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7/23/2019 CI for a Proportion http://slidepdf.com/reader/full/ci-for-a-proportion 1/24 Confidence Interval for a Proportion 1 Confidence Interval for a Proportion Example 74% of a company’s customers would like to see new product packaging. A random sample of 50 customers is taken. X  is the number of customers in the  sample who would like to see the new  packaging; then the sample proportion is n  X  p   ˆ . The mean and standard deviation for  p ˆ  are Mean of  p ˆ :  p  p   ˆ  = 0.74 Standard deviation of  p ˆ : 06203 50 26 . 0 74 . 0 1 ˆ   n  p  p  p   Since np = 50(0.74) = 37 and n(1 –   p) = 50(0.26) = 13 are both at least 10, the distribution of  p ˆ  is approximately Normal. Using the Normal approximation we know that the probability is approximately 95% that  p ˆ falls within 1.96 standard deviations of the mean. 95% of all samples have a sample proportion  p ˆ  between 0.74 –  1.96(0.06203) = 0.618 and 0.74 + 1.96(0.06203) = 0.862. As 1.96(0.06203) = 0.1216 we could equivalently state “  p ˆ  is within 0.122 of 0.74.” For this situation approximately 95% of all possible samples yield a proportion  p ˆ  within 0.122 of 0.740 (within 12.2% of 74.0%). In general, provided a Normal approximation can be used, 95% of all possible samples yield a  proportion  p ˆ  within  n  p  p   1 96 . 1  of p. An approximate 95% Confidence Interval The section begins with a description of steps that lead to a usable result. (A rigorous treatment of the issue requires a good deal of mathematical statistics.) The explanations provided below are simplified. We’ve observed the following:  For approximately 95% of all samples  p ˆ  is within  n  p  p   1 96 . 1  of p. Flip-flopping p and  p ˆ  yields the following statement (which is true): For approximately 95% of all samples  p is within  n  p  p  ˆ 1 ˆ 96 . 1    of  p ˆ . In this second statement the interval  is random. Different samples yield different values for  p ˆ , which result in different intervals.
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Confidence Interval for a Proportion 1

Confidence Interval for a Proportion

Example

74% of a companys customers would like to see new product packaging. A random sample of

50 customers is taken.Xis the number of customers in thesamplewho would like to see the new

packaging; then the sample proportion is nXp . The mean and standard deviation for p are

Mean of p : pp = 0.74

Standard deviation of p :

0620350

26.074.01

n

ppp

Since np= 50(0.74) = 37 and n(1p) = 50(0.26) = 13 are both at least 10, the distribution of p

is approximately Normal. Using the Normal approximation we know that the probability is

approximately 95% that p falls within 1.96 standard deviations of the mean.

95% of all samples have a sample proportion p between 0.741.96(0.06203) = 0.618

and 0.74 + 1.96(0.06203) = 0.862.

As 1.96(0.06203) = 0.1216 we could equivalently state p is within 0.122 of 0.74.

For this situation approximately 95% of all possible samples yield a proportion p within

0.122 of 0.740 (within 12.2% of 74.0%).

In general, provided a Normal approximation can be used, 95% of all possible samples yield a

proportion p within npp 196.1 ofp.

An approximate 95% Confidence Interval

The section begins with a description of steps that lead to a usable result. (A rigorous treatment

of the issue requires a good deal of mathematical statistics.) The explanations provided below are

simplified.

Weve observed the following:

For approximately 95% of all samples p is within npp 196.1 ofp.

Flip-floppingpand p yields the following statement (which is true):

For approximately 95% of all samplespis within npp 196.1 of p .

In this second statement the intervalis random. Different samples yield different values for p ,

which result in different intervals.

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Confidence Interval for a Proportion 2

The Resulta 95% Confidence Interval for p

For one sampleyielding a result p , the interval

Ep where nppE 196.1

forms an approximate 95% confidence intervalforp. p is the point estimate ofp;Eis the error

margin associated with the estimate.

That is: (approximately) 95% of all random samples (of size n) produce an interval includingp

within the bounds. When we obtain the random sample and compute the interval from it, we no

longer have anything random. At this point we state that we are (approximately) 95% confident

thatpis within the interval bounds.

A couple restrictions are on this formula.

1. It should be applied to situations where units are randomly selected.

2.

If sampling is without replacement, check the 20 Times Rulethe population must be at

least 20 times the sample size to use this result. (If not, and you know the population size,

you can use the adjustment described a bit later in this section. However: small

populations are uncommon, and the adjustment is rarely needed. You should recognize

that when the population is notat least 20 times the sample size, then our recipe above for

error margin does not work.1)

3. The actual counts of SuccessesXand Failures (nX) must both be at least 10.2(If not,

either you have too small a sample, or the Success probability is likely too close to 0 or 1,

for the Normal to be a decent approximation.) If this is not the case, you must seek

alternative strategies. (Minitab, and most other statistical software, can obtain theconfidence interval by an exactmethod that doesnt require the Normal distribution.

When the countsXand (nX) both at least 10, the exact method and the approximate

interval given here will be quite similar.)

We state such an interval in one of four equivalent styles:

EppEp EpEp , EpEp to Ep

The first is preferred, as it indicates what is being estimated:p. In the first through third versions,

the lowervalue is always stated first. Every confidence interval should be accompanied by an

interpretation that states the confidence level (here 95%).

1In fact our formula gives too large an error margin. So in essence you can be more than 95% confident in our result

when its misapplied to situations where the population is small. Probably not the worse error in the world.2Some sources use 5 in place of 10. 10 is better. 5 is somewhat OK, but if either of these values is between 5 and 9,

youd be better off getting some help from a statistician, rather than using our methods.

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Confidence Interval for a Proportion 3

Example

A simple random sample of 1000 adults finds that 343 approve of the President. Obtain a 95%

confidence interval for the proportion of all adults who approve of the President.

This is a random sample wherep= the proportion of all adults who approve of the President (p

is an unknown, but fixed and unvarying, quantity). The population size is hugemuch larger

than 20(1,000) = 20,000. The number of Successes and Failures are 343 and 657 respectively

both are well above 10). We can obtain and then interpret a confidence interval with the

presented method.

x= 343 out of n= 1000 trials. So p = 0.3430. Then

0294.001501.096.11000657.0343.096.1196.1 nppE .

This 95% confidence interval forp(unknown) can be written any of four ways:

0.3136

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Confidence Interval for a Proportion 4

Confidence interval for a (population) proportion p

This section summarizes our results. We extend the treatment to include a confidence interval for

a population proportionpwith any confidence level.

The typical textbook treatment of the issue takes an overblown approach to notation. In addition

to the confidence C% is the error rate (lack of confidence?) for the procedure, which is

generally denoted with , where almost always is rather small. C% = 1 - .

Then takez/2to be theZ score with /2 area in the right tail of the Normal distribution; its

oppositez/2is theZ score with /2 area in the left tail. Betweenz/2andz/2is area

(1) = C% the confidence.

The 1 Sample Z Conf idence I nterval for a Proporti on

An approximate C% confidence interval forpis

Ep

whereEis the error margin

n

ppzE

12

,

withz/2from the Standard Normal distribution, taking into consideration the confidence C%.

When should you use this formula for a confidence interval?

You have a random sample drawn from a population with unknown population

proportionp.

If sampling is without replacement, the population must be at least 20 times the size of

the sample (in most practical cases this is almost trivially true).

The sample result has at least 10 Successes and 10 Failures.

What if the population is too small or the sample is too large?

The at least 20 times rule is violated. There is a relatively simple fix, but to employ it

you must knowat least reasonably accuratelythe sizeNof the population. The error

margin becomes

1

1

2

NnN

nppzE

If you examine the adjustment (a multiplication) you can see the logic in the 20 times

rule: WhenNis more than 20 times nthis multiplier will be quite close to, and just

below, 1. In ignoring the adjustment, we end up with a slightly biggerEthan is required

so if anything we will be understating the confidence.

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Confidence Interval for a Proportion 5

What if the both at least 10 rule isnt met?

There is a method that works whether or not the both at least 5 rule is met. We wont

cover it here. (If you like, learn the Wald Interval described in textbooks. This is also

an approximate methodbut it pretty well for counts below 10.)

What if the sampling isnt random?

No statistical method can guarantee results with any particular reliability (confidence)

when sampling is not random. The situation may well be hopeless.

Example

A companys human resources department investigates the application materials submitted by 84

applicants for an entry level position over a six month period. One finding is that 15 of the

applicants falsified information in the application materials.

Assume that the 84 applicants are a random sample from a larger pool of similar applicants. Give

a 99% confidence interval for the proportion of all applicants who falsify information inapplication materials.

Solution

First check conditions. We take it for granted that the complete pool of applications is at least 20

times larger than the 84 in the sample; we are told to assume the sample is random. (In reality,

random sampling in such a circumstance might be hard to accomplish. Still, we might reasonably

assume that the sample applicants are representative of the population of all applicants.) Both the

number of falsifiers (15) and nonfalsifiers (69) exceed 10. We may use the 1 SampleZ

Confidence Interval.

For 99% confidence, 005.02 zz = 2.576. The estimated/observed proportion of falsifiers is p

= 15/84 = 0.1786. Then the error margin is

1076.00418.0576.284

8214.01786.0576.2

84

1786.011786.0576.2

12

n

ppzE

Then 0.179 0.108 is the 99% confidence interval: 0.071

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Confidence Interval for a Proportion 6

statistical software. 2) The formulaalong with trial and error and some examplescan

assist with your understanding of properties of confidence intervals.

Fact worth knowing

95% is the standard confidence level for scientific polls published in the media and online. If a

poll does not publish an error margin, you may assume that sampling is not randomthe poll isnot a scientific one. Keep in mind also that many polls with stated error margins are not done

properly. You should have less than 95% confidence in results from such polls.

Nomenclature

Theres some terminology that goes with each of these quantities. (The terminology is useful

because it is generalized to other situations.)

The confidence level (or just confidence) is C. Usually C= 0.90, 0.95 or 0.99generally

we prefer to have a high amount of confidence in our statements. However: There is

nothing illegal or necessarily wrong about a 50% CI. (Its just that 50% CIs miss the

target quantity half of the time.)

Dont confuse the confidence level (or just plain confidence) with the confidence

interval. The confidence interval is the interval of values you obtain.

p is the (point) estimate ofp. Its a single good estimate forpfrom the sample of data.

It is theprevalance of Successesin the sample.

2

z is the critical value (from the Standard Normal) that goes with C% confidence.

(Some reference materials use simpler notation likez* - and leave it to common sense

that thiszis the one that goes with the confidence C.)

The two endpoints of the interval are the bounds: lower bound and upper bound. The

width Wof a confidence interval is the distance from the lower to the upper bound. The

part in total is referred to as the error marginE. 2E= W.

n

pp 1 is often called the (estimated) standard error of p .A standard error is

essentially a standard deviation.5Recall: Standard deviation measures typical deviation

from mean. The deviation of p from its meanpis a (sampling) error. Its common to see

the abbreviation SEfor standard error. In this case:

n

pp

pSE

1

.

The error marginEfor this interval can be expressed pSEzE 2 .

5This quantity really is an estimated standard deviation, as the standard deviation of p is npp 1 .

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Confidence Interval for a Proportion 7

Mind yourp's and qs

Many textbooks make the formulas look shorter by using a second letter qto stand for Failure

rate. So q= (1p) and nxnpq 1 . When this is done

nqp

n

pp 1

.

More on interpreting the interval

If you followed the development above, you can deduce the proper interpretation of a confidence

interval. It's also possible to take the justification for granted, and come to an interpretive

understanding.

Different samples give different results. Consider all possible samples. Obtain, for each sample,

a 95% confidence interval. Some of these intervals includep, some do not. Most do. In fact: 95%

of all of them do.

In a statistical study, a single sample is drawn randomly. The data are collected and summarized,

and a 95% confidence interval is computed. We have one sample - selected at random from the

collection of all samples. Because 95% of all samples lead to an interval that coversp, we are

95% confident that the particular interval we have coversp.

We use the word confidence, rather than probability. In statistical applications where parameters

are estimated, those parameters are thought of as fixed values describing populations. They do

not vary. The parameterpis either in the interval or not. There is no probability involved.

Where did the probability go?

There was probability - beforethe sample was selected. This is similar to tossing a coin. Beforeit's tossed the probability of a Head is 1/2. But once the toss is completed, the probability - for

that toss - is either 0 or 1, depending on the outcome. In this application, the probability is either

0 (the interval covers p) or 1 (the interval doesn't), depending on whether in fact the interval does

or does not coverp. Not knowingp, we cannot tell. All we know is that 95% of all samples yield

an interval coveringp. So we are 95% confident that ours does. (Similarly, after the coin is

tossed, if you're unable to see the result, you can be 50% confident it's a Head. The word

probability doesn't apply here.)

In short: Use the word "probability" for random things that haven't yet taken place. Once they've

taken place, even if there are unknowns, use the word confidence. The unknowns merely reflect

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Confidence Interval for a Proportion 8

Properties of Confidence Intervals

Three values impact the error margin of a confidence interval.

1. the prevalence of Success (p )

2.

the sample size (n)

3. the confidence level (C)

Undertake an investigation: How do changes in each of these impact the error margin? These

issues are addressed through the exercises.

In some respects the properties you discover will convince you that statistics makes sense: The

numbers work out in ways that common sense would anticipate in advance(Common sense

would never anticipate the precise results.6But certain procedural properties do make sense.

That's what you want to discover.)

What a Confidence Interval Cannot Do

Notice that the 95% in a 95% confidence interval refers to the percent of all samples that yield an

interval that coversp. If we choose one such sample at random, were 95% confident in that

result. The error margin in a confidence interval addresses errors due to random sampling.

The error margin in a confidence interval does not include the effects of other errors. Poorly

recorded data is one source of error. Or perhaps the study didnt really sample randomly. In these

cases, quantifying sampling error is not enough.

All these other factors will lead to additional estimation errorerror that is not captured by our

formula. So while you can still use the formula when other types of error are present, it doesnt

give a 95% confidence interval. The actual confidence is unknown. For analyses involving

nonrandom data, the actual confidence will be considerably lower than 95%. Thats a real issue

in many studies.

Polling Refusals

Suppose (to oversimplify) that 88 million people approve of the President and 72 million

disapprove. So the Presidents approval rating isp= 88/160 = 0.55.

A telephone poll is taken. But: The people that approve of the President are crankier than those

that do not. They are less likely to put up with an intruding phone call. In fact, 40% of the

approvers will not respond (thats 35.2 million people). The disapprovers are more willing totake the call: only 10% of them will refuse (thats 7.2 million people). Heres the breakdown

6In fact, given the randomness involved in selecting the samples, and the various other attributes that change from

problem to problem (n,p,x, as well as the size of the population), it is remarkable that the formula we have is so

simple.

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Confidence Interval for a Proportion 9

Approve Disapprove Total

Respond 52.8 64.8 117.6

Refuse 35.2 7.2 42.4

Total 88.0 72.0 160.0The problem here is that our sample is going to reflect the views of only the responders. Of the

117.6 million responders, 52.8 million approve, for a rating of 52.8/117.6 = 0.45.

While people will be randomly called, those who refuse to respond will not be included in the

results. So: Our poll will be estimating 0.45. With a sample size of 1000, the error margin will be

around 0.03. While some samples will give results higher than 0.45, it is highly unlikely that

well get a sample that produces a confidence interval including 0.55. After all: The interval is

designed to include 0.45.

This is an example of a biasedestimation. A result is biased if it systematically [on average]

produces the wrong result. Yes: We couldget a random sample that has unusually high amountsof Approvers, and luckily gives an interval including 0.55. But we are unlikely to do so, because

on average our estimate is 0.45 not 0.55. Thats what bias is: The average result from the

sampling procedure is not equal to the intended result.

If we know that nonresponse occurred with 40% probability among Approvers, and 10% among

Disapprovers, we could adjust the survey results accordingly, and produce an unbiased estimate.

But generally nonresponse rates are unknown, and the rate changes from survey to survey,

depending on what the issue is. It is difficult to adjust results to compensate for the nonresponse

issue.

Poll results are even harder to interpret when sampling is not done randomly. Internet pollschoose subjects by convenience and interest. Only people who care enough to vote will vote.

These people may be significantly different in their views than the population of interest. No

error margin can fix up such polls. (Hopefully results are stated without an error margin.)

Statistical Software

Statistical software will compute the confidence interval forp. All you need to do is input three

values: n,x, and the confidence C, along with specifying the method the software should use.

The interval you have learned is the (approximate) 1 sample Z interval for a proportion.

Good software has other choicesthey use a different "formula" than that above. The formula

you have is approximate, and requires at least ten Successes and Failures to allow using the

Normal. For cases where this condition is not met, you may have statistical software compute the

interval using a different method/formula.7In fact, where there are at least ten Successes and

7Usually called the Exact Binomial interval or method.

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Confidence Interval for a Proportion 10

Failures, you may use the alternative method in place of the 1 sample Z interval; youll get

slightly different results.8For really large samples, these differences will be quite small.

One quirk about one method software may use: The intervals may not balanced: The estimate p

is not exactly in the middle of the interval. This is particularly noticeable for results with small

precents of either Successes or Failures. (If you have, say, 2 Successes in only 1020 trials, thenultimately the value ofpis quite small. So the distribution is clamped against the left edge of the

range of values, and has right skew. Skewness is an expression of imbance; so its no surprise

that the interval is not balanced. This is a case where you could not use the formula stated above

2 is too few Successes. The both at least 10restriction prevents use of a Normal

approximation when things arent at all close to Normal.) If bothxand innxare large, the

intervalnot matter which method is usedis nearly balanced about p , and in fact, the exact

interval and the interval from your formula will give very similar results.

Sample Size Determination

The error margin for our confidence interval is n

ppzpSEzE

1

22

. This is

equivalent to ppE

zn 1

2

2

. Suppose, prior to the study, we desire an error margin ofE. If

we can produce a reasonable educated guess for the prevalence of Successes ( p ), then an

appropriate minimum sample size for the study is ppE

zn 1

2

2

.

Example 1Suppose you want to estimate the proportion of students at a large university who are

nearsighted. The prevalence for the general population is around 0.45. Use this as a guess to

determine how many students would need to be included in a random sample if you wanted the

error margin for a 95% confidence interval to be less than or equal to 2%.

Recall that the error margin quantifies the maximum reasonable difference between the observed

value p and the population valuep.

8For really large samples, these differences will be quite small. One quirk about the Exact Binomial method: The

intervals may not balanced: The estimate is generally not exactly in the middle of the interval. This is particularly

noticeable for results with small prevalence of either Successes or Failures. (Any pother than 0.5 implies some

asymmetry, and these intervals reflect this.) If both xand nxare large, the Exact Binomial interval is nearly

symmetric about the sample propotion, and in fact, the exact interval and the interval from your formula will give

very similar results.

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Confidence Interval for a Proportion 11

Solution:The desired error margin isE= 0.02. Our guess is p = 0.45. The required sample size

is 99.237655.045.002.0

96.1 2

n . Of course we cannot sample 0.99 of a student, so we move

up to 2377.

The actual study was run and it turned out that 951 of 2377 randomly sampled students were

nearsighted. This yields a 95% confidence interval of 0.4001 0.0197.

Remark 1

If our guess is closer to 0.5 than prevalence p observed in the data, then the actual error margin

will be greater than desired. If the guess is further from 0.5, then the actual error margin will be

less than the desiredEd. (If the two are equal then someone is a very lucky guesser.)

Example 2

In a study of a new drug, the researchers assume that the cure rate for the drug is the same, 0.60,

as for the established drug. What sample size is required to obtain a 99% confidence interval

with error margin no greater than 0.05?

Solution:The desired error margin isE= 0.05. Our guess is 0.60. The required sample size is

03.63740.060.005.0

576.2 2

n . Of course one cannot sample 0.03 of a patient. To ensure a

large enough sample size, round up to 638.

When the data are collected, its found that the drug is much more effective than the established

drug: 573 of the 638 patients (thats 89.8%) are cured. The 99% confidence interval is

0.898 0.021.

Notice that the error margin is far below 0.05. But this came at a cost. If they had known that the

prevalence would be around 0.90, the researchers could have used 0.90 for a guess, and

determined a sample size of 239 (because 89.2381.09.005.0

576.2 2

n ). Notice that

239

1.09.0576.2

= 0.0500. They ended up sampling 399 more patients than necessary. This cost

them time and moneyif theyd had any clue the cure rate would be far higher, they could have

Remark 2

If the actual error margin is less than the desired value, then, while the error margin is bettered,

the expense of conducting the study was larger than necessary. A smaller sample size would

have sufficed to obtain the desired error margin.

There is no way to choose exactly the right sample size. In some cases, we may have no idea

what the prevalence is in advance. If no guess is possiblewe are completely in the dark as to

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Confidence Interval for a Proportion 12

30

40

50

60

70

80

90

100

0.02 0.03 0.04 0.05 0.06

proportion p

samplesizen

the prevalencewe can use 0.5. This guarantees that actual error margin to be no larger than

what is desired. On the other hand, it also pretty much guarantees that we will take a larger

sample than is necessary (only if the prevalence turns out to be 50% will the sample be just

large enough).

Remark 3Whenever the range of plausible guesses includes 0.5, use 0.5 as the guess. This rule works

when one has no idea what the prevalence is: the range of plausible guesses is from 0 to 1,

which certainly includes 0.5.

Most two-candidate political races are reasonably close. Pollsters9generally use 0.50 to

determine the sample size. Using a guess of 0.50 tends not to lead to dramatic oversampling

unless the result falls below 1/3 or above 2/3.

Example 3

Production line defects occur infrequently at an industrial plant. In the past the rate has generallybeen between 2% and 6% (this value would change over time as the production line, and the

employees working on it, change). What sample size is required to estimate the current rate at

90% confidence with error margin no larger than 4%?

If we assume 2%, then the required size

is 33; if 6% is assumed, the required size

is 96. Heres a plot of the relationship.

(The relationship is not exactly linear.

However: Linear interpolation would

work well here. In general, as long as the

proportion is confined to a small range

of values to one side of 0.50,

interpolation does work fine.)

You can see that 6% requires the largest n(its closest to 0.5). To cover all historical

possibilities, use n= 96. If the rate is actually less than 0.06, you will have oversampled. What if

you sample less than 96? Perhaps a good idea, but if the rate is near 0.06 you wont get the

desired error margin. And, of course, if production falls seriously out of control, you might see a

result much higher than 6%leading to an error margin considerably larger than 0.04.

Remark 4

A good idea is to produce a range of plausible guesses, and find the sample size for a number of

values within that range. Graph this relationship. If the final decision isnt yours, you can place

your graph in front of the decision maker.

9People or organizations who are paid to conduct polls.

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Confidence Interval for a Proportion 13

0

200

400

600

800

1000

0 0.2 0.4 0.6 0.8 1

proportion p

samplesiz

en

This point is illustrated by Example 3. The decision maker on sample size needs to see that

graph.

At right is a plot of the required sample

size for 95% confidence intervals having

error margin 3%. The curve has the sameshape for other confidence levels and

error margins. You can see that a

prevalence of 0.50 requires the largest

sample size.

Appendix

A general format for confi dence in tervals

The confidence interval we just studied is

p

z/2SE(p

)

Where

z/2 is the critical value, found from the Normal, taking into consideration the desired

level of confidence, and

the standard error of the estimate isSE(p ) =

n

pp 1 , yielding

the error margin

n

ppzpSEzE

1**

.

More generally, provided the conditions are right, a confidence interval is determined with

Estimate Error margin

Where

Error margin = critical value SE(estimate)

This formula is broadly applicable to all sorts of data analyses. In almost all circumstances

SE(Estimate) has the square root of the sample size(s) in the denominator.

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Confidence Interval for a Proportion 14

Exercises

1. There are 8640 students enrolled at SUNY Oswego this semester; 5146 live more than 50

miles from campus. A professor (unaware of these figures) samples 92 students and finds

that 60 of them live more than 50 miles from campus.

a)

Identify the following: i) The population proportionp; ii) The sample countX; iii) The

sample proportion p .

b) Which ofpand p is a parameter? Which is a statistic?

A student (also unaware of the whole-campus figures) is about to randomly select 142

students to estimate the proportion who live more than 50 miles away.

c) For the student: What are the mean and standard deviation for p ? Interpret this mean.

(Be sure to include the phrase allpossible samples in your statement.)

2.

The saturation rate for a particular kind of marketing via a newspaper ad is 15%. That is:15% of all newspaper buyers will read the ad. For a new ad, marketers randomly sample 30

a) Identify values forp,X, and p .

b) Which ofpand p is a parameter? Which is a statistic?

For the following exercises, when you interpret results, use the word all or population.

3. A random sample of 212 adoptive parents finds that 85 of them stated No Preference for

their childs gender. Use this sample data to construct a 95% confidence interval estimate for

the proportion of adoptive parents who state No Preference. Explicitly identify thefollowing:

a) The point estimate.

b) The critical value (Z/2).

c) The error margin.

d) Write the interval bounds in this format: p E.

e) Express the interval in this format: ________ < ________ < ________ .

f)

What confidence do you have in this result?

g) Explain whatprepresents in this situation. Is its value known?

h) p : Parameter or Statistic? p: Parameter or Statistic?

i) Interpret your interval in words. We are 95% confident that

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Confidence Interval for a Proportion 15

4. The Genetics and IVF Institute conducted a clinical trial of the XSORT method designed to

increase the probability of conceiving a girl. 325 babies were born to parents using XSORT,

and 295 of them were girls. Use this data to construct a 99% confidence interval for the

proportion of girls born to parents using XSORT. Interpret your result.

5.

Do individuals have the ability to temporarily postpone death to survive a major holiday?(The hypothesis would be that these holidays are family affairs that give a dying person

incentive to live a bit longer.) In one study, 12000 deaths, over the period from one week

before to one week after Thanksgiving, were examined. Of these, 6062 occurred in the week

before Thanksgiving. Give a 95% confidence interval for the proportion of deaths in this two

week period that occur in the earlier week. Interpret your result. Does your data conclusively

support the postpone death theory? (Hint: Check where 0.5 lands relative to your interval.)

6. Complete the small table indicating which critical value from the Standard Normal table goes

with the given levels of confidence.

C 50% 75% 90% 95% 98% 99% 99.9% 99.99%

Z

/2 1.645 1.960 3.891

7. Over a period of 11 years in Hidalgo County, Texas, 870 people were selected for grand jury

duty, and 39% of them were Mexican-American. Notice that you are toldthe value of p -

you dont have to compute it: p = 0.39. From this you can deduce that the number Xof

Mexican-Americans in thesample.Since 0.39(870) = 339.3, the number must be 339 (it

cant be 339.3 you cant select 3-10ths of a Mexican-American). The given value is

rounded for convenience: 339/870 = 0.3899 to four significant digits, and 0.390 to three,

which is sufficient for computing purposes.)

a)

Assume these data represent a random sample of jury-duty-eligible county citizens.

Obtain a 99% confidence for the percent of all county citizens that are Mexican-

b) It was determined that 79.1% of all county citizens were Mexican-American. What does

8. Perform an investigation of the relationship between confidence and error margin. Heres

how.

a) Take exercise 7, where n= 870 and p = 0.390. Youve already obtained a 99%

confidence interval: 0.390 0.043. The error margin is 0.043. Now obtain a 95%confidence interval; determine the error margin.

b) Compute intervals for each of the confidence levels specified in the table. Fill in the table

below with the error margin for the various levels of confidence.

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Confidence Interval for a Proportion 16

C 50% 75% 90% 95% 99% 99.9%

E 0.032 0.043

c) Write a sentence describing how the error margin changes as the confidence is increased

(decreased).

9. A recent survey of 4276 randomly selected households showed that 94.0% of them had

telephones.

a) How many of the 4276 households have telephones? Answer with a whole number.

b) What is the value of p to the nearest 0.0001?

c) Using these results, construct a 99% confidence interval for the proportion of households

with telephones. Interpret your result. What is the error margin for this interval?

d) Give a 99% confidence interval for the proportion of households withouttelephones.

How does the error margin compare to that in part c?10.Gregor Mendel was responsible for famous genetics experiments with peas. In one

experiment he crossed lines of peas, and the results included 428 green peas and 152 yellow

peas.

a) Find a 95% confidence interval for the proportion of all peas that are green. Interpret your

result.

b) Mendels theory of genetic propagation of inherited traits predicted that 75% of all peas

would be green. Is the theory refuted by his data?

11.Perform an investigation of the relationship between sample size and error margin. Heres

how. Take exercise 7 where p = 428/580 = 0.7393. Youve already obtained a 95%

confidence interval: 0.7393 0.0358 (keep figures to the nearest 0.0001 for this). The error

margin is 0.0358.

a) Suppose hypothetically the study had investigated four times fewer peas, but the percent

that are green is the same: 107 green and 38 yellow. Determine a 95% confidence interval

for this outcome. Place the error margin in the table below. The sample size is 4 times

smaller: How many times larger is this error margin?

b) Suppose hypothetically the study had investigated twenty-five times more peas than in

the actual study, with 10700 green and 3800 yellow. Determine a 95% confidence

interval for this outcome. Place the error margin in the table below. The sample size is 25

times larger: How many times smaller is this error margin?

n 145 580 14500

E 0.0358

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Confidence Interval for a Proportion 17

c) Write a sentence describing how the error margin changes when the sample size is k

times larger. Check the solution to make sure you have the right result in mind as you go

forward.

12.Jack conducts a student opinion poll and gets an error margin of 10% for his result. He is

not happy. He wants

3%. How must he adjust his sample size? (Assume the confidence andsample proportion remain the same.)

13.A poll of 4000 people gives an error margin of 0.01. What would the error margin be for a

similar poll of 800 people? (Assume the confidence and sample proportion remain the same.)

Summary

Here is the formula for the error margin: n

ppzE

12

. At this point you ought to be able

to look at the formula and deduce that10

:

The error margin increases when the confidence is increased. (This happens through the

valuez/2.)

The error margin decreases when the sample size is increased. (The sample size appears in

the denominator of the formula.) In particular, the relationship is that increasing (decreasing)

the sample size by a factor of kdecreases (increases) the error margin by a factor of k .

(Thats because the sample size appears in the square root.)

The error margin does not depend on what is called a Success and what is called a Failure.

(Exercise 9 parts c and d explicitly address this.)

14.Perform an investigation of the relationship between the sample prevalence of Success and

What is the relation between the socio-economic status of parents and college graduation of

their children? Different groups are sampled in order to make comparisons. For each socio-

economic status, n= 400 children are sampled and tracked through adulthood. The number of

the 400 who graduate from college is recorded.

a) Obtain a 95% CI for each socio-economic status. Determine the error margin for each

interval. Place your results in the table below.

b) How do error margins compare for the cases p = 0.10 and p = 0.90? How about p =

0.20 and p = 0.80? p = 0.30 and p = 0.70? Why does this make sense?

10Assuming that all other factors stay the same.

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Confidence Interval for a Proportion 18

c) Write a single sentence describing the relationship between the proportion p and the

error margin of the confidence interval.

Parents Status # of grads p 95% CI Error Margin

Welfare 40

Poor 80

Low Income 120

Middle Income 200

High Income 280

Wealthy 320

Super rich 360

15.

Go back to problem 10. The 95% confidence interval is 0.3575

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Confidence Interval for a Proportion 19

18.At SUNY Oswego n= 125 students are randomly selected; 100 of them are opposed to a

proposal that calls for the college to jam cell phone signals in classrooms. (This would

prevent texting in class.) You can confirm that a 90% confidence interval is

0.741

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Confidence Interval for a Proportion 20

b) Convince yourself that the answer to part a is No. For which of these countries is the

error margin for a confidence interval the largest? The smallest? (Assume the same

confidence level is used for all three results.)

22.You want a 95% confidence interval estimate with error margin 4% for the proportion of

science majors who are left handed. How many science majors do you sample?a) Describe in words the parameter you are estimating. What symbol is it given?

b) Assume you have no idea what the prevalence of lefties is for this population. Use a

guess of 0.5 to determine the required sample size.

c) In the general population, 10% of people are lefties. Use this value to determine the

sample size.

d) Which of the answers from c or d is the better choice?

e) It turns out that 24 of 217 sampled science majors are lefties. Obtain the confidence

interval. How does the error margin compare to 4%?

23.Suppose you undertook a study of the day of the week that babies are born. You are

interested in the proportion of babies born on a weekend (Saturday or Sunday). Your goal is

a 90% confidence interval with error margin no greater than 3.5%.

a) Explain why a guess of 0.50 is unreasonable.

b) What is a better value for this guess?

c) In fact, 25% is probably an adequate value for the guess. If all days are equally likely,

then 2/7 = 28.6% should be born on weekends. However, in recent years there is more of

a trend for doctors to induce pregnancy, which usually happens on a weekday! Use 0.25to obtain a sample size for this study.

d) If the actual prevalence is p = 0.25, determine the confidence interval when the sample

size from c is used. Identify the error margindoes it meet the goal of 0.035? What

would such a result say about the 2/7 hypothesis?

e) If the actual prevalence is 0.20 and the sample size from c is used, how will the error

margin compare to 0.035? Explain.

24.Consider a large citys mayoral race where there are two candidates.

a)

Determine the required sample size for a media poll to estimate the percent of peoplewho favor the Republican candidate with error margin 3%.

b) Does the required sample size depend on the population of the city?

25.What proportion of people die during summer (as officially defined)? You decide to

investigate this issue by collecting data. How many obituaries would you examine in order to

obtain a 98% confidence interval estimate with error margin of 1%?

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Confidence Interval for a Proportion 21

Solutions

1. a) i)p= 5146/8640 = 0.5956; ii)X= 60; iii) p = 60/92 = 0.6522.p = 0.5956 is a parameter;

p = 0.6522 is a statistic, c) The mean is p= 0.5956. If we examined all possible samples of 142

students, determiningfor each samplethe proportion who live more than 50 miles from

home, the mean of these proportions is 0.5956. The standard deviation of these proporitons is

1424044.05956.0 = 0.0412.

2. a)p = 0.15. This is a parameter.X= 2. p = 2/30 = 0.06667. p= 0.15 is a parameter; p is are

statistic.

3. a) p = 85/212 = 0.4009. b) 1.96. c) E = 0.0660

3.0660.0

212

5991.04009.096.1

212

212

127

212

85

96.1

E

d) 0.4009 0.0660. e) 0.3335

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Confidence Interval for a Proportion 22

a) The 95% confidence interval is 0.3575

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Confidence Interval for a Proportion 23

b) They are the same for each pair. A Success prevalence of 90% is equivalent to a Failure

prevalence of 10%, so the error margins must be the same.

c) Error margin is largest for prevalence p = 0.5 and drops (symmetrically) as the prevalence

gets further from 0.5on either side of 0.5.

15. Yellow has prevalence 10.7379 = 0.2621. So the interval is 0.2621 0.0358. Or take

10.702 = 0.298 and 10.774 = 0.226 to get (0.226, 0.298).

16. a) I am 95% confident that between 40.7% and 47.7% of allvoters lean towards D.

b) 55.8% 3.5%. c) No. We dont know how the 55.8% is split up. d) Yes. If there are, for

instance, 20% undecided, then the result for R is around 35.8%.

17. This is a media pollthe confidence is 95%. I am 95% confident that between 77% and 85%

of all Americans favor a drawdown.

18. a) The point estimate is 0.800, the error margin is 0.059. b) The interval at Penn State is

exactly the same. c) The population size does not play into this. The formula for error margindepends only upon the prevalence p and the sample size n. This is an underappreciated fact

about sampling and statistical analysis: As long as a population is large (at least 20 times

bigger than the sample), its size is pretty much immaterial. What matters in most practical

situations is the sample size.

19. a)pis the proportion of all SUNY Oswego students who stay in Oswego. Its aparameter.

Itd be difficult to get this value youd have to census virtually every student. b) 0.70 is a

statisticit describes a sample. c) For a 90% confidence interval the error margin will be

smaller. (See #7.) d) To get an error margin that is 5 times smaller will require a sample size that

is 52= 25 times larger. (See exercises 1113.) Thats 2000 students.

20. The error margin for a larger sample size will be smaller. It will not be three times smaller. It

will be 3 = 1.732 times smaller: 0.07 / 1.732 = 0.0404about 4%. (See exercises 1113.)

21. a) No. b) The error margin is smallest for Canada and largest for the United States. (Go back

and examine #14.)

22. a) The symbol isp.p= the proportion of all science majors at this university who are left

handed. b) 25.6005.05.004.0

96.1 2

n . Select 601 science majors.

c) 09.2169.01.004.0

96.1 2

n . Select 217 science majors. d) 217 is the better choice. The lefty

rate for scientists is going to be fairly close to that for the general population. (Not only that, but

the sample size is smaller!) e) 0.1106 0.0417. Pretty close to 0.04 for the error margin. It

missed a little because the actual lefty rate was slightly closer to 0.5 than the guess of 0.10 that

was used to determine the sample size.

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23. a) The weekend constitutes 2 days out of 7. We wouldnt expect half of all births to occur in

2/7th

s of all days. b) A better guess would be 2/7 = 0.286 (anything from 0.25 to 0.30 is

reasonable; anything outside of this is not).

c) 19.41475.025.0035.0

645.1 2

n , so sample 415. d) 0.035. The interval is (0.215, 0.285).

The error margin is 0.035right on the target. The interval does not include 0.286 = 2/7. So we

have some evidence that the 2/7 hypothesis is false. e) It will be smaller. When the actual result

falls further from 0.5, the error margin is smaller. (In fact, if 0.20 is the result, the error margin is

0.032.)

24. a) Its a media poll, so the confidence is 95%. The required sample size is then 1068. (This

number is well known to pollsters. Many polls have 3% error rates because they use a sample

size of around 1000.) b) Absolutely not. Exercises 18 and 21 covered this.

25. You shouldnt guess anything except 0.25. The required sample size is then

3.1014475.025.001.0

326.2

2

n , so sample 10,145 obituaries.