Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. More About Confidence Intervals...

Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc.

More About

Confidence Intervals

Chapter 12

Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 2

Recall:• A parameter is a population characteristic – value

is usually unknown. We estimate the parameter using sample information.

• A statistic, or estimate, is a characteristic of a sample. A statistic estimates a parameter.

• A confidence interval is an interval of values computed from sample data that is likely to include the true population value.

• The confidence level for an interval describes our confidence in the procedure we used. We are confident that most of the confidence intervals we compute using a procedure will contain the true population value.


12.1 Examples of Different Estimation Situations

Situation 1. Estimating the proportion falling into a category of a categorical variable.

Example research questions:What proportion of American adults believe there is extraterrestrial life? In what proportion of British marriages is the wife taller than her husband?Population parameter: p = proportion in the population

falling into that category.Sample estimate: = proportion in the sample falling

into that category. p̂


More Estimation Situations

Situation 2. Estimating the mean of a quantitative variable.

Example research questions:

What is the mean time that college students watch TV per day? What is the mean pulse rate of women?

Population parameter: (spelled “mu” and pronounced “mew”) = population mean for the variable

Sample estimate: = the sample mean for the variablex


More Estimation SituationsSituation 3. Estimating the difference between two

populations with regard to the proportion falling into a category of a qualitative variable.

Example research questions:How much difference is there between the proportions that would quit smoking if taking the antidepressant buproprion (Zyban) versus if wearing a nicotine patch?How much difference is there between men who snore and men who don’t snore with regard to the proportion who have heart disease?

Population parameter: p1 – p2 = difference between the two population proportions.Sample estimate: = difference between the two sample proportions.

21 ˆˆ pp


Situation 4. Estimating the difference between two populations with regard to the mean of a quantitative variable.

Example research questions:How much difference is there in average weight loss for those who diet compared to those who exercise to lose weight? How much difference is there between the mean foot lengths of men and women?

Population parameter: 1 – 2 = difference between the two population means.

Sample estimate: = difference between the two sample means.

More Estimation Situations

21 xx


Independent SamplesTwo samples are called independent samples when the measurements in one sample are not related to the measurements in the other sample.

• Random samples taken separately from two populations and same response variable is recorded.

• One random sample taken and a variable recorded, but units are categorized to form two populations.

• Participants randomly assigned to one of two treatment conditions, and same response variable is recorded.


Paired Data: A Special Case of One Mean

Paired data (or paired samples): when pairs of variables are collected. Only interested in population (and sample) of differences, and not in the original data.

• Each person measured twice. Two measurements of same characteristic or trait are made under different conditions.

• Similar individuals are paired prior to an experiment. Each member of a pair receives a different treatment. Same response variable is measured for all individuals.

• Two different variables are measured for each individual. Interested in amount of difference between two variables.


12.2 Standard Errors

Rough Definition: The standard error of a sample statistic measures, roughly, the average difference between the statistic and the population parameter. This “average difference” is over all possible random samples of a given size that can be taken from the population.

Technical Definition: The standard error of a sample statistic is the estimated standard deviation of the sampling distribution for the statistic.


Poll: Random sample of 935 AmericansDo you think there is intelligent life on other planets?

Standard Error of a Sample Proportion

Example 12.1 Intelligent Life on Other Planets

proportion sampleˆ ,

ˆ1ˆ)ˆ.(.

p

n

pppes

Results: 60% of the sample said “yes”, = .60

016.

935

6.16.ˆ..

pes

p̂

The standard error of .016 is roughly the average difference between the statistic, , and the population parameter, p, for all possible random samples of n = 935 from this population.

p̂


Poll: Class of 175 students. In a typical day, about how much time to you spend watching television?

Standard Error of a Sample Mean

Example 12.2 Mean Hours Watching TV

deviation standard sample ,).(. sn

sxes

Variable N Mean Median TrMean StDev SE MeanTV 175 2.09 2.000 1.950 1.644 0.124

124.175

644.1..

n

sxes


Study: n1 = n2 = 244 randomly assigned to each treatment

Standard Error of the Difference Between Two Sample Proportions

Example 12.3 Patches vs Antidepressant (Zyban)?

2

22

1

1121

ˆ1ˆˆ1ˆ)ˆˆ.(.

n

pp

n

ppppes

Zyban: 85 of the 244 Zyban users quit smoking = .348Patch: 52 of the 244 patch users quit smoking = .213

1p̂2p̂

So, 135.213.348.ˆˆ 21 pp

040.

244

213.1213.

244

348.1348.)ˆˆ.(. and 21

ppes


Study: n1 = 42 men on diet, n2 = 47 men on exercise routine

Standard Error of the Difference Between Two Sample Means

Example 12.4 Lose More Weight by Diet or Exercise?

2

22

1

21

21 ).(.n

s

n

sxxes

Diet: Lost an average of 7.2 kg with std dev of 3.7 kgExercise: Lost an average of 4.0 kg with std dev of 3.9 kg

So, kg 2.30.42.721 xx

81.0

47

9.3

42

7.3).(. and

22

21 xxes


12.3 Approximate 95% CI

For sufficiently large samples, the interval

Sample estimate 2 Standard error

is an approximate 95% confidence interval for a population parameter.

Note: The 95% confidence level describes how often the procedure provides an interval that includes the population value. For about 95% of all random samples of a specific size from a population, the confidence interval captures the population parameter.


Necessary Conditions

• For one proportion: Both and are at least 5, preferably at least 10.

• For one mean: n is greater than 30.

• For two proportions: and are at least 5 (preferably 10) for each sample.

• For two means: n1 and n2 are each greater than 30.

Sample Size Requirements:

pnˆ pn ˆ1

pnˆ pn ˆ1



• The samples are randomly selected. In practice, it is sufficient to assume that samples are representative of the population for the question of interest.

• For the confidence intervals for the difference between two proportions or two means, the two samples must be independent of each other.

Other Requirements:


Example 12.1 Intelligent Life? (cont)

Poll: Random sample of 935 AmericansDo you think there is intelligent life on other planets?

Note: For about 95% of all random samples from the population, the corresponding confidence interval captures the population parameter. We don’t know if particular interval does or does not capture the population value.

Results: 60% of the sample said “yes”, = .60

016.

935

6.16.ˆ..

pes

Approximate 95% Confidence Interval: .60 2(.016) => .60 .032 => .568 to .632

p̂


Example 12.2 Watching TV (cont)

Note: We are 95% confident that the mean time that Penn State students spend watching television per day is somewhere between 1.842 and 2.338 hours.

Approximate 95% Confidence Interval: 2.09 2(.124) => 2.09 .248 => 1.842 to 2.338 hours

Poll: Class of 175 students. In a typical day, about how much time do you spend watching television?

The sample mean was 2.09 hours and the sample standard deviation was 1.644 hours.

124.175

644.1..

n

sxes


Example 12.3 Patch vs Antidepressant (cont)

Note: Zyban had a higher success rate and the interval does not include the value 0, so it supports a difference between the success rates of the two methods.

Approximate 95% Confidence Interval: .135 2(.040) => .135 .080 => .055 to .215

Study: n1 = n2 = 244 randomly assigned to each groupZyban: 85 of the 244 Zyban users quit smoking = .348Patch: 52 of the 244 patch users quit smoking = .213

1p̂2p̂

So, 135.213.348.ˆˆ 21 pp 040.)ˆˆ.(. and 21 ppes


Example 12.4 Diet vs Exercise (cont)

Note: We are 95% confident the interval 1.58 to 4.82 kg covers the increased mean population weight loss for dieters compared to those who exercise. The interval does not cover 0, so a real difference is likely to hold for the population.

Approximate 95% Confidence Interval: 3.2 2(.81) => 3.2 1.62 => 1.58 to 4.82 kg

Study: n1 = 42 men on diet, n2 = 47 men exerciseDiet: Lost an average of 7.2 kg with std dev of 3.7 kgExercise: Lost an average of 4.0 kg with std dev of 3.9 kg

So, kg 2.30.42.721 xx kg 81.0).(. and 21 xxes


12.4 General CI for One Mean or Paired Data

A Confidence Interval for a Population Mean

where the multiplier t* is the value in a t-distribution with degrees of freedom = df = n - 1 such that the area between -t* and t* equals the desired confidence level. (Found from Table A.2.)

Conditions: • Population of measurements is bell-shaped and

a random sample of any size is measured; OR• Population of measurements is not bell-shaped,

but a large random sample is measured, n 30.

n

stxxestx ** ..


Example 12.5 Mean Forearm Length

Data: Forearm lengths (cm) for a random sample of n = 9 men

25.5, 24.0, 26.5, 25.5, 28.0, 27.0, 23.0, 25.0, 25.0

Note: Dotplot shows no obvious skewness and no outliers.

95% Confidence Interval: 25.5 2.31(.507) => 25.5 1.17 => 24.33 to 26.67 cm

507.9

52.1.. and ,52.1 ,5.25

n

sxessx

Multiplier t* from Table A.2 with df = 8 is t* = 2.31


Example 12.6 What Students Sleep More? Q: How many hours of sleep did you get last night,

to the nearest half hour?

Note: Bell-shape was reasonable for Stat 10 (with smaller n).

Notes: Interval for Stat 10 is wider (smaller sample size)Two intervals do not overlap => Stat 10 average significantly higher than Stat 13 average.

Class N Mean StDev SE MeanStat 10 (stat literacy) 25 7.66 1.34 0.27Stat 13 (stat methods) 148 6.81 1.73 0.14


Data: two variables for n individuals or pairs; use the difference d = x1 – x2.

Population parameter: d = mean of differences for the population = 1 – 2.

Sample estimate: = sample mean of the differences

Standard deviation and standard error: sd = standard deviation of the sample of differences;

Confidence interval for d: , where df = n – 1 for the multiplier t*.

Paired Data Confidence Interval

destd ..*

n

sdes d..

d


Example 12.7 Screen Time: Computer vs TV

Data: Hours spent watching TV and hours spent on computer per week for n = 25 students.

Note: Boxplot shows no obvious skewness and no outliers.

Task: Make a 90% CI for the mean difference in hours spent using computer versus watching TV.


Example 12.7 Screen Time: Computer vs TV

90% Confidence Interval: 5.36 1.71(3.05) => 5.36 5.22 => 0.14 to 10.58 hours

05.325

24.15.. and ,24.15 ,36.5

n

sdessd d

d

Multiplier t* from Table A.2 with df = 24 is t* = 1.71

Results:

Interpretation: We are 90% confident that the average difference between computer usage and television viewing for students represented by this sample is covered by the interval from 0.14 to 10.58 hours per week, with more hours spent on computer usage than on television viewing.


12.5 General CI for Difference Between Two Means (Indep)

A CI for the Difference Between Two Means(Independent Samples):

where t* is the value in a t-distribution with area between -t* and t* equal to the desired confidence

level. The df used depends on if equal population variances are assumed.

2

22

1

21*

21 n

s

n

stxx



• Two samples must be independent.

Either …

• Populations of measurements both bell-shaped, and random samples of any size are measured.

or …

• Large (n 30) random samples are measured.


Degrees of Freedom

The t-distribution is only approximately correct and df formula is complicated (Welch’s approx):

Statistical software can use the above approximation, but if done by-hand then use a conservative df = smaller of n1 – 1 and n2 – 1.


Example 12.8 Effect of a Stare on Driving

Randomized experiment: Researchers either stared or did not stare at drivers stopped at a campus stop sign; Timed how long (sec) it took driver to proceed from sign to a mark on other side of the intersection.

Task: Make a 95% CI for the difference between the mean crossing times for the two populations represented by these two independent samples.

No Stare Group (n = 14): 8.3, 5.5, 6.0, 8.1, 8.8, 7.5, 7.8, 7.1, 5.7, 6.5, 4.7, 6.9, 5.2, 4.7

Stare Group (n = 13): 5.6, 5.0, 5.7, 6.3, 6.5, 5.8, 4.5, 6.1, 4.8, 4.9, 4.5, 7.2, 5.8



Checking Conditions:

Boxplots show …

• No outliers and no strong skewness.

• Crossing times in stare group generally faster and less variable.



The 95% confidence interval for the difference between the population means is 0.14 seconds to 1.93 seconds .

Note: The df = 21 was reported by the computer package based on the Welch’s approximation formula.


Equal Variance Assumption

Often reasonable to assume the two populations have equal population standard deviations, or equivalently, equal population variances:

Estimate of this variance based on the combined or “pooled” data is called the pooled variance. The square root of the pooled variance is called the pooled standard deviation:

222

21

2

11 deviation standard Pooled

21

222

211

nn

snsnsp


Pooled Standard Error

21

21

2

2

2

1

2

21

11

11

).(. Pooled

nns

nns

n

s

n

sxxes

p

p

pp

Note: Pooled df = (n1 – 1) + (n2 – 1) = (n1 + n2 – 2).


Pooled Confidence Interval

Pooled CI for the Difference Between Two Means (Independent Samples):

where t* is found using a t-distribution with df = (n1 + n2 – 2) and sp is the pooled standard deviation.

21

*21

11

nnstxx p


Example 12.9 Male and Female Sleep Times

Data: The 83 female and 65 male responses from students in an intro stat class.

Note: We will assume equal population variances.

Task: Make a 95% CI for the difference between the two population means sleep hours for females versus males.

Q: How much difference is there between how long female and male students slept the previous night?



Notes:• Two sample standard deviations are very similar.• Sample mean for females higher than for males.• 95% confidence interval contains 0 so cannot rule out

that the population means may be equal.

Two-sample T for sleep [with “Assume Equal Variance” option]

Sex N Mean StDev SE MeanFemale 83 7.02 1.75 0.19Male 65 6.55 1.68 0.21

Difference = mu (Female) – mu (Male)Estimate for difference: 0.46195% CI for difference: (-0.103, 1.025)T-Test of difference = 0 (vs not =): T-Value = 1.62 P = 0.108 DF = 146Both use Pooled StDev = 1.72



Pooled standard deviation and pooled standard error “by-hand”:

72.1957.2

26583

68.116575.1183

2

11 dev std Pooled

22

21

222

211

nn

snsnsp

285.065

1

83

172.1

11).(. Pooled

2121

nn

sxxes p


Pooled or Unpooled? • If sample sizes are equal, the pooled and unpooled

standard errors are equal. If sample standard deviations similar, assumption of equal population variance is reasonable and pooled procedure can be used.

• If sample sizes are very different, pooled test can be quite misleading unless sample standard deviations are similar. If the smaller standard deviation accompanies the larger sample size, we do not recommend using the pooled procedure.

• If sample sizes are very different, the standard deviations are similar, and the larger sample size produced the larger standard deviation, the pooled procedure is acceptable because it will be conservative.


12.5 The Difference Between Two Proportions (Indep)

A CI for the Difference Between TwoProportions (Independent Samples):

where z* is the value of the standard normal variable with area between -z* and z* equal to the desired confidence level.

2

22

1

11*21

11

n

pp

n

ppzpp


• Condition 1: Sample proportions are available based on independent, randomly selected samples from the two populations.

• Condition 2: All of the quantities –

– are at least 5 and preferably at least 10.


22221111 ˆ1 and ,ˆ ,ˆ1 ,ˆ pnpnpnpn


Example 12.10 Snoring and Heart Attacks

Data: Of 1105 snorers, 86 had heart disease.Of 1379 nonsnorers, 24 had heart disease.

Q: Is there a relationship between snoring and risk of heart disease?

0604.ˆˆ and ,0174.1379

24ˆ ,0778.

1105

86ˆ 2121 pppp

0088.

1379

0174.10174.

1105

0778.10778.)ˆˆ.(. and 21

ppes


Example 12.10 Snoring and Heart Attacks

Note: the higher the level of confidence, the wider the interval.

5.40174.0778.

ˆˆ

2

1 pp

It appears that the proportion of snorers with heart disease in the population is about 4% to 8% higher than the proportion of nonsnorers with heart disease.

Risk of heart disease for snorers is about 4.5 times what the risk is for nonsnorers.

Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. More About Confidence Intervals...

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Transcript of Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. More About Confidence Intervals...