Lecture 9 - Sampling, Point Estimates, and …kkl13/courses/sta102F13/lec/Lec9.pdf · Lecture 9 -...

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





0 10 20 30 40 50


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


Example - Review





0 10 20 30 40 50


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


Example - Review





0 10 20 30 40 50


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 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



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].


Confidence intervals

1 Central Limit Theorem

2 Confidence intervalsConstructing a confidence interval

STA 102

Lec 9 Kenneth K. Lopiano


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.


Confidence intervals Constructing a confidence interval


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.



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



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



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.



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.



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



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.



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


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.





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).









This is incorrect since the width is a function of both the confidence leveland the standard error.








This is incorrect since the width is a function of both the confidence leveland the standard error.


This is incorrect since it is possible to make very precise statements withvery little confidence.



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


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