Download Fall 2009 Math 130 section 2

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Math 17 Section 02/ Enst 24 Section 01
—
Introduction to Statistics
Final Exam
(j
Dec. 20, 2009
Instructions:
1.
Show all work. You may receive partial credit for partially completed problems.
2.
You may use calculators and one two-sided sheet of reference notes, as well as the provided
tables. You may not use any other references or any texts.
3.
You may not discuss the exam with anyone but me.
4.
Suggestion: Read all questions before beginning and complete the ones you know best first.
Point values per problem are displayed below if that helps you allocate your time among
problems. Problems 2 and 3 share a page, as do problems 5 and 6. Problem 1 covers 2 pages.
5.
Additional info: The data sets are themed a bit snakes and lizards. A common variable in the
lizard data is snout-vent length (SVL). SVL is a standard measurement of body length for lizards.
—
In case you are wondering about it, the SVL measurement is from the tip of the nose (snout) to
the anus (vent), and excludes the tail.
6.
fl-,-—
Good luck!
-——,—..—-—-———
Problem
——..—
1
2
3
4
5
6
7
.
8
ocEar
PossibiePonts18f36iTJi7
.
——_....---_-—-.——..r--—_----•-..
9
10
Total
—
13
11
1310
-
100
-
-
1, A biologist studying lizards, specifically
Cophosaurus texanus, recorded the weight
(mass) in grams, snout-vent length (SVL) and
hind limb span (HLS) of a random sample of 25
such lizards. The biologist wants to study the
relationship between variables, looking to see
if SVL can be used to predict weight (mass)
accurately. A basic scatterplot shows the data
at right
a. Based on the scatterplot, how would you
describe the relationship between SVL and
mass?
50
63
A student working in the biologist’s lab runs a regression analysis on the data and produces the
following partial Rcmdr output:
Coefficients:
Intercept)
SVL
Si
n
3
f.
Estimate Std. Error t value Pr(>It)
-13.51551
1.22931
-10.99 1.24e-10
0.32459
0.01786
18.18 3,84e-15
codes:
0
‘‘
0.001
**‘
0.01
*
0.05
.
0.1
Pesidual standard error: 0.6986 on 23 degrees of freedom
f1uitiple R-squared: 0.9349,
Adjusted R-squared: 0.9321
F statistic: 330.5 on 1 and 23 DF,
p-value: 3.836e—15
b. What is the value of the correlation coefficient?
c. How large in magnitude, on average, were the residuals?
d. Interpret the R-squared value. Additionally, describe how well you think this model fits the data.
e. What is the equation of the least squares regression line?
______
C Obtain predictions for mass based on SVLs of 64 and 98, if appropriate. If inappropriate, explain why.
The following 2 graphs were also produced using Rcmdr.
Residuals vs Pitted
2
4
6
8
10
Fitted values
12
Normal Q-Q
14
.1
0
1
2
Theoreucal Ouanties
g. For each graph, state the assumption it is used to check, describe what it should look like if that
assumption checks out, and then decide whether or not it checks out.
h. Obtain a 99% confidence interval for the population slope. (You do not need to list assumptions.) Can
you conclude the population slope is less than 1? Explain.
2. Determine whether a hypothesis test or confidence interval from the 5 main scenarios, or some other
analysis (regression, ANOVA, chi-square) is appropriate for each research question. If you select other,
you need to specify the other procedure. Topics were chosen from most recent issue of the Journal of
Agricultural, Biological, and Environmental Statistics.
a. A study compared the toxicity on flies of four different types of selenium. The number of dead flies
was counted and the selenium type (type 1, 2, 3, or 4) was recorded for each observation. The
researchers want to know if the toxicities differ between selenium types.
Hypothesis Test
Confidence Interval
Other:
b. A study conducted in Australia measured crop yields for wheat and lupin on many different fields and
researchers want to estimate the difference in mean crop yields for the two crops.
Hypothesis Test
Confidence Interval
Other:
c. A study wanted to examine the relationship between number of Japanese beetle grubs and
percentage of organic matter in the soil for locations on a golf course in New York. The researchers want
to know if higher grub numbers are associated with lower percentages of organic matter.
Hypothesis Test
Confidence Interval
Other:
3. A study on Concho water snakes recorded the sex, age, tail length in mm (tail) and snout-vent length
in mm (svl) for a random sample of snakes. Tail length was used to determine whether each snake had a
short, medium, or long tail for the 37 female snakes.
A graduate student believes that for female snakes, 1/3 should be short, 1/2 should be medium, and 1/6
should be long in terms of tail length. Perform an appropriate analysis to determine if the graduate
students belief is valid (no need to check assumptions), filling in the parts denoted below, and expected
counts in the table above. Circle the appropriate decision (no conclusion) at a .01 significance level.
Analysis:
Null hypothesis:
Test statistic:
df:
Decision:
(C;rcle one)
p-value:
Reject the null hypothesis
Do not reject the null hypothesis
4. ‘Mysterious X” is a variable that has shown up in an
analysis run by a statistical consulting firm. Help the
firm understand the main features of this variable by
addressing the questions below.
a. What is the main feature of the histogram?
b. Two descriptive statistics lost their labels in the
table, The missing values are 9.18 and 28.21. Complete
the table with the appropriate missing values.
Mean
SD
Median
O.83
0,3
2O398.OO8
Max
‘4T0$
c. Interpret your chosen standard deviation in b.
d. Would a boxplot for “Mysterious X” show any outliers? Explain with supporting work.
e. Which descriptive statistics from the table in b. would be influenced most if a new observation vias
collected with a “Mysterious X” value of 65?
f. The consulting firm learns that these values of “Mysterious X” were collected via a SRS of 4-year
college students. Describe a more appropriate sampling strategy (give formal name and describe) if the
firm wants a second study to take class year into account.
___________
_______
__________
5. A survey of a random sample of 1000 residents in a city in the early 2000s asked “Do you recycle
regularly?” with possible answers of yes and no. A pilot study using the same question looked at a
random sample of 10 residents. Previous studies indicate that at this time and in this region. roughly
70% of residents recycle regularly, which you can assume is the true percentage.
a. What is the most appropriate distribution for X, the number of residents in the pilot study who recycle
regularly? Provide all details.
b. For the pilot study, what is the probability that exactly 9 of the 10 residents sampled say they recycle
regularly?
c. Moving to the larger study of 1000 residents, let Y denote the number of residents in the larger study
who do not recycle regularly. What is the approximate probability that Y is less than 250? Be sure to
check any conditions necessary for your approximation to be appropriate.
6. The data from the study in question 5 was analyzed further, taking age into account (2 age groups,
under 35, 35 or older). A 90% confidence interval for the difference in proportions of residents who
recycle was created (young-old), but the parts of the computation are missing. Fill in the computation
and related questions below. You may assume the assumptions to create the Cl are met.
Estimate +/- multiplier*(standard error)
=>
+/-
*
(
)
>
(
.02
,
.08
Interpret the confidence level in context.
Is there evidence to suggest that a higher proportion of younger residents recycle regularly than older
residents? Explain, and provide the significance level at which you can make your conclusion.
Researchers studying salmon in the Pacific Northwest
collected a data set with the diameter of growth rings for
first year freshwater growth (hundredths of an inch) and
type of salmon (1= Alaskan females, 2= Alaskan males, 3=
Canadian females, 4= Canadian males) along with some
other variables for a random sample of salmon. The
researchers want to know if there are differences in the
average diameter of these growth rings for the four different
types of salmon. They decide to run an ANOVA to address
that research question. Some preliminary analysis and
partial ANOVA Rcmdr output is shown below.
Df
??
??
Sum S Mean Sq F value
38591
12864
??
295
28338
a. (Circle one) This ANOVA is
PrL.•F
<2.2e
16
balanced
Type
1
2
3
Mean
96.8
104.33
13.54
S
18.34
13.49
l.)1
unbalanced.
b. In order for this ANOVA to be valid, you need 4 independent random samples which come from
normally distributed populations. Assume those assumptions are met. What is the second assumption
about the populations? Does it appear to be met? Discuss two different ways of checking it with the
output provided.
Assumption:
Method 2:
Method 2:
c. Determine the missing values for the degrees of freedom and the test
statistic.
Typedf=
Residuals df=
Fvalue=
d. What distribution was used to compute the p-value (provide all
details)?
e.
What is your best estimate of the common population variance?
f. Are multiple comparisons appropriate? Explain. If yes, summarize the
re 5 u Its.
95% famiy-wise confidence ieiei
2
1
8. More lizards! Lizard measurements of mass and snout-vent length (SVL) for 2 genera
Cnemidophorus and Sceloporus were collected in 1997 and 1999. The primary researcher wants to
know whether or not Cnemidophorus has a smaller SVL than Scelophorus, on average. Observations
were collected for a random sample of 20 Cnemidophorus and 40 Sceloporus lizards.
—
-
a. Explain in one sentence why a paired t-test is not appropriate for this data set and research question,
b. Set up appropriate hypotheses and parameter definitions to address the researcher’s question.
Null:
Alternative:
Where
c. What is the normality condition related to this inference procedure? In order to check it, what graphs
would you need? Explain if having these graphs to check the condition is more important for one genus
than the other.
Assuming the conditions checked out, the following Rcmdr output was obtained. The subtraction order
was Cneidophorus Sceloporus.
—
Welch Two Sample t-test
data:
svl by genera
=
1.5696, df = 48.187, p—value = 0.1231
alternative hypothesis: true difference in means is not equal
mean in group Cnemidophorus
mean in group Sceloporus
81.9750
759125
d. What is the p-value for your test determined in b.?
e. Interpret your p-value in context.
f. Provide an appropriate conclusion at a .1 significance level.
to 0
___________
_____—
9, Return of the water snakes! We return to take a more complete look at the Concho water snake data
to compare the males and females with regards to short or long snout-vent lengths. Short means an SVL
< 500 mm, and long is >= 500 mm. Researchers want to know if there is an association between sex and
SVL recorded as short or long. Recall that this was a random sample of these water snakes.
Sex\SVL
Female
Total
Short
25
Lri
34
I
Long
Total
I
12
37
)
0 29
32
66
a. if you randomly selected a snake from this sample, what is the probability that you would pick a male
snake?
b. If you randomly selected a snake from this sample, what is the probability it viould have a long SVL if
you knew that it was a female snake?
c. What analysis is appropriate to address the researcher’s question? Determine the appropriate
hypotheses.
Analysis:
Null:
Alternative:
d. The expected counts for this procedure are computed using the
rule.
e. List and check the conditions for your test procedure. Please fill in expected counts in the table.
f. Compute your test statistic, df, and find the p-value.
Test statistic:
d t=
p-value=
g. Decision at a .05 significance level : Reject the null hypothesis
h. What type of error might you have made?
Type 1
Do not reject the null hypothesis
Type 2
None
__________________
______
____________________
10. Sexual dimorphism is the systematic difference in form between individuals of different sex in the
same species and detecting these differences is a common goal especially when investigating new
species. A data set collected on 45 female hook-billed kites (not a new species, but an example species)
contains their wing lengths and tail lengths (measured in mm). Some preliminary analysis of the tail
lengths
is
shown.
C4
0
(‘4
0’
Co
1
170
mean
tail
193.6222
0% 25%
ad
10.98613 173 187
50% 75%
191 202
180
210
100%
n
216 45
a. Comment on what the preliminary analysis reveals to you about the distribution of tail length.
b. Male hook-billed kites have a population average tail length of 190 mm with a population standard
deviation of 10 mm. If the female hook-billed kites have the same population distribution, what would
be the distribution for the sample mean tail length for a sample of 45 female hook-billed kites? Provide
all details.
c. Assuming the females do have the same distribution as the males, how unusual are the sample results
you obtained? Provide support for your answer using a probability calculation.
d. True/False, Choose one for each statement below.
i. Standard error is the estimated standard deviation of a statistic.
ii. All statistics have a sampling distribution which is normally distributed.
ard S
APPENDIX D
Table L
Areac under the
andard.\ermai curve
F
A95
“
Second decimal place in =
).‘)8
t’
)iu
00’S
(11)4
0.12
003
uoi
U 1)ij1)
0.101 1
0.0001
————
d/—\—
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
00001
0.0001
0.0001
0.0001
o 0001
0 0001
0.0001
0.0002
0.01)02
0.0002
0.0005
0.0007
0.0010
0.0003
0.0004
0.0005
0.0007
0.0010
0.0014
0.0019
0.0026
0.0001
0.0001
1)0001
00001
0.0002
0.0001
00002
0.00)1
0 0002
0.0001
0.0001
0.0002
0 0001
0.0001
0,0002
0.0001
00001
0.0002
0.0001
0.0002
°,0002
0.0002
0.3002
0.0003
0.0004
0.0005
0.0008
0.0011
0.0003 0.0003
0.0004 0.0004
0.0006 0.0006
0.0008 0.0008
0.0011 0.0011
0.0003
0.0004
0.0006
0.0008
0.0012
0.0003 0.0003
0.0004 0.0003
0.0006 0 0006
0.0009 0.0009
0.3012 0.0013
0.0003
0.0005
0.0007
0.0009
0.0013
0.1)01)3 —3.4
3
0.0005
5.2
0.0007
31
0.0010
0.0013 —.3 U
0.0014 0.0015
0.0020 0.0021
0.0015
0.0021
0.0027 0.0028
0.0037 0.0038
0.0029
0.0039
0.0049
0.0051
0.0052
0.0016 0.0017 0.0018 0.0018
0.0023 0.0023 0.0024 0.0025
0.0031 0.0032 0.0033 0.0034
0.0041 0.0043 0.0034 0.0045
0.0055 0.0057 0.0059 00060
0.0019 —2.0
0.0026 —2.8
0.0035 —2.”
0.0047
2.e
0.0062
2.5
0.0066
0.0087
0.0113
0.0146
0.0188
0.0068 0.0069 0.0071
0.0089 0.0(391 0.0094
0.0116 0.0119 0.0122
0.0150 0.0154 0.0158
0.0192 0.0197 0.0202
0.0073
0.0075
0.0078
0.0080
0.0082
0.0096
0.0099
0.0102
0.0104
0.0107
0.0125
0.0162
0,0207
0.0129 0.0132
0.0166 0.0170
0.0212 0.0217
0.0136
0.0174
0.0222
0.0139
0.0179
0.0228
0.0233 0.0239 0.0244 0.0250 0.0256
0.0294 0.0301 0.0307 0.0314 0.0322
0.0367 0.0375 0.0384 0.0392 0.0401
0.0455 0.0465 0.0475 0.0485 0.0495
0.0262
0.0329
0.0409
0.0505
0.0559
0.0571
0.0582
0.0594
0.0606
0.0618
0.0268 0.0274
0.0336 0.0344
0.0418 0.0427
0.0516 0.0526
0.0630 0.0643
0.0281
0.0351
0.0436
0.0537
0.0655
0.0287
1.9
1.8
0.0359
0.0446 —1.7
0.0348
7 0
0.0668 —1.5
0.0681
0.0823
0.0985
0.1170
0.1379
0.0694
0.0838
0.1003
0.1190
0.1401
0.0708 0.0721
0,0853 0.0869
0.1020 0.1038
0.1210 0,1230
0.1423 0.1446
0.0735
0.0885
0.1056
0.1251
0.1469
0.0739 0.0764
0.0901 0.0918
0.1075 0.1093
0.1271 0.1292
0,1492 0.1515
0.0793
0.0951
0.1131
0.1335
0.1562
0.0808 —1.4
0.0968
7.3
1.2
0.1151
0.1357
1.1
0.1587 —1,0
0.1611
0.1867
0.2138
0.2451
0.2776
0.1635
0.1894
0.2177
0.2483
0.2810
0.1660 0.1685 0.1711 0.1736
0.1922 0.1949 0.1977 0.2005
0.2206 0.2236 02266 0.2296
0.2513 (1.2546 0.2578 0.2611
0.2843 0.2877 0.2912 0.2946
0.3121
0.3483
0.3859
0.4247
0.4641
0.3156 0.3192 0.3228
0.3520 0.3557 0,3594
0.3897 0.3936 0.3974
0.4286 0.4325 0.4364
0.468) 0.4721 0.4761
For =
—3.90, the areas are 0.0000 to four decimal places.
0.0003
0.0036
0.0048
0.0064
0.0084
0.0110
0.0143
0.0183
0.0016
0.0022
0.0030
0.0040
0.0054
0.0778
0.0934
0.1112
0.1314
0.1539
3.
o
5
-
—2.4
—2.3
—2.2
2.1
2.0
-
—
0.1762
0.1788
0.1814
0.1841
0.2033
0.2061
0.2090
0.2119
09
0.8
0.2327 0.2338
0.2643 0.2676
0.2389
0.2709
0.2420
0.2733
A
“a’
0.2981
0.3013
0.3030
0.3083
—03
0.3336
3446
03821
0.4207
0 4602
1501 3
(14
0.3264
0.3300
0.3372
0.3409
0.3632
0.4013
0.4404
0.4801
0.3669 0.3707 Q.’(745
0.4052 0.4090 0.4129
0.4443 0.4483 (1422
a48400.4880 0.4920
0 3783
(.4168
1.4562
0.391,0
1)
12
1
p
Ij
t-9G
APPEND(X 0
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0.9719
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0.959
0.960%
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((‘-1744
0,9793
1,9935
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0.41t’
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0.9’375
11.95 35
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0,9750
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0.9965
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0.9971
0.9974
0.9975
0.9977
0.9981
0.9982
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0.9977
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0.9979
0.9982
0.972
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0.9988
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9,9999
0.9992
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9 9993
0.9987
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11,9997
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2.3
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3
APPENDIX D
Tho-lail probabIlIty
One-tail probability
Table T
Valuesoft,
df
1
2
3
4
2.Ofl
1.943
1.895
1.860
IS33
2371
2447
2%5
2.306
2.2o2
1.372
1.3o3
1 156
3.350
1.345
1.812
1.796
1.782
lit’)
1.761
2.228
2.201
2.179
2.160
2.145
335
1.141
2.998
2.806
2.821
2.764
2.718
2.681
2.o50
2.624
75
10
17
18
19
1.341
1.337
1.333
1330
1.328
1.753
1746
1.740
1.734
1.729
2.131
2.120
2.770
2.101
2.093
2.602
2.583
2.5e7
2.552
2339
2.947
2.921
2.898
2.878
2.861
20
21
22
23
24
1.325
1323
1321
1.319
1318
1325
1.721
1.717
1.714
1311
2.086
2.080
2.074
2.069
2.064
2.528
2.518
2.508
2.500
2.492
2.845
2.831
2.819
2.807
2.797
25
26
27
28
29
1.316
1315
1314
1313
1311
1.708
1.706
1.703
1.701
1.699
2.060
2.056
2.052
2.048
2.045
2.485
2.479
2.473
2.467
2.4b2
2.787
2.779
2.771
2.763
2.756
30
32
35
40
45
1310
1.309
1.306
1303
1301
1.697
1.694
1.690
1.684
1.679
2.042
2.037
2.030
2.021
2.014
2.457
2.449
2.438
2.423
2.412
2.750
2.738
2.725
2.704
2.690
50
60
75
lOt)
120
1.299
1.296
1.293
1.290
1.289
1.676
1.671
1.665
1.660
1.658
2.009
2.000
1.992
1.984
1.980
2.403
2.390
2377
2.364
2.358
2.678
2.660
2.643
2.626
2.617
140
180
250
309
101)0
1.288
1.286
1.285
1.284
1282
1.656
1.653
1.651
1.649
1.646
1.977
1.973
1.969
1.966
1.962
2353
2 147
2.341
23%
2.3W)
2.611
2.603
2.596
2.588
2.551
1.282
1M5
1.960
2.326
27$
80°..
90”..
95’..
98..
I
8
9
‘.2
I
10
11
12
13
14
(
I
.
.
.
r•%•*.
-
r’
-
—-vrrnnm;::_
hisS’
nYb
4$41
3.747
4
3
2
2
41,04
4.1132
7i)7
3.4”q
3.15S
1.250
5
0
-
.4
3.169
11%
3.055
3.012
2977
1.’
.
-
32
13
74
I
•
15
I
17
lB
20
21
22
23
24
25
26
2’
28
29
I
J
I
j
3”)
32
35
40
I
I
I
I
1
I
j
I
I
‘
I.—
J.S7
497
g.t
—-—
1.4Th
1.440
1415
1.397
1.383
Confidence levels
.
o.o
0.ilfl5
11.h21
rLNh.
- . — V —
0.02
0111
12706
4.101
3.1s2
276
•
j I
.
n.114
2.Q20
2.33
2.132
r*cIab
.
005
0.i’25
ii I
3.07$
1.886
1.638
1533
,
—
*
0 10
005
-—
O
0
Onfli
0.20
0.10
-I
5
0
‘
‘
r,t..n
. -
30
oh
JiNt
12u
14”
131’
iSV
4.W,
: bf
—
A88
APPENDIX D
T 1
F in
Right4ai1 probabiHty
T’ableX
0 It
0 09
0 026
2
4.b05
6.2n1
7 779
%41
5.991
715
94H
9,024
7.975
p.348
11 133
h 63
9210
11 943
13.27
12 833
14449
16.619
1,53
19.023
15.OSo
lh.612
18,479
20.090
21.6ht,
j5S4
20.278
21.999
23,9
25 158
26 757
28 300
29.819
31.319
()
i
J
0
5
dt
I
—
5
9236
6
10645
8
Q
12.017
13.362
14 684
11070
12592
14067
15.507
16.919
lv
11
12
1.3
14
15.987
17.275
18.549
19.812
21.064
18.307
19.675
21 026
225o2
23.685
20.483
21,920
23.337
24.736
26.119
23.209
24.725
26.217
.h85
27
29.131
22.307
23.542
24.769
25.989
27.204
24.996
26.296
27.587
28.869
30.143
27.488
28.845
30.191
31.526
32.852
30.578
32.000
33.409
34.805
36.191
28.412
29.615
30.813
32.007
33.196
31.410
32.671
33.924
35.172
36.415
15
lb
17
18
19
20
21
22
23
24
37.566
38.932
30.290
49.638
32.980
39.997
31.301
32.796
44.151
45.559
43.314
45.642
46.963
48.278
59.588
46.928
48.290
49.635
50.994
52.336
50,892
63.691
76.154
88.381
100.424
53.672
66.767
79.490
91.955
104.213
112.328
124.115
R5.8]1
116.320
128.296
140.177
34.382
35.563
36.741
37.916
39.087
37.653
38.885
40.113
41.337
42.557
40.647
31.923
43.195
44.461
45.722
30
40
50
60
70
40256
51 .805
63.167
74.397
85.527
43.773
55 759
67.505
79.082
90.531
46.979
59,342
71.420
83.298
95.023
iooj
96.578
107.565
118.499
901.879
113.135
32 01
34.297
39.718
37.156
38.562
34.170
35.479
36.781
38.076
39.364
25
2h
27
28
29
80
90
10.
I2.53
14.e
106628
118.135
129563