confidence intervals unit 8 1

7/29/2019 confidence intervals Unit 8 1

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Confidence

Intervals


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Rate your confidence

0 - 100 Name my age within 10

years?

within 5 years?

within 1 year?

Shooting a basketball at awading pool, will makebasket?

Shooting the ball at a largetrash can, will make basket?

Shooting the ball at acarnival, will make basket?


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What happens to your

confidence as the interval

gets smaller?

The larger your confidence,

the wider the interval.


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Point Estimate

Use a single statistic based on

sample data to estimate a

population parameter Simplest approach

But not always very precise due to

variation in the samplingdistribution


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Confidence intervals

Are used to estimate the

unknown population mean

Formula:

estimate + margin of error


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Margin of error

Shows how accurate we believe our

estimate is

The smallerthe margin of error, the

more precise our estimate of the true

parameter

Formula:

statistictheof

deviationstandard

value

criticalm


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Confidence level

Is the success rate of the methodused to construct the interval

Using this method, ____% of the

time the intervals constructed will

contain the true populationparameter


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What does it mean to be 95%

confident?

95% chance that m is contained inthe confidence interval

The probability that the intervalcontains m is 95%

The method used to construct the

interval will produce intervals thatcontain m 95% of the time.


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Found from the confidence level

The upper z-score with probability p lying toits right under the standard normal curve

Confidence level tail area z*

.05 1.645

.025 1.96

.005 2.576

Critical value (z*)

.05

z*=1.645

.025

z*=1.96

.005

z*=2.57690%

95%99%


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Confidence interval for a

population mean:

n

zx

*

estimate

Criticalvalue

Standard

deviation of thestatistic

Margin of error


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Activity


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Steps for doing a confidence

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 known2) Calculate the interval

3) Write a statement about the interval in the

context of the problem.


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Statement: (memorize!!)

We are ________% confidentthat the true mean context lies

within the interval ______ and______.


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Assumptions:

Have an SRS of blood measurements

Potassium 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 amean of 3.2. What is a 90% confidence

interval for the mean potassium level?

3899.3,0101.33

2.645.12.3


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Assumptions:Have an SRS of blood measurements

Potassium level is normally distributed

(given) known


potassium level is between 2.97 and

3.43.

95% confidence interval?

4263.3,9737.23

2.96.12.3


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99% confidence interval?

Assumptions:

Have an SRS of blood measurements

Potassium level is normally distributed

(given) known


potassium level is between 2.90 and 3.50.

4974.3,9026.23

2.576.22.3


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What happens to the interval as the

confidence level increases?

the interval gets wider as the

confidence level increases


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How can you make the margin of

error smaller?

z* smaller

(lower confidence level)

smaller(less variation in the population)

n larger

(to cut the margin of error in half, n mustbe 4 times as big)

Really cannotchange!


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A random sample of 50 SWH students

was taken and their mean SAT scorewas 1250. (Assume = 105) What is a95% confidence interval for the mean

SAT scores of SWH students?

We are 95% confident that the truemean SAT score for SWH students

is between 1220.9 and 1279.1


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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 isbetween 1115.1 and 1270.6.


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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!


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The heights of SWH male students is

normally distributed with = 2.5inches. How large a sample is

necessary to be accurate within + .75

inches with a 95% confidenceinterval?

n = 43


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In a randomized comparative experimenton the effects of calcium on blood

pressure, researchers divided 54 healthy,white males at random into two groups,takes calcium or placebo. The paperreports a mean seated systolic blood

pressure of 114.9 with standard deviationof 9.3 for the placebo group. Assumesystolic blood pressure is normally

distributed.Can you find a z-interval for thisproblem? Why or why not?


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Students t- distribution

Developed by William Gosset

Continuous distribution

Unimodal, symmetrical, bell-shapeddensity curve

Above the horizontal axis

Area under the curve equals 1 Based on degrees of freedom


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Graph examples of t- curves vs normal

curve

Graph examples of t- curves vs normal

curveNormal:

T-Curve: 2 dfs

T-Curve: 5 dfs

T-Curve: 30 dfs


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How does t compare tonormal?

Shorter & more spread out

More area under the tails

As n increases, t-distributionsbecome more like a standard

normal distribution


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How to find t*

Use Table B fort distributions

Look up confidence level at bottom &df on the sides

df = n 1

Find these t*

90% confidence when n = 5

95% 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)


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Formula:

n

s

tx *:IntervalConfidence

estimate

Critical value

Standard

deviation ofstatistic

Margin of error


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Assumptions fort-interval

Have an SRS from population

unknown

Normal distribution Given

Large sample size

Check graph of data


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For the Ex. 4: Find a 95% confidenceinterval 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(27

3.9056.29.114


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Robust

An inference procedure is ROBUST if

the confidence level or p-value doesnt

change much if the assumptions are

violated.

t-procedures can be used with some

skewness, as long as there are nooutliers.

Larger n can have more skewness.


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Ex. 5 A medical researcher measured

the pulse rate of a random sample of 20

adults and found a mean pulse rate of72.69 beats per minute with a standard

deviation of 3.86 beats per minute.

Assume pulse rate is normallydistributed. Compute a 95% confidence

interval for the true mean pulse rates of

adults.

(70.883, 74.497)


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Another medical researcher claims that

the true mean pulse rate for adults is 72

beats per minute. Does the evidencesupport 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.


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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)


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A diet guide claims that you will get 120

calories from a serving of vanillayogurt. 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 120calories.

Note: confidence intervals tell us

if something is NOT EQUAL

never less or greater than!


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Some Cautions:

The data MUST be a SRS from the

population

The formula is not correct for morecomplex sampling designs, i.e.,

stratified, etc.

No way to correct for bias in data


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Cautions continued:

Outliers can have a large effect on

confidence interval

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

confidence intervals unit 8 1

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