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MATH 1342 Name: Spring 2020 Instructor: Ritter Exam 3 4/8/2020 1. (19 points) In baseball, a batting average is obtained by dividing hits by at-bats. Is there evidence that team batting average is positively correlated with the number of team wins? The hypotheses are H0 : ρ = 0 Ha : ρ > 0 My sample data came from the 30 Major League Baseball teams in 2016, I found that r = 0.340. Assume that the α = 0.05 significance level will be used. (a) (4 points) Where would a randomization distribution be centered? (b) (4 points) Where would a bootstrap distribution (for the confidence interval) for the correlation be centered? (c) (5 points) The p-value for the hypothesis test is 0.032. Interpret the p-value. (d) (6 points) State the conclusion of the test in context using the α = 0.05 significance level. MATH 1342 Exam 3 - Page 2 of 11 4/8/2020 2. (13 points) A study was conducted to determine whether just one session of cognitive behavioral therapy can help people with insomnia. In the study, forty people who have been diagnosed with insomnia were randomly divided into two groups of 20 each. People in one group received a one-hour cognitive behavioral therapy session while those in the other group received no treatment. Each subject was asked if their sleep improved. The results follow in the table below. Improvement No Improvement Total CBT 14 6 20 No Treatment 3 17 20 Total 17 23 40 Was the proportion who improved in the therapy group greater than the proportion in the no treatment group with statistical significance? (a) (5 points) What are the null and alternative hypotheses? (b) (4 points) Where would the randomization distribution for this test be centered? (c) (4 points) At what value would the bootstrap distribution be centered? MATH 1342 Exam 3 - Page 3 of 11 4/8/2020 3. (15 points) A statistics professor was interested in whether female patrons eat less hot wings on average than male patrons at a bar and grill he frequents. Thus, we consider the hypotheses H0 : µf = µm vs. Ha : µf < µm. he took random samples of 15 men and 15 women at the bar. the difference in sample means for the number of hot wings eaten is xf − xm = 9.33 − 14.53 = −5.2. a dotplot for an appropriate randomization distribution follows. it was created from 1000 randomized samples from the original data. (a) (5 points) based on the graph, what is the p-value for our one-tailed test? show how you got your answer. (b) (6 points) provide a conclusion in context in terms of strength of evidence. (c) (4 points) consider the test with a two-tailed alternative, h0 : µf = µm vs ha : µf 6= µm. what is the p-value for this test? math 1342 exam 3 - page 4 of 11 4/8/2020 4. (11 points) some archaeologists theorize that ancient egyptians interbred with several different immigrant populations over thousands of years. to see if there is any indication of a difference in body structure that might have resulted, they measured a random sam- ple of 30 skulls dated from 4000 b.c.e. and 30 others from 200 b.c.e. the measurements made were the skull widths in cm. (a) (5 points) what are the appropriate null and alternative hypotheses? (b) (6 points) the researchers report: “the test of difference in means results in a p- value of 0.00067.”do you think this provides sufficient evidence of change in skull width from on time period to the other or do you think our results were due to chance? explain. (the explanation can be done in one or two sentences.) math 1342 exam 3 - page 5 of 11 4/8/2020 5. (12 points) say researchers are interested in scores on the math portion of the sat for a particular school. for each graph of side-by-side dotplots, indicate which of the two samples shows stronger evidence that the mean is less than 500. if they show about the same strength of evidence, indicate that. math 1342 exam 3 - page 6 of 11 4/8/2020 6. (11 points) in a study published in the american journal of psychiatry, researchers wanted to measure the effect of alcohol on the region of the brain called the hippocampus. the researchers randomly selected 12 adolescents with alcohol use disorders to determine whether the hippocampal volumes in alcoholic adolescents were significantly different from the normal volume of 9.02 cubic centimeters (cm3). an analysis of the sample data revealed an x = 8.10 cm3. (a) (5 points) state the null and alternative hypotheses for the significance test. (b) (6 points) the researchers computed a 95% confidence interval: (7.472, 8.728). what is the conclusion in context of the hypothesis test at the α = 0.05 level? explain how you arrived at your answer. (it can be answered in one or two sen- tences.) math 1342 exam 3 - page 7 of 11 4/8/2020 7. (16 points) indicate whether each question is best assessed by using a confidence interval or a hypothesis test. for a hypothesis test, write hypotheses and use correct parameter notation. for a confidence interval, write the correct parameter notation for what we are trying to estimate. you do not need to define the parameter(s). (a) (4 points) what percent of us adults approve of president trump’s job perfor- mance? (b) (4 points) is the proportion of left-handers in the us different from the proportion of left-handers in japan? (c) (4 points) what is the average number of hours per week an elementary statistics student studies for the course? (d) (4 points) is there a difference between the mean number of hours per week that a full-time female college student spends studying and the mean number of hours that a full-time male college student spends studying? math 1342 exam 3 - page 8 of 11 4/8/2020 8. (24 points) to study whether st. john’s wort, a herbal drug, is effective in treating depression, 25 of 50 subjects with depression were randomly assigned to take a fixed daily dose of st. john’s wort, and the remaining 25 took the same fixed daily dose of a placebo. level of depression was measured before the treatment and 3 months after the treatment, and the change in depression score was obtained for each subject. at the conclusion of the study, researchers looked to see if there was a significant difference between the mean score changes for the two groups. in parts (a) to (c) below, make a conclusion in context for the test using a significance level of α = 0.05. also give a conclusion in context. (a) (4 points) the p-value was 0.768. (b) (4 points) the p-value was 0.005. (c) (4 points) the p-value was 0.035. (d) (4 points) which of the p-values in parts (a) through (c) provides the strongest evidence for the alternative hypothesis? (e) (4 points) which of the p-values in parts (a) through (c) provides the weakest evidence for the alternative hypothesis? (f) (4 points) how do your formal conclusions change in parts (a) through (c) change if the researchers make their conclusions using a significance level of α = 0.01? math 1342 exam 3 - page 9 of 11 4/8/2020 9. (15 points) recall the data set restaurant tips which we looked at in class. the two variables of interest in this data set are amount of the bill and the tip percent. we wish to test whether there significant evidence of a positive correlation between amount of the bill and the tip percent. a randomization distribution based on the data is shown below. (a) (6 points) what are the null and alternative hypotheses? (b) (3 points) what is the correct notation for the sample statistic? (c) (6 points) which of the following sample statistics would give statistically signifi- cant evidence at the α = 0.05 level for the alternative hypothesis? (-0.5, -0.20, 0, 0.05, 0.25, 0.5) math 1342 exam 3 - page 10 of 11 4/8/2020 10. (8 points) about 13% of the population is left-handed. therefore, if you select a person at random from the population, the probability that they will be left-handed is 0.13. consider the process of selecting two people from the population at random. (a) (4 points) what is the probability that both are left-handed? (b) (4 points) what is the probability that exactly one is left-handed? 11. (8 points) calculate the following: (a) (4 points) ( 10 5 ) (b) (4 points) ( 7 3 ) math 1342 exam 3 - page 11 of 11 4/8/2020 12. (8 points) a standard deck of 52 cards contains 4 suits: hearts, diamonds, clubs and spades. there are 13 cards in each suit. all 52 cards are pictured below: (a) (4 points) what is the probability of not drawing a heart? (b) (4 points) what is the probability of drawing a heart or a diamond? µm.="" he="" took="" random="" samples="" of="" 15="" men="" and="" 15="" women="" at="" the="" bar.="" the="" difference="" in="" sample="" means="" for="" the="" number="" of="" hot="" wings="" eaten="" is="" xf="" −="" xm="9.33" −="" 14.53="−5.2." a="" dotplot="" for="" an="" appropriate="" randomization="" distribution="" follows.="" it="" was="" created="" from="" 1000="" randomized="" samples="" from="" the="" original="" data.="" (a)="" (5="" points)="" based="" on="" the="" graph,="" what="" is="" the="" p-value="" for="" our="" one-tailed="" test?="" show="" how="" you="" got="" your="" answer.="" (b)="" (6="" points)="" provide="" a="" conclusion="" in="" context="" in="" terms="" of="" strength="" of="" evidence.="" (c)="" (4="" points)="" consider="" the="" test="" with="" a="" two-tailed="" alternative,="" h0="" :="" µf="µm" vs="" ha="" :="" µf="" 6="µm." what="" is="" the="" p-value="" for="" this="" test?="" math="" 1342="" exam="" 3="" -="" page="" 4="" of="" 11="" 4/8/2020="" 4.="" (11="" points)="" some="" archaeologists="" theorize="" that="" ancient="" egyptians="" interbred="" with="" several="" different="" immigrant="" populations="" over="" thousands="" of="" years.="" to="" see="" if="" there="" is="" any="" indication="" of="" a="" difference="" in="" body="" structure="" that="" might="" have="" resulted,="" they="" measured="" a="" random="" sam-="" ple="" of="" 30="" skulls="" dated="" from="" 4000="" b.c.e.="" and="" 30="" others="" from="" 200="" b.c.e.="" the="" measurements="" made="" were="" the="" skull="" widths="" in="" cm.="" (a)="" (5="" points)="" what="" are="" the="" appropriate="" null="" and="" alternative="" hypotheses?="" (b)="" (6="" points)="" the="" researchers="" report:="" “the="" test="" of="" difference="" in="" means="" results="" in="" a="" p-="" value="" of="" 0.00067.”do="" you="" think="" this="" provides="" sufficient="" evidence="" of="" change="" in="" skull="" width="" from="" on="" time="" period="" to="" the="" other="" or="" do="" you="" think="" our="" results="" were="" due="" to="" chance?="" explain.="" (the="" explanation="" can="" be="" done="" in="" one="" or="" two="" sentences.)="" math="" 1342="" exam="" 3="" -="" page="" 5="" of="" 11="" 4/8/2020="" 5.="" (12="" points)="" say="" researchers="" are="" interested="" in="" scores="" on="" the="" math="" portion="" of="" the="" sat="" for="" a="" particular="" school.="" for="" each="" graph="" of="" side-by-side="" dotplots,="" indicate="" which="" of="" the="" two="" samples="" shows="" stronger="" evidence="" that="" the="" mean="" is="" less="" than="" 500.="" if="" they="" show="" about="" the="" same="" strength="" of="" evidence,="" indicate="" that.="" math="" 1342="" exam="" 3="" -="" page="" 6="" of="" 11="" 4/8/2020="" 6.="" (11="" points)="" in="" a="" study="" published="" in="" the="" american="" journal="" of="" psychiatry,="" researchers="" wanted="" to="" measure="" the="" effect="" of="" alcohol="" on="" the="" region="" of="" the="" brain="" called="" the="" hippocampus.="" the="" researchers="" randomly="" selected="" 12="" adolescents="" with="" alcohol="" use="" disorders="" to="" determine="" whether="" the="" hippocampal="" volumes="" in="" alcoholic="" adolescents="" were="" significantly="" different="" from="" the="" normal="" volume="" of="" 9.02="" cubic="" centimeters="" (cm3).="" an="" analysis="" of="" the="" sample="" data="" revealed="" an="" x="8.10" cm3.="" (a)="" (5="" points)="" state="" the="" null="" and="" alternative="" hypotheses="" for="" the="" significance="" test.="" (b)="" (6="" points)="" the="" researchers="" computed="" a="" 95%="" confidence="" interval:="" (7.472,="" 8.728).="" what="" is="" the="" conclusion="" in="" context="" of="" the="" hypothesis="" test="" at="" the="" α="0.05" level?="" explain="" how="" you="" arrived="" at="" your="" answer.="" (it="" can="" be="" answered="" in="" one="" or="" two="" sen-="" tences.)="" math="" 1342="" exam="" 3="" -="" page="" 7="" of="" 11="" 4/8/2020="" 7.="" (16="" points)="" indicate="" whether="" each="" question="" is="" best="" assessed="" by="" using="" a="" confidence="" interval="" or="" a="" hypothesis="" test.="" for="" a="" hypothesis="" test,="" write="" hypotheses="" and="" use="" correct="" parameter="" notation.="" for="" a="" confidence="" interval,="" write="" the="" correct="" parameter="" notation="" for="" what="" we="" are="" trying="" to="" estimate.="" you="" do="" not="" need="" to="" define="" the="" parameter(s).="" (a)="" (4="" points)="" what="" percent="" of="" us="" adults="" approve="" of="" president="" trump’s="" job="" perfor-="" mance?="" (b)="" (4="" points)="" is="" the="" proportion="" of="" left-handers="" in="" the="" us="" different="" from="" the="" proportion="" of="" left-handers="" in="" japan?="" (c)="" (4="" points)="" what="" is="" the="" average="" number="" of="" hours="" per="" week="" an="" elementary="" statistics="" student="" studies="" for="" the="" course?="" (d)="" (4="" points)="" is="" there="" a="" difference="" between="" the="" mean="" number="" of="" hours="" per="" week="" that="" a="" full-time="" female="" college="" student="" spends="" studying="" and="" the="" mean="" number="" of="" hours="" that="" a="" full-time="" male="" college="" student="" spends="" studying?="" math="" 1342="" exam="" 3="" -="" page="" 8="" of="" 11="" 4/8/2020="" 8.="" (24="" points)="" to="" study="" whether="" st.="" john’s="" wort,="" a="" herbal="" drug,="" is="" effective="" in="" treating="" depression,="" 25="" of="" 50="" subjects="" with="" depression="" were="" randomly="" assigned="" to="" take="" a="" fixed="" daily="" dose="" of="" st.="" john’s="" wort,="" and="" the="" remaining="" 25="" took="" the="" same="" fixed="" daily="" dose="" of="" a="" placebo.="" level="" of="" depression="" was="" measured="" before="" the="" treatment="" and="" 3="" months="" after="" the="" treatment,="" and="" the="" change="" in="" depression="" score="" was="" obtained="" for="" each="" subject.="" at="" the="" conclusion="" of="" the="" study,="" researchers="" looked="" to="" see="" if="" there="" was="" a="" significant="" difference="" between="" the="" mean="" score="" changes="" for="" the="" two="" groups.="" in="" parts="" (a)="" to="" (c)="" below,="" make="" a="" conclusion="" in="" context="" for="" the="" test="" using="" a="" significance="" level="" of="" α="0.05." also="" give="" a="" conclusion="" in="" context.="" (a)="" (4="" points)="" the="" p-value="" was="" 0.768.="" (b)="" (4="" points)="" the="" p-value="" was="" 0.005.="" (c)="" (4="" points)="" the="" p-value="" was="" 0.035.="" (d)="" (4="" points)="" which="" of="" the="" p-values="" in="" parts="" (a)="" through="" (c)="" provides="" the="" strongest="" evidence="" for="" the="" alternative="" hypothesis?="" (e)="" (4="" points)="" which="" of="" the="" p-values="" in="" parts="" (a)="" through="" (c)="" provides="" the="" weakest="" evidence="" for="" the="" alternative="" hypothesis?="" (f)="" (4="" points)="" how="" do="" your="" formal="" conclusions="" change="" in="" parts="" (a)="" through="" (c)="" change="" if="" the="" researchers="" make="" their="" conclusions="" using="" a="" significance="" level="" of="" α="0.01?" math="" 1342="" exam="" 3="" -="" page="" 9="" of="" 11="" 4/8/2020="" 9.="" (15="" points)="" recall="" the="" data="" set="" restaurant="" tips="" which="" we="" looked="" at="" in="" class.="" the="" two="" variables="" of="" interest="" in="" this="" data="" set="" are="" amount="" of="" the="" bill="" and="" the="" tip="" percent.="" we="" wish="" to="" test="" whether="" there="" significant="" evidence="" of="" a="" positive="" correlation="" between="" amount="" of="" the="" bill="" and="" the="" tip="" percent.="" a="" randomization="" distribution="" based="" on="" the="" data="" is="" shown="" below.="" (a)="" (6="" points)="" what="" are="" the="" null="" and="" alternative="" hypotheses?="" (b)="" (3="" points)="" what="" is="" the="" correct="" notation="" for="" the="" sample="" statistic?="" (c)="" (6="" points)="" which="" of="" the="" following="" sample="" statistics="" would="" give="" statistically="" signifi-="" cant="" evidence="" at="" the="" α="0.05" level="" for="" the="" alternative="" hypothesis?="" (-0.5,="" -0.20,="" 0,="" 0.05,="" 0.25,="" 0.5)="" math="" 1342="" exam="" 3="" -="" page="" 10="" of="" 11="" 4/8/2020="" 10.="" (8="" points)="" about="" 13%="" of="" the="" population="" is="" left-handed.="" therefore,="" if="" you="" select="" a="" person="" at="" random="" from="" the="" population,="" the="" probability="" that="" they="" will="" be="" left-handed="" is="" 0.13.="" consider="" the="" process="" of="" selecting="" two="" people="" from="" the="" population="" at="" random.="" (a)="" (4="" points)="" what="" is="" the="" probability="" that="" both="" are="" left-handed?="" (b)="" (4="" points)="" what="" is="" the="" probability="" that="" exactly="" one="" is="" left-handed?="" 11.="" (8="" points)="" calculate="" the="" following:="" (a)="" (4="" points)="" (="" 10="" 5="" )="" (b)="" (4="" points)="" (="" 7="" 3="" )="" math="" 1342="" exam="" 3="" -="" page="" 11="" of="" 11="" 4/8/2020="" 12.="" (8="" points)="" a="" standard="" deck="" of="" 52="" cards="" contains="" 4="" suits:="" hearts,="" diamonds,="" clubs="" and="" spades.="" there="" are="" 13="" cards="" in="" each="" suit.="" all="" 52="" cards="" are="" pictured="" below:="" (a)="" (4="" points)="" what="" is="" the="" probability="" of="" not="" drawing="" a="" heart?="" (b)="" (4="" points)="" what="" is="" the="" probability="" of="" drawing="" a="" heart="" or="" a="">