Confidence Intervals

Rate your confidenceRate your confidence0 - 1000 - 100

• Name my 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 mean

• 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

criticalm

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 is contained in the confidence interval

• The probability that the interval contains is 95%

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

• 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%

Confidence interval for a Confidence interval for a population mean:population mean:

n

zx

*

estimate

Critical value

Standard deviation of the statistic

Margin of error

Steps for doing a confidence Steps for doing a confidence interval:interval:1) Assumptions –

• SRS from population• Sampling distribution is normal (or approximately

normal)• Given (normal)• Large sample size (approximately normal)• Graph data (approximately normal)

• is known

2) Calculate the interval

3) Write a statement about the interval in the context of the problem.

Statement: Statement: (memorize!!)(memorize!!)

We are ________% confident that the true mean context lies within the interval ______ and ______.

Assumptions:Have an SRS of blood measurementsPotassium level is normally distributed (given) known

We are 90% confident that the true mean potassium level is between 3.01 and 3.39.

A test for the level of potassium in the blood is not perfectly precise. Suppose that repeated measurements for the same person on different days vary normally with = 0.2. A random sample of three has a mean of 3.2. What is a 90% confidence interval for the mean potassium level?

3899.3,0101.33

2.645.12.3

Assumptions:Have an SRS of blood measurementsPotassium level is normally distributed (given) known

We are 95% confident that the true mean potassium level is between 2.97 and 3.43.

95% confidence interval?

4263.3,9737.23

2.96.12.3

99% confidence interval?

Assumptions:Have an SRS of blood measurementsPotassium level is normally distributed (given) known

We are 99% confident that the true mean potassium level is between 2.90 and 3.50.

4974.3,9026.23

2.576.22.3

What happens to the interval as the What happens to the interval as the confidence level increases?confidence level increases?

the interval gets wider as the confidence level increases

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

(lower confidence level)

• smaller(less variation in the population)

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

must be 4 times as big)

Really cannot change!

A random sample of 50 SWH students was taken and their mean SAT score was 1250. (Assume = 105) What is a 95% confidence interval for the mean SAT scores of SWH students?

We are 95% confident that the true mean SAT score for SWH students is between 1220.9 and 1279.1

Suppose that we have this random sample of SAT scores:

950 1130 1260 1090 1310 1420 1190

What is a 95% confidence interval for the true mean SAT score? (Assume = 105)

We are 95% confident that the true mean SAT score for SWH students is between 1115.1 and 1270.6.

Find a sample size:Find a sample size:

n

zm

*

• If a certain margin of error is wanted, then to find the sample size necessary for that margin of error use:

Always round up to the nearest person!

The heights of SWH male students is normally distributed with = 2.5 inches. How large a sample is necessary to be accurate within + .75 inches with a 95% confidence interval?

n = 43

In a randomized comparative experiment on the effects of calcium on blood pressure, researchers divided 54 healthy, white males at random into two groups, takes calcium or placebo. The paper reports a mean seated systolic blood pressure of 114.9 with standard deviation of 9.3 for the placebo group. Assume systolic blood pressure is normally distributed.

Can you find a z-interval for this Can you find a z-interval for this problem? Why or why not?problem? Why or why not?

Some Cautions:Some Cautions:

• The data MUST be a SRS from the population

• The formula is not correct for more complex sampling designs, i.e., stratified, etc.

• No way to correct for bias in data

Cautions continued:Cautions continued:

• Outliers can have a large effect on confidence interval

• Must know to do a z-interval – which is unrealistic in practice