Confidence Confidence Intervals with Intervals with

MeansMeans

Rate your confidenceRate your confidence0 - 1000 - 100

Name my age within 10 years?Name my age within 10 years? within 5 years?within 5 years? within 1 year?within 1 year?

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

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

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

What happens to your confidence as the interval gets smaller?

The larger your confidence, the wider the interval.

Point EstimatePoint Estimate

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

SimplestSimplest approach approach But not always very precise But not always very precise

due to variation in the due to variation in the sampling distributionsampling distribution

Confidence intervalsConfidence intervals

Are used to Are used to estimateestimate the the unknown population meanunknown population mean

Formula:Formula:

estimate estimate ++ margin of error margin of error

Margin of errorMargin of error

Shows howShows how accurateaccurate we believe we believe our estimate isour estimate is

TheThe smallersmaller the margin of error, the margin of error, thethe more precisemore precise our estimate of our estimate of the true parameterthe true parameter

Formula:Formula:

statistic theof

deviation standard

value

criticalm

Confidence levelConfidence level

Is the success rate of theIs the success rate of the methodmethod used to construct the used to construct the intervalinterval

Using this method, ____% of Using this method, ____% of the time the intervals the time the intervals constructedconstructed willwill containcontain the the true population parametertrue population parameter

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

95% chance that µ is contained 95% chance that µ is contained in the confidence intervalin the confidence interval

The probability that the The probability that the interval contains µ is 95%interval contains µ is 95%

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

Found from the confidence levelFound from the confidence level The upper z-score with probability p The upper z-score with probability p

lying to its right under the standard lying to its right under the standard normal curvenormal curve

Confidence levelConfidence level tail areatail area z*z*.05.05 1.6451.645.025.025 1.961.96.005.005 2.5762.576

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

.05

z*=1.645

.025

z*=1.96

.005

z*=2.576

90%95%99%

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

n

zx

*

estimatestimatee

Critical Critical valuevalue

Standard Standard deviation of deviation of thethe statisticstatistic

Margin of errorMargin of error

ActivityActivity

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

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

approximately normal)approximately normal) Given (normal)Given (normal) Large sample size (approximately Large sample size (approximately

normal)normal) Graph data (approximately normal)Graph data (approximately normal)

• σσ is known is known2)2) Calculate the intervalCalculate the interval3)3) Write a statement about the interval Write a statement about the interval

in the context of the problem.in the context of the problem.

Statement: Statement: (memorize!!)(memorize!!)We are __________% We are __________% confident that the true confident that the true mean of (context) lies mean of (context) lies within the interval within the interval _______ and ______._______ 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 A test for the level of potassium in the blood is not perfectly precise. Suppose blood is not perfectly precise. Suppose that repeated measurements for the same that repeated measurements for the same person on different days vary normally person on different days vary normally with with σσ = 0.2. A random sample of three = 0.2. A random sample of three has a mean of 3.2. What is a 90% has a mean of 3.2. What is a 90% confidence interval for the mean confidence interval for the mean potassium level?potassium level?

3899.3,0101.33

2.645.12.3

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

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

95% confidence interval?95% confidence interval?

4263.3,9737.23

2.96.12.3

99% confidence interval?99% confidence interval?

Assumptions:Have an SRS of blood measurementsPotassium level is normally distributed (given)s 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 What happens to the interval as the confidence level increases?the confidence level increases?

the interval gets wider as the the interval gets wider as the confidence level increasesconfidence level increases

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

(lower confidence level)(lower confidence level)

σσ smaller smaller(less variation in the (less variation in the

population)population)

n largern larger(to cut the margin of error in (to cut the margin of error in

half, n must be 4 times as big)half, n must be 4 times as big)

Really cannot change!

A random sample of 50 AHS A random sample of 50 AHS students was taken and their students was taken and their mean SAT score was 1250. mean SAT score was 1250. (Assume (Assume σσ = 105) What is a = 105) What is a 95% confidence interval for the 95% confidence interval for the mean SAT scores of AHS mean SAT scores of AHS students?students?We are 95% confident that the true mean SAT score for AHS students is between 1220.9 and 1279.1

Suppose that we have this Suppose that we have this random sample of SAT scores:random sample of SAT scores:

950 1130 1260 1090 1310 1420 1190

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

We are 95% confident that the true mean SAT score for AHS 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 If a certain margin of error is wanted, then to find the sample wanted, then to find the sample size necessary for that margin of size necessary for that margin of error use:error use:

Always round up to the nearest person!

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

n = 43

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

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

Student’s t- distributionStudent’s t- distribution

Developed by William GossetDeveloped by William Gosset Continuous distributionContinuous distribution Unimodal, symmetrical, bell-Unimodal, symmetrical, bell-

shaped density curveshaped density curve Above the horizontal axisAbove the horizontal axis Area under the curve equals 1Area under the curve equals 1 Based on degrees of freedomBased on degrees of freedom

Graph examples of t- Graph examples of t- curves vs normal curvecurves vs normal curve

How does How does tt compare to compare to normal?normal?Shorter & more spread outShorter & more spread outMore area under the tailsMore area under the tailsAs n increases, t-As n increases, t-

distributions become distributions become more like amore like a standard standard normal distributionnormal distribution

How to find How to find tt**

Use Table B for t distributionsUse Table B for t distributions Look up confidence level at Look up confidence level at

bottom & df on the sidesbottom & df on the sides df = n – 1df = n – 1

Find these Find these tt**90% confidence when n = 590% confidence when n = 595% confidence when n = 1595% confidence when n = 15

t* =2.132

t* =2.145

Can also use invT on the calculator!

Need upper t* value with 5% is above – so 95% is below

invT(p,df)

Formula:Formula:

n

stx * :Interval Confidence

estimate

Critical value

Standard deviation of statistic

Margin of errorMargin of error

Assumptions for Assumptions for tt--inferenceinferenceHave an SRS from population σ unknownNormal distribution

–Given

–Large sample size

–Check graph of data

For the Ex. 4: Find a 95% confidence interval for the true mean systolic blood pressure of the placebo group.

Assumptions:

• Have an SRS of healthy, white males

• Systolic blood pressure is normally distributed (given).

• is unknown

We are 95% confident that the true mean systolic blood pressure is between 111.22 and 118.58.

)58.118,22.111(273.9

056.29.114

RobustRobust

An inference procedure is ROBUST if the confidence level or p-value doesn’t change much if the assumptions are violated.

t-procedures can be used with some skewness, as long as there are no outliers.

Larger n can have more skewness.

Ex. 5 – A medical researcher measured the pulse rate of a random sample of 20 adults and found a mean pulse rate of 72.69 beats per minute with a standard deviation of 3.86 beats per minute. Assume pulse rate is normally distributed. Compute a 95% confidence interval for the true mean pulse rates of adults.

(70.883, 74.497)

Another medical researcher claims that the true mean pulse rate for adults is 72 beats per minute. Does the evidence support or refute this? Explain.The 95% confidence interval contains the claim of 72 beats per minute. Therefore, there is no evidence to doubt the claim.

Ex. 6 – Consumer Reports tested 14 randomly selected brands of vanilla yogurt and found the following numbers of calories per serving:

160 200 220 230 120 180 140

130 170 190 80 120 100 170

Compute a 98% confidence interval for the average calorie content per serving of vanilla yogurt.

(126.16, 189.56)

A diet guide claims that you will get 120 calories from a serving of vanilla yogurt. What does this evidence indicate?

Since 120 calories is not contained within the 98% confidence interval, the evidence suggest that the average calories per serving does not equal 120 calories.

Note: confidence intervals tell us if something is NOT EQUALNOT EQUAL

– never less or greater than!

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 σ to do a z-interval – which is unrealistic in practice