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Summary STA1000 Module 5
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Comprehensive Summary and additional notes for module 5.
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Module 5: Hypothesis TestingWork Unit 1: Introducing Probability
Population
- Everything that is being measured, is measured individually.
- The quantity of interest in the population: parameter.
- Generally unknown but what we want to find out through the sample statistics.
- Mean = μ. Variance = σ 2.
Sample
- Every single thing is not measured, a part (sample) of the population is measured to gain an idea
of the population trends.
- The quantity of interest in a sample: statistic.
- Used as estimate of population parameter.
- Mean = x̅. Variance = s2.
In order to make a legitimate inference about the population, our sample must be:
- Representative: similar factors making up sample as population.
- Random: every member of population has equal chance of being selected.
Sampling Distribution
- the probability distribution of a statistic (plotting the various means of different random sample
statistics).
n
- ∑ xi N (nμ , n σ 2)
i=1
- E [ x̅ ] =μ
σ2
- Var [ x̅ ]=
n
σ2
So, X̅ ̴ N ( μ , )
n
As samples are smaller, their means are likely to vary (not as accurate) – small n means big Var.
Central Limit Theorem
- The average (sum) of a large number independent, identically distributed random variables
always has a normal distribution.
The mean of the data at a sample size of about 30 or more will follow a normal distribution.
Underlying data may not be normal but the mean of many data points is normally distributed.
Calculating Probabilities with X́
, X́ −μ
- Z=
σ ÷ √n
Work Unit 2: Confidence Intervals
Point Estimate
- No information regarding uncertainty of estimate
- Estimate which consists of a single value
Interval
- Range of values used to communicate how much precision or uncertainty is present in the
estimate
Confidence Interval Method
- Pr (−1,96 < z< 1,96 )=0,95
- Replace z with its formula in terms of X̅ and such:
σ σ
- Pr ( X́ −1,96 < μ< X́ +1,96 )
√n √n
We call this a 95% confidence interval for μ.
Width of Confidence Intervals
- This is determined by the z value and the standard error of the sample mean.
α
( )
2 σ
- If we denote this as the product of the two, z × = L, we can see that the confidence
√n
interval width of any variable will be 2xL
- If you increase n, the confidence interval gets narrower
- If you increase z, the interval gets wider
Large samples yield better results than small samples
Within how many units do we want our estimate to lie? - L
How confident do you want to be of your estimate? - z
How variable is the population you are studying? - σ
To Calculate the Sample Size needed for optimal efficient and meaningful results:
2
z×σ
n=
L( )
If we don’t know μ we can construct an interval that we are 100 ( 1−α ) % confident will contain it:
α α
( ) ( ) σ =1−α
Pr ( X́ −z 2
σ
√n
< μ< X́ + z 2
√n )
The sample mean will always lie at the centre of the confidence interval.
“Our best estimate of the unknown mean is X́ and we are
α
2 σ
100 ( 1−α ) % confident that it lies within the interval X́ ± z × .”
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