課程名稱︰統計導論
課程性質︰數學系選修
課程教師︰江金倉
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2016.11.7
考試時限（分鐘）：110
試題 :
Introduction to Statistics (Midterm)
1.(20%) Explain the following terms:
(1a) interval measurement scale variable. (1b) systematic sampling.
(1c) outlier. (1d) ogive. (1e) matched-paired design.
2.(20%) Identify each of the studies in the appendix as a survey, an
experimental study, a prospective observational study, a retrospective
observational study, a case-control design in an observational study, or a
case study.
3.(10%) Compute the interquartile of the grouped data with f_1,f_2,...,f_g
being the frequencies of the g classes [L,L+h), [L+h,L=2h),..., [L+(g-1)h,U],
where U=L+gh.
4.(10%) Consider the data of thirteen ninth-year students in Sydney using
conventional instructional meterial.The number of seconds, required to
arrive at a solution, are recorded for the students. Each student was
allotted at most 5 minutes to answer a question. What is the median time for
these students to arrive at a solution?
Data: 130, 139, 146, 150, 161, 177, 228, 242, 265, 300+, 300+, 300+, 300+.
(Note: t+ represents the required time more than t seconds.)
5.(10%) Let r_XY be the Pearson correlation coefficient of the sample
{(X_1,Y_1),...,(X_n,Y_n)}. Show that |r_XY|≦1.
6.(5%)(5%)(5%) Give data with three points (X_1,Y_1), (X_2,Y_2), and (X_3,Y_3)
so that the corresponding sample correlation coefficients are 0, 1, and -1,
respectively.
7.(7%)(8%) Assume that the GRE score has a symmetric distribution with mean
of 497 and standard deviation of 115. Find the median GRE score and the Z-
score for the GRE score of 450.
◎ Appendix
1. Bumpus Data on natural selection -An observational study
In a 1989 biology lecture at Woods Hole, Massachusetts, Hermon Bumpus remained
his audience that the process of natural selection of evolutionary change was
an unproved theory. As evidence in support selection, he presented
measurements on house sparrows brought to the Anatomical Laboratory of Brown
University after an uncommonly severe winter storm.
Research interest: Do humerus (arm bone) lengths tend to be different for
survivors than for those that perished? If so, how large is the difference?
Data collection: The humerus lengths for the 24 adult male sparrows that
perished and for the 35 adult males that survived were recorded.
2. Anatomical Abnormalities associated with Schizophrenia
Early studies, based largely on postmortem analysis, suggest that the size of
certain areas of the brain may be different in persons afflicted with
schizophrenia than in others. The researchers used magnetic resonance imaging
(MRI) to measure the volume of several regions inside the twin's brain.
Research interest: What is the magnitude of the difference in volume of the
left hippocampus between the unaffected and the affected individuals? Can the
observed difference be attributed to chance?
Data collection: In a 1990 article, researchers conducted a study that
controlled for gentic and socioeconomic differences by examining 15 pairs of
monozygotic twins, where one of the twins was schizophrenic and other was not.
3. Bird keeping and lung cancer
Research interest: A 1972-1981 health survey in Hague, Netherlands, discovered
an association between keeping pet birds and increased risk of lung cancer. To
investigate bird keeping as risk of factor, researchers conducted a case-
control study of patients in 1985 at four hospitals in Hague.
Data collection: They identified 49 cases of lung cancer patients who were
registered with a general praceice, who were 65 or younger, and who had resided
in the city since 1965. They also selected 98 controls from a population of
residents having the same general age structure.
4. Diet Restriction and Longevity
349 female mice were randomly assigned to one of the following six treatment
groups.
┌────────────────┐ ┌────────────────┐
│N/N85 │ │NP │
│Normal (Control) │ │Ate as much as they pleased of a│
│Normal 85 kcal/wk │ (e) │non-purified,standard diet │
│ (57)│───│ (49)│
└────────────────┘ └────────────────┘
│
│(a)
│
┌────────────────┐ ┌────────────────┐
│N/R50 │ │N/R50 lopro │
│Before weaning: normal diet │ │Normal │
│After weaning: 50 kcal/wk │ (d) │50 kcal/wk & decreasing dietary │
│(reduced-calorie) (56)│───│protein content (56)│
└────────────────┘ └────────────────┘
│ ＼
│(b) ＼(c)
│ ＼
┌────────────────┐ ┌────────────────┐
│R/R50 │ │N/R40 │
│50 kcal/wk │ │Normal │
│50 kcal/wk │ │40 kcal/wk │
│ (56)│ │(severely reduced diet) (60)│
└────────────────┘ └────────────────┘
(a) Does reduce from 85 to 50 kcal/wk increase lifespan?
(b) Is there an effect of pre-wearning diet?
(c) Does further reduction from 50 to 40 kcal/wk increase lifespan more?
(d) Does reduction in protein, with same calories, change lifetime
distribution?
(e) Does control mice have same lifetimes as the the laboratory mice?