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Judgement of 22 pages for the course Higher Certificate in mathematics and Statistics at Higher Certificate in mathematics and Statistics
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Practice Exam Questions; Statistics 301; Professor WardropChapters 1, 12, 2, and 3
1. Measurements are collected from 100 subjects from each of two sources. The data
yield the following frequency histograms. The number above each rectangle is its height.
Source 1
9 10 10 9 8
7 8 7
5 5 4
2 3 4 3 2
1 1 1 1
5 6 7 8 9 10 11 12 13 14 15 16 17 18
21
Source 2 16
13
8 8
5 6 5
4
1 1
5 6 7 8 9 10 11 12 13 14 15 16 17 18
Each sample has the same mean, 10.00. In order to answer (b) and (c) below, refer to the empirical
rule for interpreting s, taking into account the shape of the histogram. Do not try to calculate s
because you do not have enough information to do so. In addition, you will receive no credit for
simply identifying the correct s; you must provide an explanation.
(a) What is the most precise correct state- 2. The mean and median of Al’s n = 3 observa-
ment that you can make about the numer- tions both equal 10. The mean and median of
ical value of the median of the data from Bev’s n = 5 observations both equal 18.
source 2? Do not explain your answer.
Hint: Here is a correct statement: The me- (a) Carol combines Al’s and Bev’s data into
dian is between 0 and 20. This statement one collection of n = 8 observations.
is not precise enough to receive any credit. Can the mean of Carol’s data be calculated
from the information given? If you think
(b) Among the possibilities 1.50, 2.00 and
not, just say that. If you think it can, then
2.50, which is the numerical value of s for
calculate Carol’s mean.
the data from source 1? Explain your an-
swer. (b) Refer to part (a). Demonstrate, by an ex-
plicit example, that there is not enough
(c) Among the possibilities 1.00, 1.50 and
information to determine Carol’s median.
2.00, which is the numerical value of s for
Hint: Find two sets of data sets that sat-
the data from source 2? Explain your an-
isfy Al’s and Bev’s conditions, yet, when
swer.
combined, give different medians.
1
,3. A sample of size 40 yields the following sorted x P (X = x) P (X ≤ x) P (X ≥ x)
data. Note that I have x-ed out x(39) (the sec- −0.6667 0.0001 0.0001 1.0000
ond largest number). This fact will NOT pre- −0.5278 0.0024 0.0025 0.9999
vent you from answering the questions below. −0.3889 0.0242 0.0267 0.9975
−0.2500 0.1104 0.1371 0.9733
−0.1111 0.2588 0.3959 0.8629
14.1 46.0 49.3 53.0 54.2 54.7 54.7 0.0278 0.3220 0.7179 0.6041
54.7 54.8 55.4 57.6 58.2 58.3 58.7 0.1667 0.2094 0.9273 0.2821
58.9 60.8 60.9 61.0 61.1 63.0 64.3 0.3056 0.0652 0.9925 0.0727
65.6 66.3 66.6 67.0 67.9 70.1 70.3 0.4444 0.0075 1.0000 0.0075
72.1 72.4 72.9 73.5 74.2 75.3 75.4
(a) Find the P-value for the first alternative
75.9 76.5 77.0 x 88.9
(p1 > p2 ) if a = 6.
(a) Calculate range, IQR, and median of these (b) Find the P-value for the third alternative
data. (p1 6= p2 ) if x = −0.2500.
(c) Determine both the P-value and x that sat-
(b) Given that the mean of these data is
isfy the following condition: The data are
63.50 (exactly) and the standard devia-
statistically significant but not highly sta-
tion is 12.33, what proportion of the data
tistically significant for the second alter-
lie within one standard deviation of the
native (p1 < p2 ).
mean?
(c) How does your answer to (b) compare to 5. Sarah performs a CRD with a dichotomous re-
the empirical rule approximation? sponse and obtains the following data.
(d) Ralph decides to delete the smallest obser- Treatment S F Total
vation, 14.1, from these data. Thus, Ralph 1 a b 22
has a data set with n = 39. Calculate the 2 c d 16
range, IQR, and median of Ralph’s new Total 8 30 38
data set.
(e) Refer to (d). Calculate the mean of Next, she obtains the sampling distribution of
Ralph’s new data set. the test statistic for Fisher’s test for her data; it
is given below.
4. Sarah performs a CRD with a dichotomous re-
sponse and obtains the following data. x P (X = x) P (X ≤ x) P (X ≥ x)
−0.5000 0.0003 0.0003 1.0000
Treatment S F Total −0.3920 0.0051 0.0054 0.9997
1 a b 18 −0.2841 0.0378 0.0432 0.9946
2 c d 12 −0.1761 0.1376 0.1808 0.9568
Total 8 22 30 −0.0682 0.2722 0.4530 0.8192
0.0398 0.3016 0.7546 0.5470
0.1477 0.1831 0.9377 0.2454
Next, she obtains the sampling distribution of
0.2557 0.0558 0.9935 0.0623
the test statistic for Fisher’s test for her data; it
0.3636 0.0065 1.0000 0.0065
is given below.
(a) Find the P-value for the first alternative
(p1 > p2 ) if a = 6.
2
, (b) Find the P-value for the third alternative 7. Consider an unbalanced study with six subjects,
(p1 6= p2 ) if x = −0.1761. identified as A, B, C, D, E and G. In the actual
study,
(c) Determine both the P-value and x that sat-
isfy the following condition: The data are • Subjects A and B are assigned to the first
statistically significant but not highly sta- treatment, and the other subjects are as-
tistically significant for the second alter- signed to the second treatment.
native (p1 < p2 ). • There are exactly two successes, obtained
by A and C.
6. Consider a balanced study with six subjects,
identified as A, B, C, D, E and G. In the actual This information is needed for parts (a)–(c) be-
study, low.
(a) Compute the observed value of the test
• Subjects A, B and C are assigned to the statistic.
first treatment, and the other subjects are
(b) Assume that the Skeptic is correct. Deter-
assigned to the second treatment.
mine the observed value of the test statis-
• There are exactly four successes, obtained tic for the assignment that places D and E
by A, D, E and G. on the first treatment, and the remaining
subjects on the second treatment.
This information is needed for parts (a)–(c) be- (c) We have obtained the sampling distribu-
low. tion of the test statistic on the assumption
that the Skeptic is correct. It also is possi-
(a) Compute the observed value of the test ble to obtain a sampling distribution of the
statistic. test statistic if the Skeptic is wrong pro-
(b) Assume that the Skeptic is correct. Deter- vided we specify exactly how the Skeptic
mine the observed value of the test statis- is in error. These new sampling distribu-
tic for the assignment that places C, D and tions are used in the study of statistical
E on the first treatment, and the remaining power which is briefly described in Chap-
subjects on the second treatment. ter 7 of the text. Assume that the Skeptic
is correct about subjects A and G, but in-
(c) We have obtained the sampling distribu-
correct about subjects B, C, D and E.
tion of the test statistic on the assumption
that the Skeptic is correct. It also is possi- For the assignment that puts D and G on
ble to obtain a sampling distribution of the the first treatment, and the other subjects
test statistic if the Skeptic is wrong pro- on the second treatment, determine the re-
vided we specify exactly how the Skeptic sponse for each of the six subjects.
is in error. These new sampling distribu- 8. A comparative study is performed; you are
tions are used in the study of statistical given the following information.
power which is briefly described in Chap-
ter 7 of the text. Assume that the Skeptic • The total number of subjects equals 33.
is correct about subjects C, D and E, but • The observed value of the test statistic is
incorrect about subjects A, B and G. greater than 0.
For the assignment that puts D, E and G I used the website to obtain the exact P-value for
on the first treatment, and the other sub- Fisher’s test for each of the three possible alter-
jects on the second treatment, determine natives. These three P-values are below along
the response for each of the six subjects. with three bogus P-values.
3
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