Section 10.1Section 10.1Confidence IntervalsConfidence Intervals

The BasicsThe Basics

Confidence Intervals!Confidence Intervals!A level C confidence level for a parameter is an interval computed from sample data by a method that has probability C of producing an interval containing the true value of the parameter.

A confidence interval is found by the following formula:

estimate ± margin of error

The margin of error shows how accurate we believe our guess is, based on the variability of the estimate.

Confidence Intervals!Confidence Intervals!Any confidence interval has two

parts:–An interval computed from the data –A confidence level giving the

probability that the method produces an interval that covers the parameter.

Critical Values!Critical Values!The number z* with The number z* with

probability probability pp lying to lying to its right under the its right under the standard normal standard normal curve is called the curve is called the upper upper pp critical critical valuevalue of the of the standard normal standard normal distribution.distribution.

*

Make sure you know how to read the table of critical values.

Conditions for Inference about a Conditions for Inference about a Population Mean Population Mean

1.1. RandomRandom--The data is an SRS from the population or --The data is an SRS from the population or from a randomized experimentfrom a randomized experiment

2.2. Observations from the population have a Observations from the population have a normalnormal distribution with an distribution with an unknownunknown mean ( mean (μμ) and standard ) and standard deviation (deviation (σσ))

3.3. IndependenceIndependence is assumed for the individual is assumed for the individual observations when calculating a confidence interval. observations when calculating a confidence interval. When we are sampling without replacement from a When we are sampling without replacement from a finite population, it is sufficient to verify that the finite population, it is sufficient to verify that the population is at least 10 times the sample size in population is at least 10 times the sample size in order to use our st. dev formula.order to use our st. dev formula.

CAUTIONCAUTIONBe sure to check that the Be sure to check that the conditions for constructing a conditions for constructing a confidence interval for the confidence interval for the population mean are satisfied population mean are satisfied before you perform any before you perform any calculations.calculations.

Here’s how to find the level Here’s how to find the level CC confidence interval!confidence interval!

Any normal curve has probability Any normal curve has probability CC between point z* between point z* standard deviations below its mean and the point z* standard deviations below its mean and the point z* standard deviations above its mean.standard deviations above its mean.The standard deviation of the sampling distribution ofThe standard deviation of the sampling distribution of is is /√/√nn and its mean is the population mean and its mean is the population mean . So there . So there is probability is probability C C that the observed sample mean that the observed sample mean is between is between

- - zz*(*(/√/√nn) and ) and + + zz*(*(/√/√nn) )

Whenever this happens, the population mean Whenever this happens, the population mean is contained is contained betweenbetween

- - zz*(*(/√/√nn) and ) and + + zz*(*(/√/√nn)) This is the confidence interval. The estimate of the unknownThis is the confidence interval. The estimate of the unknown is is , and the margin of error is , and the margin of error is zz*(*(/√/√nn) )

x

x

x x

x

Possible Statements About a Possible Statements About a Confidence Interval:Confidence Interval:

““I am __% confident that the true (parameter) is between __ I am __% confident that the true (parameter) is between __ and __ because when I computed the endpoints of the interval and __ because when I computed the endpoints of the interval I followed a procedure that creates intervals containing the I followed a procedure that creates intervals containing the true (parameter) about __% of the time it is followed.”true (parameter) about __% of the time it is followed.”““This __% confidence interval is one result of a method that This __% confidence interval is one result of a method that had a __% chance of producing an interval capturing the true had a __% chance of producing an interval capturing the true (parameter).”(parameter).”““This __% confidence interval is one observed result of a This __% confidence interval is one observed result of a process that, were it to be repeated over many random process that, were it to be repeated over many random samples, would produce intervals containing the population samples, would produce intervals containing the population parameter __% of the time.”parameter __% of the time.”““I am 100% confident that the sampling procedure I used has I am 100% confident that the sampling procedure I used has a __% chance of obtaining a sample with a (statistic) such that a __% chance of obtaining a sample with a (statistic) such that the interval constructed around the (statistic) contains the the interval constructed around the (statistic) contains the (parameter).”(parameter).”

Coach Adams’ Standard CommentCoach Adams’ Standard CommentKEEP IN MIND, I change how I say things, but I never KEEP IN MIND, I change how I say things, but I never change the fact that I must comment on what I am change the fact that I must comment on what I am confident about, why I have that level of confidence, confident about, why I have that level of confidence, and INCLUDE CONTEXT!and INCLUDE CONTEXT!

Since we don’t want to forget context, imagine that Since we don’t want to forget context, imagine that when I conducted a 99% confidence interval on the when I conducted a 99% confidence interval on the Capture the Mean Activity that I got (76.7, 91.7).Capture the Mean Activity that I got (76.7, 91.7).

Here is what I might say:Here is what I might say:

““I am 99% confident that the actual course average for I am 99% confident that the actual course average for students in the Fall 2009 AP Statistics class was students in the Fall 2009 AP Statistics class was between a 76.7 and a 91.7 between a 76.7 and a 91.7 because the methods I because the methods I used results in intervals such that 99% of all intervals used results in intervals such that 99% of all intervals created will capture this true average grade.”created will capture this true average grade.”

Helping Us Understand Helping Us Understand Confidence IntervalsConfidence Intervals

The confidence interval mentioned on the The confidence interval mentioned on the previous slide (76.7, 91.7) was calculated with previous slide (76.7, 91.7) was calculated with 99% confidence. Essentially, this particular 99% confidence. Essentially, this particular interval is based on one sample so it may be interval is based on one sample so it may be one of the 99% of all intervals that captures one of the 99% of all intervals that captures or it may be one of the 1% of all intervals that or it may be one of the 1% of all intervals that fails to capture fails to capture ..

The confidence level is the probability that the The confidence level is the probability that the METHOD gives an interval that captures the METHOD gives an interval that captures the true parameter.true parameter.

REMEMBERREMEMBER

Our “confidence” is in the Our “confidence” is in the PROCEDURE used to PROCEDURE used to generate the interval.generate the interval.

INFERENCE TOOLBOX (p 631)INFERENCE TOOLBOX (p 631)

1—1—PPARAMETER—Identify the population of interest ARAMETER—Identify the population of interest and the parameter you want to draw a conclusion and the parameter you want to draw a conclusion about.about.2—2—CCONDITIONS—Choose the appropriate inference ONDITIONS—Choose the appropriate inference procedure. VERIFY conditions (procedure. VERIFY conditions (Random, Normal, Random, Normal, Independent) Independent) before using it.before using it.3—3—CCALCULATIONS—If the conditions are met, carry ALCULATIONS—If the conditions are met, carry out the inference procedure.out the inference procedure.4—4—IINTERPRETATION—Interpret your results in the NTERPRETATION—Interpret your results in the context of the problem. CONCLUSION, context of the problem. CONCLUSION, CONNECTION, CONTEXT (meaning that our CONNECTION, CONTEXT (meaning that our conclusion about the parameter connects to our work conclusion about the parameter connects to our work in part 3 and includes appropriate context)in part 3 and includes appropriate context)

Steps for constructing a CONFIDENCE INTERVAL:

What does a confidence interval What does a confidence interval look like?look like?

There are two commonly used formats for There are two commonly used formats for expressing a confidence intervalexpressing a confidence interval– Estimate Estimate ± margin of error± margin of error

Example: 57.39 ± 3.97Example: 57.39 ± 3.97

– Interval notationInterval notationExample: (53.42, 61.36)Example: (53.42, 61.36)

How confidence intervals behave!How confidence intervals behave!

Margin of error gets smaller whenMargin of error gets smaller when– z* gets smaller. Smaller z* is the same as smaller z* gets smaller. Smaller z* is the same as smaller

confidence level confidence level CC. To obtain a smaller margin of . To obtain a smaller margin of error, you must be willing to accept lower error, you must be willing to accept lower confidence.confidence.

– gets smaller. The standard deviation gets smaller. The standard deviation measures measures the variation in the population. It is easier to pin the variation in the population. It is easier to pin down down when when is smaller. is smaller.

– nn gets larger. Increasing the sample size gets larger. Increasing the sample size n n reduces reduces the margin of error for any fixed confidence interval. the margin of error for any fixed confidence interval. Because Because n n appears under the square root sign, we appears under the square root sign, we must take four times as many observations in order must take four times as many observations in order to cut the margin of error in half.to cut the margin of error in half.

Sample size for desired margin of error!Sample size for desired margin of error!

To determine the sample size To determine the sample size n n that will yield a confidence that will yield a confidence interval for a population mean with a specific margin of error interval for a population mean with a specific margin of error mm, set the expression for the margin of error to be less than or , set the expression for the margin of error to be less than or equal to equal to m m and solve for and solve for n:n:

*z mn

Ultimately, the population standard deviation is being given to us, so this doesn’t change.

And we are typically given a set confidence level, C, which directly determines z*.

So, ultimately, we can only change our sample size to achieve the desired SMALL INTERVAL WITH A HIGH LEVEL OF CONFIDENCE!

ALWAYS ROUND UP to the next whole number value of n to ensure that your sample is large enough to give desired margin of error (or smaller).

Sample Size ExampleSample Size ExampleAgain, consider our Capture the Mean activity. If we Again, consider our Capture the Mean activity. If we want to estimate the true mean within 5 points, how want to estimate the true mean within 5 points, how big of a sample do we need?big of a sample do we need?

σσ=6.4551; 99% confident=6.4551; 99% confidentz*=2.576z*=2.576

2

6.45512.576 5

2.576 6.4551

5

2.576 6.4551

5

11.0601

n

n

n

n

So we need at least 12 grades in our sample to bring the margin of error to within 5 points. (Notice that we rounded 11.06 up to 12.)

Some cautions!Some cautions!The data must be an SRS sample from the population (or from a The data must be an SRS sample from the population (or from a randomized experiment).randomized experiment).The formula is not correct for probability sampling designs more The formula is not correct for probability sampling designs more complex than an SRS.complex than an SRS.There is no correct method for inference from data haphazardly There is no correct method for inference from data haphazardly collected with bias of unknown size.collected with bias of unknown size.Because is strongly influenced by a few extreme observations, Because is strongly influenced by a few extreme observations, outliers can have a large effect on the confidence interval. Search for outliers can have a large effect on the confidence interval. Search for outliers and try to correct them or justify their removal before outliers and try to correct them or justify their removal before computing the interval.computing the interval.If the sample size is small and the population is not normal, the true If the sample size is small and the population is not normal, the true confidence interval will be different from the value confidence interval will be different from the value CC used in used in computing the interval. If ncomputing the interval. If n≥ 15 then the confidence interval will not be ≥ 15 then the confidence interval will not be greatly disturbed by nonnormal populations.greatly disturbed by nonnormal populations.You must know the standard deviation of the population.You must know the standard deviation of the population.

x

Technology DisclaimerTechnology Disclaimer

As always, you will be allowed unrestricted use of your As always, you will be allowed unrestricted use of your calculator on quizzes and tests (as well as the actual calculator on quizzes and tests (as well as the actual AP Exam). For this reason, ALWAYS be certain to AP Exam). For this reason, ALWAYS be certain to write down the values of key numbers that are being write down the values of key numbers that are being used (means, standard deviations, degrees of used (means, standard deviations, degrees of freedom, significance levels, etc.) along with results of freedom, significance levels, etc.) along with results of the calculator procedures in order to receive full credit.the calculator procedures in order to receive full credit.

To use your calculator: STAT: TESTS: Z IntervalTo use your calculator: STAT: TESTS: Z Interval

Make sure you are comfortable with difference Make sure you are comfortable with difference between when you have data versus statsbetween when you have data versus stats

Plug in exactly what you are asked for.Plug in exactly what you are asked for.