatively betterd. All of the exam scores are relatively equivalent22. The statement βIf there is sufficient evidence to reject a null hypothesis at the 10%significance level, then there is sufficient evidence to reject it at the 5% significance levelβ is:Please select the best answer of those provided below.a. Always Trueb. Never Truec. Sometimes True; the p-value for the statistical test needs to be provided for a conclusiond. Not Enough Information; this would depend on the type of statistical test used-7- PageACTM-Statistics
ARKANSAS COUNCIL OF TEACHERS OF MATHEMATICS2017 STATISTICS REGIONAL EXAMNAME:23. Assuming weights of female athletes are normally distributed with a mean of 140 lbs and astandard deviation of 15 lbs, what is the probability that a randomly selected female athleteweighs more than 170 lbs? Round to 3 decimal places.Also, is the probability above the same as the probability that a randomly selected sample ofsize π (where π 1) has a mean weight more than 170 lbs?a. 0.023; yes, these two probabilities would be the sameb. 0.023; no, these two probabilities would not be the samec. 0.977; yes, these two probabilities would be the samed. 0.977; no these two probabilities would not be the sameFor Questions 24β25, refer to the relevant results from a regression analysis provided below.A simple random sample of 5k race times for 32 competitive male runners aged 15-24 yearsold resulted in a mean 5k race time of 16.79 minutes. The simple linear regression equationthat fit the sample data was obtained and found to be π¦Μ 21.506 0.276π₯ where π₯represents the age of the runner in years and π¦ represents the 5k race time for a competitivemale runner in minutes. When testing the claim that there is a linear correlation between ageand 5k race times of competitive male runners, an observed test statistic of (π‘ 7.87)resulted in an approximate p-value of 0.0001.24. The proportion of variation in 5k race times that can be explained by the variation in the ageof competitive male runners was approximately 0.663. What is the value of the sample linearcorrelation coefficient? Round to 3 decimal places.a. 0.663b. 0.814c. -0.814d. 0.44025. Using all of the results provided, is it reasonable to predict the 5k race time (minutes) of acompetitive male runner 73 years of age?a. Yes; linear correlation between age and 5k race times is statistically significantb. Yes; both the sample linear regression equation and an age in years is providedc. No; linear correlation between age and 5k race times is not statistically significantd. No; the age provided is beyond the scope of our available sample data-8- PageACTM-Statistics
ARKANSAS COUNCIL OF TEACHERS OF MATHEMATICS2017 STATISTICS REGIONAL EXAMNAME:-Tie Breaker Question 1Assume the following probabilities for two events, π΄ and π΅:π(π΄) 0.50, π(π΅) 0.70, π(π΄ π΅) 0.85Are the events, π΄ and π΅, independent in this situation?You must provide reasoning for your answer.-9- PageACTM-Statistics
ARKANSAS COUNCIL OF TEACHERS OF MATHEMATICS2017 STATISTICS REGIONAL EXAMNAME:-Tie Breaker Question 2The table below relates to a study where infants listened to three types of βmusicβ in utero andtheir advancement to crawling/walking was then observed and categorized as either early, ontime, or late. The goal of the study was to determine if there was a statistically significantassociation between music in utero and time of advancement to crawling/walking in infants.Advancement to Crawling/WalkingRow Sample SizesMusic In UteroEarlyOn TimeLateNMozart (Piano Sonata)50.8%30.2%19.0%63Philip Glass (minimalist music)40.0%38.3%21.7%60White Noise and Silence17.9%21.1%61.0%44The table above provides row percentages and sample sizes. For example, 50.8% of βMozartβinfants have βEarlyβ advancement to crawling/walking and there are 63 total βMozartβ infants.a. Based upon the row percentages provided, fill in the contingency table below with theapproximate cell counts, rounding to the nearest whole number.Advancement to Crawling/WalkingMusic In UteroEarlyOn TimeLateMozart (Piano Sonata)Philip Glass (minimalist music)White Noise and Silenceb. Do the data indicate an association between music in utero and time of advancement tocrawling/walking in infants? Conduct an appropriate hypothesis test to answer thisquestion using a 0.05 significance level. Provide the hypotheses, test statistic(s), pvalue(s), and a formal conclusion.- 10 - P a g eACTM-Statistics
ARKANSAS COUNCIL OF TEACHERS OF MATHEMATICS2017 STATISTICS REGIONAL EXAMNAME:-Tie Breaker Question 3For π, π, π β such that 0 π π π, is the following combinatorial statement true?π ππ π π( )( ) ( )()π ππ π πYou must provide reasoning for your answer.- 11 - P a g eACTM-Statistics
ACTM STATISTICSMULTIPLE CHOICE & TIE BREAKER KEY2017 REGIONAL EXAMMultiple Choice Key1. c2. d3. d4. a5. d6. b7. b8. d9. c10. a11. b12. b13. d14. a15. a16. b17. d18. b19. c20. d21. c22. c23. b24. c25. d-1- PageACTM-Statistics
ACTM STATISTICSMULTIPLE CHOICE & TIE BREAKER KEY2017 REGIONAL EXAM-Tie Breaker Question 1Assume the following probabilities for two events, π΄ and π΅:π(π΄) 0.50, π(π΅) 0.70, π(π΄ π΅) 0.85Are the events, π΄ and π΅, independent in this situation?You must provide reasoning for your answer.Solution:Yes. π΄ and π΅ are independent.Reasoning (One Possible Answer):Notice that π(π΄ π΅) π(π΄) π(π΅) π(π΄ π΅)So, 0.85 0.50 0.70 π(π΄ π΅)Solving for π(π΄ π΅), we find that π(π΄ π΅) 0.35If π΄ and π΅ are independent, π(π΄ π΅) π(π΄) π(π΅)π(π΄ π΅) 0.35 0.50 0.70 π(π΄) π(π΅)Thus, π΄ and π΅ are independentRubric:0 pts β Answer that π΄ and π΅ are NOT independent1 pts β Answer that π΄ and π΅ are independent but with no reasoning or improper reasoning2 pts β Answer that π΄ and π΅ are NOT independent but with an arithmetic error in proper reasoning3 pts β Answer that π΄ and π΅ are independent with proper reasoning*Note that βproper reasoningβ implies that the response includes a valid probabilistic definition ofindependent events such as π(π΄ π΅) π(π΄) π(π΅) or π(π΄ π΅) π(π΄) or π(π΅ π΄) π(π΅) etc.-2- PageACTM-Statistics
ACTM STATISTICSMULTIPLE CHOICE & TIE BREAKER KEY2017 REGIONAL EXAM-Tie Breaker Question 2Advancement to Crawling/WalkingRow Sample SizesMusic In UteroEarlyOn TimeLateNMozart (Piano Sonata)50.8%30.2%19.0%63Philip Glass (minimalist music)40.0%38.3%21.7%60White Noise and Silence17.9%21.1%61.0%44Solution Provided in Red:a. Based upon the row percentages provided, fill in the contingency table below with theapproximate cell counts, rounding to the nearest whole number.Advancement to Crawling/WalkingMusic In UteroEarlyOn TimeLateMozart (Piano Sonata)321912Philip Glass (minimalist music)242313White Noise and Silence8927b. Do the data indicate an association between music in utero and time of advancement tocrawling/walking in infants? Conduct an appropriate hypothesis test to answer thisquestion using a 0.05 significance level. Provide the hypotheses, test statistic(s), pvalue(s), and a formal conclusion. HypothesesH0 : Music in Utero and Advancement are Independent (there is no association){H1 : Music in Utero and Advancement are Dependent (there is association) Test Statisticπ 2 27.516, ππ 4 P-Valueπ π£πππ’π 0.0001 0 Formal Conclusion at πΆ π. ππ (in terms of π―π )Reject the null hypothesis at the 5% significance level. There is sufficient evidence tosupport the claim that Music in Utero and advancement to crawling/walking areassociated/dependent.Rubric: 0 pts to 5 pts Possible1 point for each completely correct answer and 0 points for each incorrect answer of thefollowing tie breaker components: (1) approximate cell counts, (2) hypotheses, (3) test statistic,(4) p-value, and (5) formal conclusion-3- PageACTM-Statistics
ACTM STATISTICSMULTIPLE CHOICE & TIE BREAKER KEY2017 REGIONAL EXAM-Tie Breaker Question 3For π, π, π β such that 0 π π π, is the following combinatorial statement true?π ππ π π( )( ) ( )()π ππ π πYou must provide reasoning for your answer.Solution:Yes. The statement is true.Reasoning (One Possible Arithmetic Based Proof):Starting with the left hand side (LHS), we will work to get the right hand side (RHS)π ππ!π!( )( ) π ππ! (π π)! π! (π π)!Notice that an π! can be canceled π!π! (π π)! (π π)! (π π)!π! π! (π π)! (π π)! (π π)!Multiply both numerator and denominator by (π π)! π!(π π)! π! (π π)! (π π)! (π π)!Notice that this is the RHSπ π π ( )()π π πThus, the statement is true.Rubric:0 pts β Answer the statement is NOT true with no reasoning1 pts β Answer the statement is true with no reasoning2 pts β Answer the statement is NOT true but with error in an attempt at proper reasoning3 pts β Answer the statement is true with proper reasoning-4- PageACTM-Statistics
AP Cal BC 85.5% 71.0% 93.2% 93.0% 74.8% . a. Two-tailed two-sample paired/dependent t-test of means . b.oTw -tailed two-sample independent t-test of means c. Two-tailed two-sample independent z-test of means . standard deviation of 3.5 An economics exam grade of 67; the mean
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