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Lecture

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McGill University

Mathematics & Statistics (Sci)

MATH 203

Patrick Reynolds

Fall

Description

Descriptive data response of just a sample (summarize & present)
CHAPTER 1-
1) Identify population or sample (collection of exp.units)
Statistical inference an estimate, or prediction about a population based on a sample (generalization) 2) Identify variable(s)
Population set of units we are interested in studying; “set of all *insert sample+” 3) Collect data
Sample subset of the units in a population; a certain # 4) Describe data
Quantitative data numeric data
Qualitative data can be classified into classes TYPES OF DATA Inferential data utilize data to make inferences about a pop.
Representative sample exhibits characteristics typical of those possessed by the target population
1) Identify population (collection of all experimental units)
Random sample of n experimental units; sample selected from the population in such a way that every
sample of size n has a chance of being selected 2) Identify variables
3) Collect sample data (subset of pop.)
4) Inference about population based on sample
5) Measure of reliability of inferences
EX: 1) Poll on war in Iraq: inference of interest: to estimate the proportion who believed the war was justified
Problems with random samples: Methods of data collection:
(based on 2000 individuals called(sample), population= all US citizens)
2) Does massage enable faster recovery for tired athletes? 8 boxers(experimental units)
in designed experiment. Inference drawn= having a massage does not enable muscles of boxers to recover faster. 1) Selection bias Observational
2) Non-responsive bias Publishes source
No change in mean HR or blood levels can’t make this inference about all athletes (population) can only 3) Measurement error Survey
make inferences that relates to the sample used Designed the experiment
CHAPTER 2-
Class frequency # of observations in the data set that falls into a particular cBar Graph (qual) Pareto diagram (qual) Stem and leaf (quan) Histogram (quan)
Class relative frequency class freq/n Summation notation divided into classes des. order cannot be divided into
Note: ∑ 2= distinct classes
Class % (Class relative frequency)x100 𝑛 x + x …
2 2
𝑋𝑖
𝑖=
2 2
Sample variance (S ) units= x mode most frequently occurring value Skeweness right skewed left skewed symmetric
𝑛 2 ∑ 𝑖= 𝑥 𝑖 2
Standard deviation (S) units=x ∑ 𝑖= 𝑥𝑖 median (middle #) arranged in ascending
𝑛 𝑆 2
𝑆 𝑆2 𝑛 order. Even #= mean of middle 2 #s. odd #
= middle number mean sum/n
2 2 2 2 2 2 1 2 2
EX: 1) variance , ∑ =1.53 + 1.50 + 1.37- + 1.51 + 1.55 + 1.42 + 1.41 + 1.48 =17.34 note: population mean =µ, pop. Std variance = σ,pop. Std
variance = σ , therefore (µ±k σ).--> for every unit in a population
2 7 – 2
S = 7 =0.0033ppm S=0.058ppm Chebyshev’s Rule works for all datasets regardless of shape 𝑥 𝑥̅ 𝑥 µ
𝑧 𝑠𝑎𝑚𝑝𝑙𝑒 𝑧 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛
1) Very few measurements fall within 1 std.devs ( ̅ ̅ 𝑆 𝜎
Empirical rule only works for symmetric mound shapes 2) At least 3/4 of the measurements fall within 2 std. dev ̅ ̅ Z-score represents
1) Approximately 68% of the measurements fall distance between given
within 1 std dev. 𝑥̅𝑠 ̅𝑥 𝑠 3) At least 8/9 of the measurements fall within 3 std. devs ( ̅ ̅ measurement and the
2) Approximately 95% of the measurements fall 4) For any #, K>1, at least (1-1/k ) of the measurements fall within k std. devs ( ̅ mean (given in std. dev),
with 2 std. dev 𝑥̅ 𝑠 𝑥 𝑠 smaller z score =more likely
3) Approximately 99.7% of the measurements fall Ex: 1) given the largest value is 765 and the smallest is 135m estimate the std. dev to contribute to that group
760− 5
within 3 std. dev. 𝑥̅ 𝑠 ̅𝑥 𝑠 Using chebyshev’s rule at least 8/9 of the measurements fall within 3 std. dev = 104=
6 𝑠𝑡𝑑 𝑑𝑒𝑣 EX: looking for students in
S rd
760− 5 top 2.5% (GPA) 3 line
Using empirical rule (given its mound shaped) 95% fall within 2 std dev 𝑠𝑡𝑑 𝑑𝑒𝑣= 156=S 𝑥−2 7
z score =2, = 0 5 ,
using 4 std de. Because missing 5% x= 3.7 (GPA), top 16%
2)in units anaesthesia mean = 79, std dev=23. Assuming mound shape, what % of dentist use less nd
than 102 units? 79+23=102 +1 std deviation need to still uinclude all those less 50% + +2.5% + 13.5% 2 line, z-
𝑥−2 7
34% (̅𝑥 𝑠 = 84% **MEASURES OF DISPERSION:RANGE+IQR+VARIANCE+STD score = 1, = ,x= 3.2
0 5

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