chapter 7: inferences based on a single sample: estimation with confidence interval statistics

Chapter 7: Inferences Based on a Single Sample: Estimation with Confidence Interval

Statistics

McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample:

Estimation by Confidence Intervals

2

Where We’ve Been

Populations are characterized by numerical measures called parameters

Decisions about population parameters are based on sample statistics

Inferences involve uncertainty reflected in the sampling distribution of the statistic



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Where We’re Going

Estimate a parameter based on a large sample

Use the sampling distribution of the statistic to form a confidence interval for the parameter

Select the proper sample size when estimating a parameter

7.1: Identifying the Target Parameter

The unknown population parameter that we are interested in estimating is called the target parameter.



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Parameter Key Word or Phrase Type of Data

µ Mean, average Quantitative

p Proportion, percentage, fraction, rate

Qualitative

7.2: Large-Sample Confidence Interval for a Population Mean

A point estimator of a population parameter is a rule or formula that tells us how to use the sample data to calculate a single number that can be used to estimate the population parameter.



5




6

Suppose a sample of 225 college students watch an average of 28 hours of television per week, with a standard deviation of 10 hours. What can we conclude about all college

students’ television time?




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Assuming a normal distribution for television hours, we can be 95%* sure that

31.128)67(.96.128

2251096.128

96.1

n

x

*In the standard normal distribution, exactly 95% of the area under the curve is in the interval

-1.96 … +1.96




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An interval estimator or confidence interval is a formula that tell us how to use sample data to calculate an interval that estimates a population parameter.

xzx




9

The confidence coefficient is the probability that a randomly selected confidence interval encloses the population parameter.

The confidence level is the confidence coefficient expressed as a percentage.(90%, 95% and 99% are very commonly used.)

95% sure

xzx




10

The area outside the confidence interval is called α

95 % surexzx

So we are left with (1 – 95)% = 5% = α uncertainty about µ




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Large-Sample (1-α)% Confidence Interval for µ

If is unknown and n is large, the confidence interval becomes

nzxzx axa

2/2/

nszxszx axa 2/2/




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If n is large, the sampling distribution of the sample mean is normal, and s is a good estimate of

7.3: Small-Sample Confidence Interval for a Population Mean

Large Sample Sampling Distribution on

is normal Known or large n

Standard Normal (z) Distribution

Small Sample



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Sampling Distribution on is unknown

Unknown and small n

Student’s t Distribution (with n-1 degrees of freedom)

nzx a

2/nstx a 2/


Large Sample Small Sample



14

nzx a

2/nstx a 2/




15* If not, see Chapter 14




16

Suppose a sample of 25 college students watch an average of 28 hours of television per week, with a standard deviation of 10 hours. What can we conclude about all college

students’ television time?




17

Assuming a normal distribution for television hours, we can be 95% sure that

128.428)67(.064.2282510064.228

064.2

nsx

7.4: Large-Sample Confidence Interval for a Population Proportion



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p̂

npq

p ˆ

13ˆ0 ˆ pp

pq 1




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Sampling distribution of




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nqpzp

npqzpzpp p

ˆˆˆˆˆ 2/2/ˆ2/

nxp ˆ pq ˆ1ˆ where and

We can be 100(1-α)% confident that


A nationwide poll of nearly 1,500 people … conducted by the syndicated cable television show Dateline: USA found that more than 70 percent of those surveyed believe there is intelligent life in the universe, perhaps even in our own Milky Way Galaxy.



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What proportion of the entirepopulation agree, at the 95% confidence level?

023.70.)012)(.96.1(70.

1500)30)(.70(.96.170.

ˆˆˆ 2/

pp

p

nqpzpp a




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If p is close to 0 or 1, Wilson’s adjustment for estimating p yields better results

where

42~

4)~1(~~

2/

nxp

nppzp




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032.0289.4100

)0289.1(0289.96.10289.

0289.410021

42~

4)~1(~~

2/

p

p

nxp

nppzpp

Suppose in a particular year the percentage of firms declaring bankruptcy that had shown profits the previous year is .002. If 100 firms are sampled and one had declared bankruptcy, what is the 95% CI on the proportion of profitable firms that will tank the next year?

7.5: Determining the Sample Size

To be within a certain sampling error (SE) of µ with a level of confidence equal to 100(1-α)%, we can solve

for n:



24

SEn

z

2/

2

222/ )(SEzn


The value of will almost always be unknown, so we need an estimate: s from a previous sample approximate the range, R, and use R/4

Round the calculated value of n upwards to be sure you don’t have too small a sample.



25




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Suppose we need to know the mean driving distance for a new composite golf ball within 3 yards, with 95% confidence. A previous study had a standard deviation of 25 yards. How many golf balls must we test?




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Suppose we need to know the mean driving distance for a new composite golf ball within 3 yards, with 95% confidence. A previous study had a standard deviation of 25 yards. How many golf balls must we test?

26778.26632596.1

)(

2

22

2

222/

n

n

SEzn




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For a confidence interval on the population proportion, p, we can solve

for n:

SEnpqz 2/

2

22/ )()(SE

pqzn

To estimate p, usethe sample proportion from a prior study, oruse p = .5.

Round the value of n upward to ensure the sample size is large enough to produce the required level of confidence.




29

How many cellular phones must a manufacturer test to estimate the fraction defective, p, to within .01 with 90% confidence, if an initial estimate of .10 is used for p?




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How many cellular phones must a manufacturer test to estimate the fraction defective, p, to within .01 with 90% confidence, if an initial estimate of .10 is used for p?

24364.2435)01(.

)9)(.1(.)645.1(

)()()(

2

2

2

22/

2/

n

n

SEpqzn

npqzSE

chapter 7: inferences based on a single sample: estimation with confidence interval statistics

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