Confidence Intervals for Proportions

Confidence Intervals for Proportions - Transcript

Chapter 19 Confidence Intervals for Proportions
Far better an approximate answer to the right question than an exact answer to the wrong question John W Tukey

Standard Error
To

find the standard error

SE p



pq n

Because

the sampling distribution model is

Normal


68 of all samples will be within 95 of all samples will be within 99 5 of all samples will be within

p 1SE



p 2 SE
p 3SE



Confidence Interval
One proportion

z interval




Putting a number on the probability that this interval covers the true proportion Our best guess of where the parameter is and how certain we are that it s within some range

Margin of Error
The

p is called the margin of error ME estimate ME

extent of the interval on either side of

In

general confidence intervals are written as is a conflict between certainty and precision

There


Choose a confidence level the data does not determine the confidence level

Assumptions and Conditions
Independence

Assumption The data values are assumed to be independent from each other


Plausible independence condition
Do the data values somehow affect each other Dependent on knowledge of the situation




Randomization condition
Where data sampled at random or generated from a properly randomized experiment Proper randomization helps ensure independence




10 condition
Samples are always drawn without replacement Samples size should be less than 10 of the population


Assumptions and Conditions
Sample
The

Size Assumption Based upon the Central Limit Theory CLT
sample must be large enough to make the sampling model for the sampling proportions approximately Normal More data is needed as the proportion gets closer to either extreme 0 or 1 Success failure condition expect at least 10 successes and 10 failures



One proportion z interval
When

the conditions are met we are ready to find the confidence interval for the population proportion p Since the standard error of the proportion is estimated by

pq SE p the confidence interval is n p z SE p The critical value z depends





on the particular confidence level C that you specify

TI 83 Tips
TI 83

can calculate a confidence interval for a population proportion STAT TESTS A 1 PROPZInt

TI 83 Tips
Enter

the number of successes observed and the sample size a confidence level and then Calculate

Specify

Caution Caution Caution
Don t
Do

mistake what the interval means

not suggest that the parameter varies

The

population parameter is fixed the interval varies from sample to sample

Do

not claim that other samples will agree with this sample
The

interval isn t about sample proportions it is about the population proportion

Don t
We

be certain about the parameter

can t be absolutely certain that the population proportion isn t outside the interval just pretty sure

Caution Caution Caution
Don t

forget it s a parameter

The

confidence interval is about the unknown population parameter p

Don t Take

claim too much
your confidence statement about your sample

Write

responsibility

Confidence

intervals are about uncertainty You are uncertain however not the parameter

Margin of Error Too Large to be Useful
Think

about the margin of error during design of the study Choose a larger sample to reduce variability in the sample proportion To cut the standard error and the ME in half quadruple the sample size Remember though that bigger samples cost more money and effort

Margin of Error An Example
Suppose

a candidate is planning a poll and wants to estimate voter support within 3 with 95 confidence How large a sample is needed


pq pq ME z 0 03 1 96 n n Worst case largest sample size p 5 0 03 1 96 n 1 96

5 5
n

0 03 n 1 96 32 67

5 5
2

5 5

0 03 Round up so sample size needs to be 1068 to keep the margin of error as small as 3 with a confidence level of 95

n 32 67

1067 1

Violation of Assumptions
Watch

out for biased samples
potential sources of bias
on voluntary response of the population bias

Check

Relying

Undercoverage Nonresponse Response

bias

Think

about independence