Confidence Intervals

Chapter 19

Rate your confidenceRate your confidence0 - 1000 - 100

• Name Mr. Holloway’s age within 10 years?• within 5 years?• within 1 year?

• Shooting a basketball at a wading pool, will make basket?

• Shooting the ball at a large trash can, will make basket?

• Shooting the ball at a carnival, will make basket?

What happens to your confidence as the interval gets smaller?

The larger your confidence, the wider the interval.

Point Estimate

• Use a singlesingle statistic based on sample data to estimate a population parameter

• Simplest approach

• But not always very precise due to variationvariation in the sampling distribution

Confidence intervalsConfidence intervals

• Are used to estimate the unknown population parameter

• Formula:

estimate + margin of error

Margin of errorMargin of error

• Shows how accurate we believe our estimate is

• The smaller the margin of error, the more precisemore precise our estimate of the true parameter

• Formula:

statistic theof

deviation standard

value

criticalME

• Found from the confidence level• The upper z-score with probability p lying to

its right under the standard normal curve

Confidence level tail area z*

.05 1.645

.025 1.96

.005 2.576

Critical value (z*)Critical value (z*)

.05

z*=1.645

.025

z*=1.96

.005

z*=2.57690%95%99%

For the sampling distribution of ,

and for large* n the sampling distribution of p is approximately normal.

pp ˆ

n

ppp

)1(ˆ

* np 10 and n(1-p) 10

Recall

p̂

Confidence interval for a Confidence interval for a population proportion:population proportion:

n

ppzp

)1(*ˆ

estimate

Critical value

Standard deviation of the statistic

Margin of error

Standard Error

n

ppp

)1(ˆ

n

ppSE

)ˆ1(ˆ

• The standard error of a statistic is the estimated standard deviation of the statistic, using the sample values since we don’t know the true population values.

• For sample proportions, the standard deviation of the sampling distribution is

• This means that the standard error of the sample proportion is

So our confidence interval is So our confidence interval is actually:actually:

n

ppzp

)ˆ1(ˆ*ˆ

estimate

Critical value

Standard error

Margin of error

Assumptions:

• Assumptions for inference with proportions:– Data values must be independent– Large enough sample for the sampling

distribution to be approximately normal

• We can’t actually check all of the assumptions, so we check related conditions

Conditions:

• SRS

• n<10% of the population

• At least10 Successes/Failures

10)ˆ1( and 10ˆ pnpn

Example• For a project, a student randomly sampled 182 other

students at a large university to determine if the majority of students were in favor of a proposal to build a parking garage. He found that 75 were in favor of the proposal. Use a 95% confidence interval to estimate the proportion of the student body in favor of the proposal.

• Define the Parameter of interest

– Let p = the true proportion of all students at the university that favor the proposal.

Example - Conditions

• We are told that he took a random sample.

• 182 students is certainly less than 10% of all students at a large university

• Our sampling distribution is approximately normal because

1075182

75182ˆ

pn

10107182

107182)ˆ1(

pn

Example - Calculations

• We will use a 1-proportion z-interval to approximate the true proportion of students who favor the proposal.

• A 95% confidence interval for p can be found using

4121.0182

75ˆ p

Error ofMargin Estimate

SEzp *ˆ

Example – Calculations (continued)

SEzp *ˆ

n

ppp

)ˆ1(ˆ96.1ˆ

182

)4121.01(4121.096.14121.0

07151.04121.0

The 95% confidence interval for p is

( 0.3406 , 0.4836 )

Example – Conclusion

We are 95% confident that the true proportion of students at this university that support the proposal to build a new parking garage on campus is between 0.3406 and 0.4836.

Confidence levelConfidence level

• Is the success rate of the method used to construct the interval

• Using this method, ____% of the time the intervals constructed will contain the true population parameter

What does it mean to be 95% What does it mean to be 95% confident?confident?

• 95% chance that p is contained in the confidence interval

• The probability that the interval contains p is 95%

• The method used to construct the interval will produce intervals that contain p 95% of the time.

Example – Interpretation of the confidence level

If we were to repeat this process many times, 95% of the confidence intervals we constructed would capture the true proportion of students at this university that support the proposal to build a new parking garage on campus.

Note: You only need to interpret the confidence level if it you are specifically asked to.

Interpreting a confidence interval:Interpreting a confidence interval:

We are ________% confident that the true proportion of context is between ______ and ______.

Interpreting the confidence level:Interpreting the confidence level:

If we were to repeat this process many times, ________% of the confidence intervals we constructed would capture the true proportion of context.

A May 2000 Gallup Poll found that 38% of a random sample of 1012 adults said that they believe in ghosts. Find a 95% confidence interval for the true proportion of adults who believe in ghosts.

Conditions:

•Have an SRS of adults

•1012 adults is certainly <10% of adults

•We have at least 10 successes & 10 failures, so the sampling distribution is approximately normal

Step 2: check conditions!

p = the true proportion of all adults that believe in ghosts Step 1: define parameter!

1044.627)62(.1012)ˆ1(

1056.384)38(.1012ˆ

pn

pn

41,.35.

1012

)62(.38.96.138.

ˆ1ˆ*ˆ

n

ppzp

We are 95% confident that the true proportion of adults who believe in ghosts is between .35 and .41

Step 3: do the calculations

Step 4: conclusion in context

Another Gallop Poll is taken in order to measure the proportion of adults who approve of attempts to clone humans. What sample size is necessary to be within + 0.04 of the true proportion of adults who approve of attempts to clone humans with a 95% Confidence Interval?

To find sample size:

However, since we have not yet taken a sample, we do not know a p-hat (or p) to use!

n

ppz

ˆ1ˆ*ME

What p-hat (p) do you use when trying to find the sample size for a given margin of error?

.1(.9) = .09

.2(.8) = .16

.3(.7) = .21

.4(.6) = .24

.5(.5) = .25

By using .5 for p-hat, we are using the worst-case scenario and using the largest SD in our calculations.

Another Gallop Poll is taken in order to measure the proportion of adults who approve of attempts to clone humans. What sample size is necessary to be within + 0.04 of the true proportion of adults who approve of attempts to clone humans with a 95% Confidence Interval?

60125.600

25.

96.1

04.

5.5.

96.1

04.

5.5.96.104.

1*ME

2

n

n

n

n

n

ppz

Use p-hat = .5

Divide by 1.96

Square both sides

Round up on sample size

How can you make the margin of How can you make the margin of error smaller?error smaller?• z* smaller

(lower confidence level)

• p smaller

• n larger(to cut the margin of error in half, n

must be 4 times as big)

Really cannot change!