Post on 29-Jul-2018
Lecture 9 - Sampling, Point Estimates, and ConfidenceIntervals
STA 102
Kenneth K. Lopiano
September 23, 2013
Example - Review
To the right is a plot of a population distribution.Match each of the following descriptions to one ofthe three plots below.
(1) a single random sample of 100 observations fromthis population(2) a distribution of 100 sample means from randomsamples with size 7
(3) a distribution of 100 sample means from random
samples with size 49
0 10 20 30 40 50
Populationµ = 10σ = 7
Plot A4 6 8 10 12 14 16 18
05
1015202530
Plot B0 5 10 15 20 25 30 35
05
1015202530
Plot C8 9 10 11 12
0
5
10
15
20
STA 102 (Kenneth K. Lopiano) Lec 9 September 23, 2013 2 / 13
Example - Review
To the right is a plot of a population distribution.Match each of the following descriptions to one ofthe three plots below.
(1) a single random sample of 100 observations fromthis population(2) a distribution of 100 sample means from randomsamples with size 7
(3) a distribution of 100 sample means from random
samples with size 49
0 10 20 30 40 50
Populationµ = 10σ = 7
Plot A4 6 8 10 12 14 16 18
05
1015202530
Plot B0 5 10 15 20 25 30 35
05
1015202530
(1)
Plot C8 9 10 11 12
0
5
10
15
20
STA 102 (Kenneth K. Lopiano) Lec 9 September 23, 2013 2 / 13
Example - Review
To the right is a plot of a population distribution.Match each of the following descriptions to one ofthe three plots below.
(1) a single random sample of 100 observations fromthis population(2) a distribution of 100 sample means from randomsamples with size 7
(3) a distribution of 100 sample means from random
samples with size 49
0 10 20 30 40 50
Populationµ = 10σ = 7
Plot A4 6 8 10 12 14 16 18
05
1015202530
(2)
Plot B0 5 10 15 20 25 30 35
05
1015202530
(1)
Plot C8 9 10 11 12
0
5
10
15
20
STA 102 (Kenneth K. Lopiano) Lec 9 September 23, 2013 2 / 13
Example - Review
To the right is a plot of a population distribution.Match each of the following descriptions to one ofthe three plots below.
(1) a single random sample of 100 observations fromthis population(2) a distribution of 100 sample means from randomsamples with size 7
(3) a distribution of 100 sample means from random
samples with size 49
0 10 20 30 40 50
Populationµ = 10σ = 7
Plot A4 6 8 10 12 14 16 18
05
1015202530
(2)
Plot B0 5 10 15 20 25 30 35
05
1015202530
(1)
Plot C8 9 10 11 12
0
5
10
15
20
(3)STA 102 (Kenneth K. Lopiano) Lec 9 September 23, 2013 2 / 13
Central Limit Theorem
1 Central Limit Theorem
2 Confidence intervalsConstructing a confidence interval
STA 102
Lec 9 Kenneth K. Lopiano
Central Limit Theorem
Central Limit Theorem for Sample Means
Central limit theorem
The distribution of the sample mean is well approximated by a normal model:
x̄ ∼ N
(mean = µ, SE =
σ√n
)If σ is unknown, use s.
Assumptions/conditions:
1 Independence: Sampled observations must be independent.
random sampling/assignment is used
2 Sample size/skew: the population distribution must be nearly normalor n > 30 and the population distribution is not extremely skewed.
We can check it using the sample data
STA 102 (Kenneth K. Lopiano) Lec 9 September 23, 2013 3 / 13
Central Limit Theorem
Central Limit Theorem for Proportions
Central limit theorem for proportions
The distribution of the sample proportion is well approximated by a normalmodel:
p̂ ∼ N
(mean = p,SE =
√p (1− p)
n
)If p is unknown use p̂
Assumptions/conditions:1 Independence: Observations must be independent
Random sample
2 Normality: At least 15 successes [np ≥ 15] and 15 failures[n(1− p) ≥ 15].
STA 102 (Kenneth K. Lopiano) Lec 9 September 23, 2013 4 / 13
Confidence intervals
1 Central Limit Theorem
2 Confidence intervalsConstructing a confidence interval
STA 102
Lec 9 Kenneth K. Lopiano
Confidence intervals
Point Estimates
Continuous random variable - Population mean µ, populationstandard deviation σ
Independently sample n observations from the population; x1, x2, . . . , xnPoint estimate x̄ = 1
n
∑ni=1 xi
Bernoulli random variable - Population proportion (probability ofsuccess) p
Independently sample n observations from the population;x1, x2, . . . , xn. Note xi is either 0 (failure) or 1 (success)Point estimate p̂ = 1
n
∑ni=1 xi = X
n where X is the number of successes.
STA 102 (Kenneth K. Lopiano) Lec 9 September 23, 2013 5 / 13
Confidence intervals Constructing a confidence interval
Confidence intervals
A plausible range of values for the population parameter is called aconfidence interval.
Using only a point estimate to estimate a parameter is like fishing in amurky lake with a spear, and using a confidence interval is like fishingwith a net.
We can throw a spear where we saw a
fish but we are more likely to miss. If we
toss a net in that area, we have a better
chance of catching the fish.
If we report a point estimate, we probably will not hit the exactpopulation parameter. If we report a range of plausible values – aconfidence interval – we have a good shot at capturing the parameter.
STA 102 (Kenneth K. Lopiano) Lec 9 September 23, 2013 6 / 13
Confidence intervals Constructing a confidence interval
Confidence Interval for µ
Confidence interval, a general formula
Confidence level (1− α)× 100%
point estimate ± Z1−α/2 × SE
x̄ ± Z1−α/2s√n
Z1−α/2 × SE is the Margin of Error
Assumptions/conditions:1 Independence: Observations in the sample must be independent
random sample/assignment
2 Sample size / skew: the population distribution must be nearlynormal or n ≥ 30 and distribution not extremely skewed
STA 102 (Kenneth K. Lopiano) Lec 9 September 23, 2013 7 / 13
Confidence intervals Constructing a confidence interval
Confidence Interval for p
Confidence interval, a general formula
Confidence level (1− α)× 100%
point estimate ± Z1−α/2 × SE
p̂ ± Z1−α/2
√p̂(1− p̂)
n
Z1−α/2 × SE is the Margin of Error
Assumptions/conditions:1 Independence: Observations must be independent
Random sample
2 Normality: At least 15 successes and 15 failures
STA 102 (Kenneth K. Lopiano) Lec 9 September 23, 2013 8 / 13
Confidence intervals Constructing a confidence interval
Changing the confidence level
point estimate ±Margin of Error
x̄ ± Z1−α/2s√n
p̂ ± Z1−α/2
√p̂(1− p̂)
n
In order to change the confidence level all we need to do is adjustZ1−α/2 in the above formula. We do this by adjusting α where theconfidence level is equal to 1− α× 100%Commonly used confidence levels in practice are 90%, 95%, 98%, and99%.For a 95% confidence interval (i.e., α = 0.05), Z1−α/2 = Z.975 = 1.96.Using the Z table it is possible to find the appropriate Z1−α/2 for anyconfidence level.
STA 102 (Kenneth K. Lopiano) Lec 9 September 23, 2013 9 / 13
Confidence intervals Constructing a confidence interval
Example - Calculating Z1−α/2
What is the appropriate value for Z1−α/2 when calculating a 98%confidence interval? This implies α = 0.02.
STA 102 (Kenneth K. Lopiano) Lec 9 September 23, 2013 10 / 13
Confidence intervals Constructing a confidence interval
Example - Calculating Z1−α/2
What is the appropriate value for Z1−α/2 when calculating a 98%confidence interval? This implies α = 0.02.
-3 -2 -1 0 1 2 3
0.98z = -2.33 z = 2.33
0.01 0.01
STA 102 (Kenneth K. Lopiano) Lec 9 September 23, 2013 10 / 13
Confidence intervals Constructing a confidence interval
Width of an interval
If we want to be very certain that we capture the population parameter,i.e. increase our confidence level, should we use a wider interval or asmaller interval?
A wider interval.
Can you see any drawbacks to using a wider interval?
If the interval is too wide it may not be very informative.
STA 102 (Kenneth K. Lopiano) Lec 9 September 23, 2013 11 / 13
Confidence intervals Constructing a confidence interval
Width of an interval
If we want to be very certain that we capture the population parameter,i.e. increase our confidence level, should we use a wider interval or asmaller interval?
A wider interval.
Can you see any drawbacks to using a wider interval?
If the interval is too wide it may not be very informative.
STA 102 (Kenneth K. Lopiano) Lec 9 September 23, 2013 11 / 13
Confidence intervals Constructing a confidence interval
Width of an interval
If we want to be very certain that we capture the population parameter,i.e. increase our confidence level, should we use a wider interval or asmaller interval?
A wider interval.
Can you see any drawbacks to using a wider interval?
If the interval is too wide it may not be very informative.
STA 102 (Kenneth K. Lopiano) Lec 9 September 23, 2013 11 / 13
Confidence intervals Constructing a confidence interval
Width of an interval
If we want to be very certain that we capture the population parameter,i.e. increase our confidence level, should we use a wider interval or asmaller interval?
A wider interval.
Can you see any drawbacks to using a wider interval?
If the interval is too wide it may not be very informative.
STA 102 (Kenneth K. Lopiano) Lec 9 September 23, 2013 11 / 13
Confidence intervals Constructing a confidence interval
Width of an interval
If we want to be very certain that we capture the population parameter,i.e. increase our confidence level, should we use a wider interval or asmaller interval?
A wider interval.
Can you see any drawbacks to using a wider interval?
If the interval is too wide it may not be very informative.STA 102 (Kenneth K. Lopiano) Lec 9 September 23, 2013 11 / 13
Confidence intervals Constructing a confidence interval
Common Misconceptions
1 The confidence level of a confidence interval is the probability that theinterval contains the true population parameter.
2 A narrower confidence interval is always better.
3 A wider interval means less confidence.
STA 102 (Kenneth K. Lopiano) Lec 9 September 23, 2013 12 / 13
Confidence intervals Constructing a confidence interval
Common Misconceptions
1 The confidence level of a confidence interval is the probability that theinterval contains the true population parameter.
This is incorrect, CIs are part of the frequentist paradigm and as such thepopulation parameter is fixed but unknown. Consequently, the probabilityany given CI contains the true value must be 0 or 1 (it does or does not).
2 A narrower confidence interval is always better.
3 A wider interval means less confidence.
STA 102 (Kenneth K. Lopiano) Lec 9 September 23, 2013 12 / 13
Confidence intervals Constructing a confidence interval
Common Misconceptions
1 The confidence level of a confidence interval is the probability that theinterval contains the true population parameter.
This is incorrect, CIs are part of the frequentist paradigm and as such thepopulation parameter is fixed but unknown. Consequently, the probabilityany given CI contains the true value must be 0 or 1 (it does or does not).
2 A narrower confidence interval is always better.
This is incorrect since the width is a function of both the confidence leveland the standard error.
3 A wider interval means less confidence.
STA 102 (Kenneth K. Lopiano) Lec 9 September 23, 2013 12 / 13
Confidence intervals Constructing a confidence interval
Common Misconceptions
1 The confidence level of a confidence interval is the probability that theinterval contains the true population parameter.
This is incorrect, CIs are part of the frequentist paradigm and as such thepopulation parameter is fixed but unknown. Consequently, the probabilityany given CI contains the true value must be 0 or 1 (it does or does not).
2 A narrower confidence interval is always better.
This is incorrect since the width is a function of both the confidence leveland the standard error.
3 A wider interval means less confidence.
This is incorrect since it is possible to make very precise statements withvery little confidence.
STA 102 (Kenneth K. Lopiano) Lec 9 September 23, 2013 12 / 13
Confidence intervals Constructing a confidence interval
What does 95% confident mean?
Suppose we took many samples and built a confidence interval fromeach sample using the equation point estimate ± 2× SE .Then about 95% of those intervals would contain the true populationmean (µ) or proportion (p).
For example, 25 95% confidenceintervals created from 25different samples. Of the 25 CIs,24 contain the true populationparameter (µ or p) denoted bythe dotted line.
µ = 3.207
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It does not mean there is a 95% probability the CI contains the truevalue
STA 102 (Kenneth K. Lopiano) Lec 9 September 23, 2013 13 / 13