Chapter 211 What Is a Confidence Interval?. Chapter 212 Thought Question 1 Suppose that 40% of a certain population favor the use of nuclear power for

Chapter 21 1

Chapter 21

What Is a Confidence Interval?

Chapter 21 2

Thought Question 1

Suppose that 40% of a certain population favor the use of nuclear power for energy.

(a) If you randomly sample 10 people from this population, will exactly four (40%) of them be in favor of the use of nuclear power? Would you be surprised if only two (20%) of them are in favor? How about if none of the sample are in favor?

Chapter 21 3

Thought Question 2

(b) Now suppose you randomly sample 1000 people from this population. Will exactly 400 (40%) of them be in favor of the use of nuclear power? Would you be surprised if only 200 (20%) of them are in favor? How about if none of the sample are in favor?

Suppose that 40% of a certain population favor the use of nuclear power for energy.

Chapter 21 4

Thought Question 3

A 95% confidence interval for the proportion of adults in the U.S. who have diabetes extends from .07 to .11, or 7% to 11%. What does it mean to say that the interval from .07 to .11 represents a 95% confidence interval for the proportion of adults in the U.S. who have diabetes ?

Chapter 21 5

Thought Question 4

Would a 99% confidence interval for the proportion described in Question 3 be wider or narrower than the 95% interval given? Explain. (Hint: what is the difference between a 68% interval and a 95% interval?)

Chapter 21 6

Thought Question 5

In a May 2006 Zogby America poll of 1000 adults, 70% said that past efforts to enforce immigration laws have been inadequate. Based on this poll, a 95% confidence interval for the proportion in the population who feel this way is about 67% to 73%. If this poll had been based on 5000 adults instead, would the 95% confidence interval be wider or narrower than the interval given? Explain.

Chapter 21 7

Recall from previous chapters:Parameter

fixed, unknown number that describes the population

Statisticknown value calculated from a samplea statistic is used to estimate a parameter

Sampling Variabilitydifferent samples from the same population may yield different values of the sample statisticestimates from samples will be closer to the true values in the population if the samples are larger

Chapter 21 8

Recall from previous chapters:

Sampling Distributiontells what values a statistic takes and how often it takes those values in repeated sampling.

Example:sample proportions ( ’s) from repeated sampling would have a normal distribution with a certain mean and standard deviation.

p̂

Example:The amount by which the proportion obtained from the sample ( ) will differ from the true population proportion (p) rarely exceeds the margin of error.

p̂

Chapter 21 9

Case Study

Science News, Jan. 27, 1995, p. 451.

Comparing Fingerprint Patterns

Chapter 21 10

Case Study: Fingerprints Fingerprints are a “sexually dimorphic trait…

which means they are among traits that may be influenced by prenatal hormones.”

It is known…– Most people have more ridges in the fingerprints

of the right hand. (People with more ridges in the left hand have “leftward asymmetry.”)

– Women are more likely than men to have leftward asymmetry.

Compare fingerprint patterns of heterosexual and homosexual men.

Chapter 21 11

66 homosexual men were studied.• 20 (30%) of the homosexual men showed

left asymmetry.

186 heterosexual men were also studied• 26 (14%) of the heterosexual men showed

left asymmetry.

Case Study: FingerprintsStudy Results

Chapter 21 12

Case Study: FingerprintsA Question

Assume that the proportion of all men

who have leftward asymmetry is 15%.

Is it unusual to observe a sample of 66 men with a sample

proportion ( ) of 30% if the true population proportion (p) is 15%?

p̂

Chapter 21 13

Twenty Simulated Samples (n=66)

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Sample Size

Obs

erve

d P

rop o

r tio

n

Chapter 21 14

The Rule for Sample Proportions

If numerous simple random samples of size n are taken from the same population, the sample proportions from the various samples will have an approximately normal distribution. The mean of the sample proportions will be p (the true population proportion). The standard deviation will be:

n

pp )1(

)ˆ( p

Chapter 21 15

Rule Conditions and Illustration

For rule to be valid, must have Random sample ‘Large’ sample size

Chapter 21 16

s.d.) ( 0440 66

)1501(150)1(

66 mean); ( 150

.

..n

pp

n.p

Case Study: FingerprintsSampling Distribution

Chapter 21 17

Case Study: FingerprintsAnswer to Question

Where should about 95% of the sample proportions lie? mean plus or minus two standard deviations

0.15 2(0.044) = 0.0620.15 + 2(0.044) = 0.238

95% should fall between 0.062 & 0.238

Chapter 21 18

Simulated Data: p=0.15

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Proportion of Successes

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044066

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66150

...

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Chapter 21 19


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approximately 95% of sample proportions fall in this interval(0.062 to 0.238).

Is it likely we would observea sample proportion 0.30?

Chapter 21 20


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Chapter 21 21


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approximately 95% of sample proportions fall in this interval.

Is it likely we would observea sample proportion 0.30?

Chapter 21 22

Confidence Interval for a Population Proportion

An interval of values, computed from sample data, that is almost sure to cover the true population proportion.

“We are ‘highly confident’ that the true population proportion is contained in the calculated interval.”

Statistically (for a 95% C.I.): in repeated samples, 95% of the calculated confidence intervals should contain the true proportion.

Chapter 21 23

since we do not know the population proportion p (needed to calculate the standard deviation) we will use the sample proportion in its place.

Formula for a 95% Confidence Interval for the Population

Proportion (Empirical Rule) sample proportion plus or minus

two standard deviations ofthe sample proportion:

p̂

n)p(p

p̂ 1

2

Chapter 21 24

n

ppp

)ˆ1(ˆ2ˆ

standard error (estimated standard deviation of )p̂

Formula for a 95% Confidence Interval for the Population

Proportion (Empirical Rule)

Chapter 21 25

Margin of Error

nn

..

n

p̂p̂

1)501(50

)1(

2

2

(plus or minus part of C.I.)

Chapter 21 26

Formula for a C-level (%) Confidence Interval for the Population Proportion

npp

p z )ˆ1(ˆˆ *

where z* is the critical value of the standard normal distribution for confidence level C

Chapter 21 27

Common Values of z*Confidence Level

CCritical Value

z*50% 0.67

60% 0.84

68% 1

70% 1.04

80% 1.28

90% 1.64

95% 1.96 (or 2)

99% 2.58

99.7% 3

99.9% 3.29

Chapter 21 28

Case Study

Brown, C. S., (1994) “To spank or not to spank.” USA Weekend, April 22-24, pp. 4-7.

Parental Discipline

What are parents’ attitudes and practices on discipline?

Chapter 21 29

Case Study: Survey

Parental Discipline Nationwide random telephone survey of

1,250 adults.– 474 respondents had children under 18

living at home– results on behavior based on the smaller

sample reported margin of error

– 3% for the full sample– 5% for the smaller sample

Chapter 21 30

Case Study: Results

Parental Discipline “The 1994 survey marks the first time a

majority of parents reported not having physically disciplined their children in the previous year. Figures over the past six years show a steady decline in physical punishment, from a peak of 64 percent in 1988”– The 1994 proportion who did not spank or

hit was 51% !

Chapter 21 31

Case Study: Results

Parental Discipline Disciplining methods over the past year:

– denied privileges: 79%– confined child to his/her room: 59%– spanked or hit: 49%– insulted or swore at child: 45%

Margin of error: 5%– Which of the above appear to show a true

value different from 50%?

Chapter 21 32

Case Study: Confidence Intervals

Parental Discipline denied privileges: 79%

– : 0.79– standard error of : – 95% C.I.: .79 2(.019) : (.752, .828)

confined child to his/her room : 59%– : 0.59– standard error of : – 95% C.I.: .59 2(.023) : (.544, .636)

0190474)791(79 ...

0230474)591(59 ...

p̂p̂

p̂p̂

Chapter 21 33


Parental Discipline spanked or hit: 49%


insulted or swore at child: 45%– : 0.45– standard error of : – 95% C.I.: .45 2(.023) : (.404, .496)

0230474)491(49 ...

0230474)451(45 ...

p̂p̂

p̂p̂

Chapter 21 34

Case Study: Results

Parental Discipline Asked of the full sample (n=1,250):

“How often do you think repeated yelling or swearing at a child leads to long-term emotional problems?”– very often or often: 74%– sometimes: 17%– hardly ever or never: 7%– no response: 2%

Margin of error: 3%

Chapter 21 35


Parental Discipline hardly ever or never: 7%


Few people believe such behavior is harmless, but almost half (45%) of parents engaged in it!

00701250)071(07 ...

p̂p̂

Chapter 21 36

Key Concepts (1st half of Ch. 21)

Different samples (of the same size) will generally give different results.

We can specify what these results look like in the aggregate.

Rule for Sample Proportions Compute and interpret Confidence

Intervals for population proportions based on sample proportions

Chapter 21 37

Inference for Population MeansSampling Distribution, Confidence Intervals

The remainder of this chapter discusses the situation when interest is in making conclusions about population means rather than population proportions– includes the rule for the sampling distribution

of sample means ( )– includes confidence intervals for one mean

or a difference in two means

s'X

Chapter 21 38

Thought Question 6(from Seeing Through Statistics, 2nd Edition, by Jessica M. Utts, p. 316)

Suppose the mean weight of all women at a university is 135 pounds, with a standard deviation of 10 pounds.

• Recalling the material from Chapter 13 about bell-shaped curves, in what range would you expect 95% of the women’s weights to fall? 115 to 155 pounds

Chapter 21 39

Thought Question 6 (cont.)

• If you were to randomly sample 10 women at the university, how close do you think their average weight would be to 135 pounds?

• If you randomly sample 1000 women, would you expect the average to be closer to 135 pounds than it would be for the sample of 10 women?

Chapter 21 40

Thought Question 7

A study compared the serum HDL cholesterol levels in people with low-fat diets to people with diets high in fat intake. From the study, a 95% confidence interval for the mean HDL cholesterol for the low-fat group extends from 43.5 to 50.5...

a. Does this mean that 95% of all people with low-fat diets will have HDL cholesterol levels between 43.5 and 50.5? Explain.

Chapter 21 41

Thought Question 7 (cont.)

… a 95% confidence interval for the mean HDL cholesterol for the low-fat group extends from 43.5 to 50.5. A 95% confidence interval for the mean HDL cholesterol for the high-fat group extends from 54.5 to 61.5.

b. Based on these results, would you conclude that people with low-fat diets have lower HDL cholesterol levels, on average, than people with high-fat diets?

( ) ( )40 45 50 55 60 65

Chapter 21 42

Thought Question 8

The first confidence interval in Question 7 was based on results from 50 people. The confidence interval spans a range of 7 units. If the results had been based on a much larger sample, would the confidence interval for the mean cholesterol level have been wider, more narrow or about the same? Explain.

Chapter 21 43

Thought Question 9

In Question 7, we compared average HDL cholesterol levels for two diet groups by computing separate confidence intervals for the two means. Is there a more direct value (and single C.I.) to examine in order to make the comparison between the two groups?

Chapter 21 44

Case Study

Weights of Females at a Large University

Suppose the mean weight of all women is =135 pounds with a

standard deviation of =10 pounds and the weight values follow a bell-

shaped curve.

Hypothetical(from Seeing Through Statistics, 2nd Edition, by Jessica M. Utts, p. 316)

Chapter 21 45

What about the mean (average) of a sample of n women? What values would be expected?

Case Study: WeightsQuestions

Where should 95% of all women’s weights fall? mean plus or minus two standard deviations

135 2(10) = 115

135 + 2(10) = 155 95% should fall between 115 & 155

Chapter 21 46

Twenty Simulated Samples (n=1000)

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1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96Sample Size

Obs

erve

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ean

We i

ght

1 500 1000

Chapter 21 47

The Rule for Sample Means

If numerous simple random samples of size n

are taken from the same population, the sample

means from the various samples will have

an approximately normal distribution. The

mean of the sample means will be (the

population mean). The standard deviation will

be: ( is the population s.d.)n

)(X

Chapter 21 48

Conditions for the Rule for Sample Means

Random sample Population of measurements…

– Follows a bell-shaped curve

- or -

– Not bell-shaped, but sample is ‘large’

Chapter 21 49

)X for s.d. ( 3.1610

10

10

)population for s.d. ( 10

)X and population for mean ( 135

nσ

n

σ

μ

Case Study: Weights Sampling Distribution

(for n = 10)

Chapter 21 50

Where should 95% of the sample mean weights fall (from samples of size n=10)? mean plus or minus two standard deviations

135 2(3.16) = 128.68 135 + 2(3.16) = 141.32

95% should fall between 128.68 & 141.32

Case Study: Weights Answer to Question

(for n = 10)

Chapter 21 51

Sampling Distribution of Mean (n=10)Simulated Data: Sample Size=10

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Chapter 21 52

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(for n = 25)

Chapter 21 53


135 2(2) = 131 135 + 2(2) = 139

95% should fall between 131 & 139


(for n = 25)

Chapter 21 54


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Chapter 21 55

1100

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μ


(for n = 100)

Chapter 21 56


135 2(1) = 133 135 + 2(1) = 137

95% should fall between 133 & 137


(for n = 100)

Chapter 21 57


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Sample Means

Chapter 21 58

Case Study

Hypothetical

Exercise and Pulse Rates

Is the mean resting pulse rate of adult subjects who regularly exercise different

from the mean resting pulse rate of those who do not regularly exercise?

Find Confidence Intervals for the means

Chapter 21 59

n mean std. dev. Nonexercisers 31 75 9.0 Exercisers 29 66 8.6

Case Study: Results

Exercise and Pulse RatesA random sample of n1=31 nonexercisers yielded a sample

mean of =75 beats per minute (bpm) with a sample standard deviation of s1=9.0 bpm. A random sample of

n2=29 exercisers yielded a sample mean of =66 bpm

with a sample standard deviation of s2=8.6 bpm.

1X

2X

Chapter 21 60

The Rule for Sample Means

If numerous simple random samples of size n

are taken from the same population, the sample

means from the various samples will have

an approximately normal distribution. The

mean of the sample means will be (the

population mean). The standard deviation will

be:n

)(X

We do not know the value of !

Chapter 21 61

Standard Error of the (Sample) Mean

SEM = standard error of the mean

(standard deviation from the sample) = divided by

(square root of the sample size)

= ns

Chapter 21 62

Case Study: Results

Exercise and Pulse Rates n mean std. dev. std. err. Nonexer. 31 75 9.0 1.6 Exercisers 29 66 8.6 1.6

Typical deviation of an individual pulse rate(for Exercisers) is s = 8.6

Typical deviation of a mean pulse rate(for Exercisers) is = 1.6

ns

298.6

Chapter 21 63


Exercise and Pulse Rates

Nonexercisers: 75 ± 2(1.6) = 75 ± 3.2 = (71.8, 78.2)

Exercisers: 66 ± 2(1.6) = 66 3.2 = (62.8, 69.2)

Do you think the population means are different?

95% C.I. for the population mean: sample mean 2 (standard error)

X ns

2

Yes, because the intervals do not overlap

Chapter 21 64

Formula for a C-level (%) Confidence Interval for the Population Mean

* sxn

z

where z* is the critical value of the standard normal distribution for confidence level C

Chapter 21 65

Careful Interpretation of a Confidence Interval

“We are 95% confident that the mean resting pulse rate for the population of all exercisers is between 62.8 and 69.2 bpm.” (We feel that plausible values for the population of exercisers’ mean resting pulse rate are between 62.8 and 69.2.)

** This does not mean that 95% of all people who exercise regularly will have resting pulse rates between 62.8 and 69.2 bpm. **

Statistically: 95% of all samples of size 29 from the population of exercisers should yield a sample mean within two standard errors of the population mean; i.e., in repeated samples, 95% of the C.I.s should contain the true population mean.

Chapter 21 66

Exercise and Pulse Rates 95% C.I. for the difference in population

means (nonexercisers minus exercisers): (difference in sample means)

2 (SE of the difference) Difference in sample means: = 9 SE of the difference = 2.26 (given) 95% confidence interval: (4.48, 13.52)

– interval does not include zero ( means are different)

1 2X X


Chapter 21 67

An Experiment Testing a Vaccine for Those with Genital Herpes

Case Study

Adler, T., (1994) “Therapeutic vaccine fights herpes.” Science News, Vol. 145, June 18, p. 388.

Does a new vaccine prevent the outbreak of herpes in people already

infected?

Chapter 21 68


Case Study: Sample

98 men and women aged 18 to 55 Experience between 4 and 14

outbreaks per year Experiment

– Double-blind experiment– Randomized to vaccine or placebo

Chapter 21 69


Case Study: Report

“The vaccine was well tolerated. gD2 recipients reported fewer recurrences per month than placebo recipients (mean 0.42 [sem 0.05] vs 0.55 [0.05]…)…”

Chapter 21 70



95% C.I. for population mean recurrences:– Vaccine group: 0.42 2(0.05) : (.32, .52)– Placebo group: 0.55 2(0.05) : (.45, .65)

95% C.I. for the difference in population means:– Difference = -0.13, SE = 0.07 (given)

– C.I.: (-0.27, 0.01) (contains 0 means not different)

Chapter 21 71

Key Concepts (2nd half of Ch. 21)

Rule for Sample Means Compute confidence intervals for means

based on one sample Compute confidence intervals for means

based on two samples

Interpret Confidence Intervals for Means

Documents

Chapter 211 What Is a Confidence Interval?. Chapter 212 Thought Question 1 Suppose that 40% of a certain population favor the use of nuclear power for