1. Quantitative Methods
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2. Sampling Distributions
Concept
The Central Limit Theorem
The Sampling Distribution of the Sample Mean
The Sampling Distribution of the Sample Proportion
The Sampling Distribution of the Difference Between Two Sample Means
The Sampling Distribution of the Difference Between Two Sample Proportions
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3. Sample
Population
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4. A number describing a population
Parameter
A number describing a population
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5. A number describing a sample
Statistic
A number describing a sample
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6. Random Sample
Every unit in the population has an equal probability of being included in the sample

7. Common Sense Thing #1
A random sample should represent the population well, so sample statistics from a random sample should provide reasonable estimates of population parameters

8. Common Sense Thing #2
All sample statistics have some error in estimating population parameters

9. Common Sense Thing #3
If repeated samples are taken from a population and the same statistic (e.g. mean) is calculated from each sample, the statistics will vary, that is, they will have a distribution

10. sampling distribution.
The probability distribution of a statistic is called its
sampling distribution.
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11. Mean and Standard Deviation of

12. Distribution of when sampling from a normal distribution
has a normal distribution with
mean =
and
standard deviation =

13. The Central Limit Theorem (CLT
Whatever be the probability distribution of the population from which sample is drawn the probability distribution of sample mean will approximately normal if size of sample is large ;generally larger than 30
Strictly speaking CLT was derived only for sample means but it applies also for sample totals and sample proportions
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14. Central Limit Theorem
If the sample size (n) is large enough, has a normal distribution with
mean =
and
standard deviation =
regardless of the population distribution

15. What is Large Enough?
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16. Does have a normal distribution?
Is the population normal?
Yes No
is normal
Is ?
Yes No
is considered to be
normal
may or may not be
considered normal
(We need more info)
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17. The Sampling Distribution of the Sample Mean
Suppose X is a random variable with mean µ and standard deviation  .
(i) What is the the mean (expected value) and standard deviation of sample mean
x ¯ ?
Answer: μx ¯ = E(x ¯ ) = µ
σ x ¯ = S.D.(x¯ ) =  /√n
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18. (ii)What is the sampling distribution of the sample mean x ¯?
Answer: x¯ has a normal distribution with mean µ and standard deviation  /√n ,
Equivalently
Z =(x ¯ − μx¯ )/σ x ¯
Z = (x¯ − µ ) /  /√n
provided n is large (i.e. n ≥ 30)
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19. Example.
Consider a population, X, with mean μ = 4 and standard deviation σ = 3.
A sample of size 36 is to be selected.
(i) What is the mean and standard deviation of X¯?
(ii) Find P(4 <X ¯ <5),
(iii) Find P(X ¯ >3.5), (exercise)
(iv) Find P(3.5 ≤ X ¯ ≤ 4.5). (exercise)
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20. The Sampling Distribution of the Sample Proportion
Suppose the distribution of X is binomial with parameters n and p.
(ii) What is the the mean (expected value) and standard deviation of sample proportion p ¯ ?
Answer:μP ¯ ˆ = E(p ¯ ) = p
σ P ¯ = S.E.(p ¯ ) = √pq/n
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21. What is the sampling distribution of the sample proportion p ¯?
Answer: p ¯ has a normal distribution with mean p and standard deviation √pq /n ,
Equivalently
Z =(p ¯ − μ P ¯ )/σ P ¯
Z = (p ¯ − p) / √pq/n
provided n is large (i.e. np ≥ 5, and nq ≥ 5)
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22. Example.
It is claimed that at least 30% of all adults favour brand A versus brand B.
To test this theory a sample n = 400 is selected. Suppose 130 individuals indicated preference for brand A.
DATA SUMMARY: n = 400, x = 130, p = .30,
p ¯ = 130/400 = .325
(i) Find the mean and standard deviation of the sample proportion p ¯.
Answer:
μp ¯ = p = .30
σp ¯ =√pq/n = .023
(ii) Find Prob( p ¯ > 0.30)
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23. Comparing Two Sample Means
E(X ¯ 1 − X ¯ 2) = μ1 − μ2
σX ¯ 1−X ¯ 2 =√(σ21/n1 +σ22/n2)
( X ¯ 1 − X ¯ 2) − (μ1 − μ2)
Z =
√ σ12/n1+ σ22/n2
provided n1, n2 ≥ 30.
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24. Comparing Two Sample Proportions
E( P ¯ 1 - P ¯ 2) = p1 - p2
σ2P ¯ 1- P ¯ 2 = p1q1/n1+p2q2/n2
(P ¯ 1 - P ¯ 2) - (p1 - p2) Z=
√ p1q1/n1+ p2q2/n2
provided n1 and n2 are large.
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