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S3 Revision Notes
Combinations of random variables
Key Words: random variables, expectation, variance
 For the random variables X and Y
o
𝐸(𝑎𝑋 ± 𝑏𝑌) = 𝑎𝐸(𝑋) ± 𝑏𝐸(𝑌)
o
𝑉𝑎𝑟(𝑎𝑋 ± 𝑏𝑌) = 𝑎2 𝑉𝑎𝑟(𝑋) + 𝑏2 𝑉𝑎𝑟(𝑌) (Independent variables!)
 If 𝑋~𝑁(𝜇1 , 𝜎1 2 ) and 𝑌~𝑁(𝜇2 , 𝜎2 2 )then:
o
𝑎𝑋 ± 𝑏𝑌~𝑁(𝑎𝜇1 ± 𝑏𝜇2 , 𝑎2 𝜎1 2 + 𝑏2 𝜎2 2 )
o
𝑋1 + 𝑋2 + ⋯ + 𝑋 𝑛 + 𝑌1 + 𝑌2 + ⋯ + 𝑌 𝑚 ~𝑁(𝑛𝜇1 ± 𝑚𝜇2, 𝑛𝜎1 2 + 𝑚𝜎2 2 ) (Note no square!)
Sampling
Key Words: census, sampling, simple random sampling, lottery sampling, systematic sampling, stratified sampling,
sampling without replacement, quota sampling, primary and secondary data
 Census: observes every member of a population
o
Used if: population is small or if large accuracy required
o
ADV: accurate (of full picture)
o
DIS: time consuming, expensive, cannot be used when testing involves destroying articles,
info is difficult to process as large volume
 Sampling: observes part of a population
o
Cheap but doesn’t give full picture (size depends on accuracy required)
 Simple random sampling: Every member of the population must have an equal chance of being selected
o
ADV: free from biased, easy, fair
o
DIS: not suitable when sample size is large
 Lottery sampling: randomly drawn numbers from a container (from a sampling frame)
o
ADV: random, easy, each ticket has known chance of selection
o
DIS: not suitable for large pop, sampling frame needed (ordered list of sample selected)
 Systematic Sampling: ordered list and selected every nth member from the list (randomly select 1st
number)
o
Use when: the population is too large for simple random number sampling
o
ADV: simple, suitable for large samples
o
DIS: only random if ordered list is randomly listed, can have bias
 Stratified sampling: taking a proportion of the sampling strata relative to the size of the strata in the
population
...
g if sample of 50 taken from 500 take 1/10th of each strata
o
Use when: sample is large, population divides naturally into mutually exclusive groups
o
ADV: more accurate estimates (than simple random when clear strata)
o
DIS: if bias within strata then will be bias in the sample, strata need to be clearly defined

All above involved sampling without replacement, sampling with replacement is called unrestricted
random sampling (items can be selected more than once)
 Quota Sampling: First decide on groups into which the population is divided and a number from each
group to be interviewed to form quotas
...
If someone refuses to answer or belongs to a quota which is already full then ignore that
persons reply and continue interviewing until all quotas are full
...
g
o
𝐸(𝑋) = 𝜇, or 𝐸(𝑚𝑜𝑑𝑒) = 𝑒𝑥𝑝𝑒𝑐𝑒𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑚𝑜𝑑𝑒
 If 𝑇 is a biased estimator of 𝜃 then: (where 𝑥 is the bias)
o
𝜃 = 𝐸(𝑇) − 𝑥
𝜎
 Standard error is the standard deviation of a sample distribution:
√𝑛



Central Limit Theorem states
o
If a sample of 𝑋1 , 𝑋2 , … , 𝑋 𝑛 has a population mean, 𝜇, and variance, 𝜎 2 , then, for large n,
𝑋 ≈ ~𝑁 (𝜇,



𝜎2
𝑛

)

Confidence intervals provide a range in which the mean is likely to lie
𝜎

o
o


95% confidence limits are 𝑥̅ ± 1
...
6449 ×

𝜎
√𝑛

(other limits found by tables)

o
If 𝜇 lies outside the range of the CI limits then there is sufficient evidence to reject/question
χ²-distribution
o
Used to test 2 lists of frequencies (observed against expected)
o
If frequency of a column is < 5 then combine (collect) two columns until ≥ 5
o



√𝑛

𝜒2 = ∑

(𝑂−𝐸)2
𝐸

When testing χ²against a particular distribution calculate the expected values as follows:
o
Uniform: 𝑛𝑝 (e
...
dice 120 roles if fair each expected 20 times)
o
Continuous uniform: 𝑛𝑝 (be aware to account for class boundaries as continuous)
o
Normal: account for class boundaries as continuous, then (using tables and standardizing)
calculate 𝑃(𝑋 = 𝑥1 ) = 𝑃(𝑋 < 𝑥1 ) then for 𝑃(𝑋 = 𝑥2 ) = 𝑃(𝑋 < 𝑥2 ) − 𝑃(𝑋 < 𝑥1 ) etc

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