Powerpoint Biostat

7/28/2019 Powerpoint Biostat

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Chapter Topics Estimation process

Point estimates

Interval estimates Confidence interval estimation for the

mean ( known)

Determining sample size Confidence interval estimation for the

mean ( unknown)

Confidence interval estimation for theproportion


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Estimation Process

Mean, , isunknown

Population Random Sample

MeanX = 50

Sample

I am 95%

confident that

is between 40& 60.


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Point EstimatesEstimate Population

Parameters with Sample

Statistics

Mean

Proportion

Variance

Difference

p

2

1 2

X

SP

2S

1 2X X


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Interval Estimates Provides range of values

Take into consideration variation in

sample statistics from sample tosample

Based on observation from 1 sample

Give information about closeness tounknown population parameters

Stated in terms of level of confidence

Never 100% sure


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Confidence Interval Estimates

Mean

Unknown

Confidence

Intervals

Proportion

Known


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Elements of

Confidence Interval Estimation Level of confidence

Confidence in which the interval will

contain the unknown populationparameter

Precision (range)

Closeness to the unknown parameter

Cost

Cost required to obtain a sample of

size n


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Level of Confidence Denoted by

A relative frequency interpretation

In the long run, of all theconfidence intervals that can beconstructed will contain the unknown

parameterA specific interval will either contain or

not contain the parameter

No probability involved in a specific interval

100 1 %

100 1 %


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Interval and Level of

Confidence

Confidence Intervals

Intervals

extend from

to of intervalsconstructedcontain ;

donot.

_Sampling Distribution of the Mean

XX Z

X

/ 2/ 2

XX

1

XX Z

100 1 %

100 %

/ 2 XZ

/ 2 XZ


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Confidence Interval for( Known)

Assumptions

Population standard deviation is

known Population is normally distributed

If population is not normal, use large

sample

Confidence interval estimate

/ 2 / 2X Z X Zn n


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Assumptions

Population standard deviation is unknown

Population is normally distributed

If population is not normal, use largesample

Use Students t Distribution Confidence Interval Estimate

Confidence Interval for( Unknown)

/ 2, 1 / 2, 1n n

S S

X t X tn n


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Degrees of Freedom (df) Number of observations that are free

to vary after sample mean has been

calculated Example

Mean of 3 numbers is 2

degrees of freedom= n -1

= 3 -1

= 2

1

2

3

1 (or any number)

2 (or any number)

3 (cannot vary)

X

X

X


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Studentst Table

Upper Tail Area

df .25 .10 .05

1 1.000 3.078 6.314

20.817 1.886

2.920

3 0.765 1.638 2.353

t0 2.920tValues

Let: n = 3

df = n - 1 = 2

= .10

/2 =.05

/ 2 = .05


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ExampleA random sample of 400 Voters showed 32preferred Candidate A. Set up a 95%

confidence interval estimate for p.

/ /

1 1

.08 1 .08 .08 1 .08.08 1.96 .08 1.96400 400

.053 .107

s s s s

s s

p p p pp Z p p Z

n n

p

p


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ExampleA random sample of 25 has 50 and 8.

Set up a 95% confidence interval estimate for

n X S

/ 2, 1 / 2, 1

8 850 2.0639 50 2.063925 25

46.69 53.30

n n

S SX t X t

n n


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Confidence IntervalEstimate for Proportion

Assumptions

Two categorical outcomes

Population follows binomial distribution

Normal approximation can be used ifand

Confidence interval estimate

5np 1 5n p

/ 2 / 2

1 1S S S SS S

p p p pp Z p p Z

n n

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