Spring'2014'Math'243'Final'Exam''' ' ' Name_______________________________'''Instructions:'Show'ALL'work.'Simplify'wherever'reasonable.'Clearly'indicate'your'final'answer.'''''''''Problem'Number' Points'Possible' Score'

1' 25' '

2' 25' '

3' 25' '

4' 25' '

5' 25' '

6' 25' '

Total' 150' '''''

'''''''''

''''

1) Indicate'whether'each'of'the'following'statements'about'hypothesis'testing'are'true'or'false'(no'explanation'necessary):''

a. If'the'test'statistic'falls'in'the'rejection'region,'there'is'sufficient'evidence'to'reject'the'null'hypothesis.''

''''

b. If'the'test'statistic'does'not'fall'in'the'rejection'region,'there'is'sufficient'evidence'to'accept'the'null'hypothesis.''

''''

c. If'you'incorrectly'reject'the'null'hypothesis,'you'are'making'a'type'I'error.''''''

d. An'example'of'a'type'I'error'would'be'concluding'that'a'drug'increases'strength,'when'in'fact'the'drug'does'not.'

''''

e. The'probability'of'committing'a'type'I'error'is'1−α '''''''

f. In'a'2Rtailed'test,'the'area'of'each'rejection'region'is'equal'to'α .'''''''

g. The'method'that'you'choose'(classical'or'pRvalue)'will'never'change'whether'or'not'you'reject'the'null'hypothesis.'

'''''

h. I'am'99%'confident'that'I'will'pass'this'class.'''

2) In'order'to'see'if'tall'people'are'less'coordinated,'I'paid'someone'to'spend'years'watching'randomly'chosen'people'of'different'heights.'I'tallied'the'total'amount'of'times'that'they'stumbled.'That'data'is'shown'below'

'Height'(in'inches)' 60' 63' 66' 69' 72' 75' 78' 81'Stumbles' 14' 12' 13' 17' 19' 27' 46' 77''

a. Calculate'the'least'squares'regression'line,'and'interpret'the'coefficients'in'the'context'of'this'example.'Make'sure'you'use'the'right'explanatory'and'response'variables.'

Hint:'One'of'the'coefficients'won’t'make'sense'in'this'context,'but'state'its'meaning'anyways.'''''''

b. Using'the'residual'plot'shown'below'(impressed'that'I'was'able'to'get'that'picture'on'this'exam?),'comment'on'any'possible'issues'with'our'linear'regression'model.'What'suggestion'would'you'give'to'correct'the'issue?'

''''

c. Calculate'and'interpret'both' r 'and' r2 .''''''''''

d. The'75'inch'tall'person'in'the'model'was'me!'What'is'the'residual'of'that'observation?'What'does'this'mean'in'the'context'of'the'problem?'

'''''''

3) Suppose'that'the'amount'of'hours'I'sleep'at'night'is'approximately'normally'distributed'with'a'mean'of'8'hours'and'a'standard'deviation'of'1'hour.'Answer'the'following'questions'by'stating'the'shape,'center'and'spread,'then'sketching'the'distribution'in'question'and'labeling'your'answer.'''

a. What'is'the'probability'that'in'a'random'sample'of'30'nights,'on'average'I'sleep'more'than'8.5'hours?'

''''''''''''''

b. The'probability'that'in'a'random'sample'of'30'nights,'I'sleep'more'than' x0 'hours'is'10%.'What'is' x0 ?'

''''''''''''''

c. Now'suppose'that'on'60%'of'mornings'I'wake'up'refreshed.'What'is'the'probability'that'in'a'random'sample'of'50'mornings,'I'woke'up'refreshed'more'than'65%'of'the'time?'''''''

'''

4) In'a'sample'of'60'randomly'chosen'college'students,'35'reported'that'they'were'frequently'tired.'Answer'the'following'questions'by'stating'the'shape,'center'and'spread,'then'sketching'the'distribution'in'question'and'labeling'your'answer.'''

a. Create'a'99%'confidence'interval'for'the'proportion'of'all'college'students'that'are'frequently'tired.'

'''''''''''''''

b. Suppose'that'you'wanted'your'confidence'interval'from'part'A'to'have'a'margin'of'error'of'10%.'What'is'the'least'amount'of'college'students'you'could'include'in'your'sample?'

''''''''''''

c. Now'suppose'that'in'the'same'sample'of'60'college'students,'the'average'amount'of'times'per'week'that'a'student'napped'was'2.2'with'a'standard'deviation'of'0.5.'Create'a'95%'confidence'interval'for'the'number'of'times'per'week'that'all'college'students'nap.'

''''''''

5) Suppose'that'I'somehow'know'that'20%'of'all'people'take'naps.'In'a'random'sample'of'120'Americans,'18'took'a'nap.''Test'the'claim'that'Americans'nap'less'than'average,'with'90%'confidence'by'following'the'steps'below''

a. State'the'null'and'alternative'hypotheses.'''''

b. State'/'justify'the'shape,'center'and'spread'of'the'distribution.''''''''

c. Draw'the'appropriate'picture'for'classical'hypothesis'testing.''''''''''

d. Draw'the'appropriate'picture'for'pRvalue'hypothesis'testing.'''''''''

e. Use'either'of'the'pictures'above'to'draw'your'conclusion.'''''''''

f. Redraw'one'of'the'pictures'(from'part'c'or'd)'under'the'assumption'that'my'claim'was'that'“Being'American'affects'how'often'someone'naps”,'then'state'your'new'conclusion.'

'''

6) Below'is'some'data'showing'the'average'hours'slept'per'night'for'20'people.''

' 1' 2' 3' 4' 5' 6' 7' 8' 9' 10'Male' 8' 9' 6' 11' 7' 8' 10' 10' 12' 9'Female' 7' 7' 7' 8' 8' 7' 6' 9' 7' 8''

a. Suppose'that'the'data'was'collected'by'selecting'10'married'couples,'and'recording'the'sleep'for'both'the'husband'and'the'wife.'Test'the'claim'that'sleep'differs'between'the'genders,'with'98%'confidence.'

i. State'the'null'and'alternative'hypothesis.''''

ii. Sketch'the'appropriate'picture'using'either'the'classical'or'pRvalue'method.''''''''''

iii. State'your'conclusion.''''''

b. Now'suppose'that'the'data'is'comprised'of'10'randomly'selected'males,'and'10'randomly'selected'females.'Test'the'claim'that'men'sleep'more,'with'98%'confidence.'

i. State'the'null'and'alternative'hypothesis.'''''

ii. Sketch'the'appropriate'picture'using'either'the'classical'or'pRvalue'method.'''''''''''

iii. State'your'conclusion.'