Copyright 2011 by W. H. Freeman and Company. All rights reserved.1 Introductory Statistics: A...

Copyright 2011 by W. H. Freeman and Company. All

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Introductory Statistics: A Problem-Solving Approachby Stephen Kokoska

Chapter 8: Confidence Intervals Based on a Single

Sample

Introduction

A single value of a statistic computed from a sample conveys little information about confidence and reliability.

Alternative method: use a single value to construct an interval in which we are fairly certain the true value lies.


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Point Estimation

Point estimate of a population parameter: a single number computed from a sample which serves as a best guess for the parameter.


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Estimator

An estimator (statistic) is a rule used to produce a point estimate of a population parameter.


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Estimator and Estimate

Estimator: a statistic of interest, a random variable. An estimator has a distribution, a mean, a variance, and a standard deviation.

Estimate: a specific value of an estimator.

Unbiased Estimator


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Unbiased and Biased Estimators


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Copyright 2011, W.H.Freeman and Company, all rights

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Some Unbiased Estimators• The sample mean is an unbiased statistic for

estimating the population mean.

• The sample proportion is an unbiased statistic for estimating the population proportion.

• The sample variance is an unbiased statistic for estimating the population variance.


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Minimum-Variance Unbiased Estimator

If several statistics from which to choose, select the one with the smallest possible variance.

If one of these statistics has the smallest possible variance, it is called the minimum-variance unbiased estimator (MVUE).

Remarks

If the underlying population is normal, the sample mean is the MVUE for estimating the population mean.

So, if the population is normal, the sample mean is a really good statistic to use for estimating the populations mean.


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Confidence Interval (CI)

A confidence interval (CI) for a population parameter is an interval of values constructed so that, with a specified degree of confidence, the value of the population parameter lies in this interval.


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Confidence Coefficient

The confidence coefficient is the probability the CI encloses the population parameter in repeated samplings.


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Confidence Level

The confidence level is the confidence coefficient expressed as a percentage.


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Critical Value

zα/2 is a critical value. It is a value

on the measurement axis in a

standard normal distribution such

that

P(Z ≥ zα/2) = α/2.


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Remarks

1. The subscript on z could be any variable, or letter.

2. Z /2: a z value such that there is /2 of the area (probability) to the right of z /2.

z /2 : the negative critical value.

Remarks (cont.)

3. Critical values are always defined in terms of right-tail probability.


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Remarks (cont.)

4. zα/2 critical values are easy to find by using the complement rule and working backward.

P(Z ≥ z /2) = /2

P(Z ≤ z /2) = 1 /2

Work backward in Table 3 to find z

/2.Copyright 2011 by W. H.

Freeman and Company. All rights reserved. 17

Critical Values


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Example: Critical Values for 95% Confidence Level


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Constructing a General 100(1 )% Confidence Interval for a Population Mean

1. Find a symmetric interval about 0 such that the probability Z lies in this interval is 1 .

P(zα/2 < Z < zα/2) = 1 .


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2. Substitute for Z.


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12/2/ z

n

XzP


3. Manipulate this equation to obtain the probability statement.


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12/2/n

zXn

zXP

Critical Values


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Illustration of a (1 ) 100% CI


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Illustration of 95% CI


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How to Find a 100(1 α)% CI for µ When σ is Known

Given a random sample of size n from a population with mean µ, if

1. the underlying population distribution is normal and/or n is large, and

2. the population standard deviation σ is known, then a 100(1 α)% confidence interval for µ has as endpoints the values

nzx

2


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Remarks

1. This CI for µ can only be used if σ is known.

2. If n large and σ unknown, some statisticians substitute s for σ. This produces an approximate confidence interval.Next section: an exact confidence interval for when is unknown.

3. As confidence coefficient increases (n, constant), CI is wider.


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Concepts to Remember

1. The population parameter, µ, is fixed.

The confidence interval varies from sample to sample.

Correct statement: We are 95% confident the interval captures the true mean µ.

Incorrect statement: We are 95% confident µ lies in the interval.


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Concepts to Remember (cont.)

2. Confidence coefficient: a probability, a long-run limiting relative frequency.

In repeated samples, the proportion of confidence intervals that capture the true value of µ approaches the confidence coefficient, in this case, 0.95.

Cannot be certain about any one specific confidence interval.

The confidence is in the long-run process.


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Example: Tire Weight

The total weight of a filled tire can dramatically affect the performance and safety of an automobile. Some transportation officials argue that mechanics should check the tire weights of every vehicle as part of an annual inspection.


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Example (cont.)

Suppose the weight of a 185/60/14 filled tire is normally distributed

with standard deviation 1.25 pounds. In a random sample of 15 filled tires, the sample mean weight was 18.75 pounds. Find a 95% confidence interval for the true mean weight of 185/60/14 tires.


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Example (cont.)

nzx

025.0


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Example (cont.)


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63.075.1815

25.1)96.1(75.18

Example (cont.)

(18.12, 19.38) is a 95% confidence interval for the true mean weight (in pounds) of 185/60/14 tires.


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Necessary Sample Size

1. Suppose n is large (unknown), is known, confidence level is 100(1 )%.

2. Desired width W. B = W/2: bound on the error of estimation.B is half the width of the confidence interval.

Necessary Sample Size (cont.)

3. Confidence interval endpoints:


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nzx

2/



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nzB

2/


4. Solve for n and the resulting formula is:


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2

2/

B

zn


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A Confidence Interval for a Population Mean When σ Is Unknown

Confidence interval for µ based on Z: valid only when σ is known which is unrealistic.

Theorem


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Comparison of Density Curves


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Properties of t Distribution


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Critical Value

tα is a critical value related to a t

distribution with df degrees of

freedom. If T has a t distribution

with df degrees of freedom then

P(T ≥ t α) = α.

Illustration of Critical Values


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Example: Visualization of t0.01 = 2.5176.


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Remarks

tα:a t value such that there is of

the area to the right of tα

tα: the negative critical value Because the t distribution is

symmetric, P(T ≤ tα) = P(T ≥ tα) = α.

How to Find a 100(1 α)% CI for µ When σ is Unknown


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Remarks

1. This CI can be used with any sample size n (≥ 2).This produces an exact CI for µ.

2. This CI for µ is valid only if the underlying population is normal.

A Large-Sample CI for a Population Proportion

Let p = true population proportion of a success.

Use the sample proportion to construct a CI for p.


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A Large-Sample CI for a Population Proportion

Sample of n individuals.

X = number of individuals with the characteristic or number of successes.


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Sample Proportion


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Standardize the Sample Proportion


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A Large-Sample Confidence Interval for a Population Proportion


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Example: Americans with Hypertension

High blood pressure, or hypertension, occurs when the force of blood against the artery walls is too strong. In a recent survey, 1100 adult Americans were randomly selected and examined for high blood pressure. The number of patients classified with hypertension

is 319. Copyright 2011 by W. H. Freeman and Company. All

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Example (cont.)

Find a 95% confidence interval for the true proportion of adult Americans with

hypertension.


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1100

)29.01(29.096.129.0

)ˆ1(ˆˆ 2/

n

ppzp

Example (cont.)

= 0.29 ± 0.0268 = (0.2632, 0.3168)

(0.2632, 0.3168) is a 95% confidence interval for the true proportion, p, of adult Americans with hypertension.


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Necessary Sample Size

1. Suppose n is large (unknown), confidence level is 100(1 )%, bound on the error of estimation is B.


2. The bound on the error of estimation is the step in each direction:


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n

ppzB

)ˆ1(ˆ2/


3. Solve for n and the resulting formula is:


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2

2/)ˆ1(ˆ

B

zppn


Problem: sample proportion is unknown. Don't know sample proportion until we have n.


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Solutions:1. Use a reasonable estimate for sample

proportion from previous experience.2. If no prior information is available, use

0.5 as the estimate of the sample proportion.

This produces a very conservative, large value of n.Copyright 2011 by W. H.



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A Confidence Interval for a Population Variance

1. Seems reasonable to use S 2 as an estimator for σ 2.

2. A CI for σ 2 is based on a new standardization and a chi-square distribution.

3. Chi-square (2) distribution:a) Positive probability only for non-negative

values.b) Focus on the properties and a method for

finding critical values.


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Properties of a Chi-Square Distribution

1. A chi-square distribution is completely determined by one parameter, df, the number of degrees of freedom, a positive integer (df = 1, 2, 3, 4,…). There is a different chi-square distribution corresponding to each value of df.


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Properties of a Chi-Square Distribution (cont.)

2. If X has a chi-square distribution with df degrees of freedom, denoted X ~df

2, then µX = df and σX

2= 2df.

The mean of X is df, the number of degrees of freedom, and the variance is 2 twice the number of degrees of freedom.


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Properties of a Chi-Square Distribution (cont.)

3. Suppose X ~ df2. The density

curve for X is positively skewed (not symmetric), and as x increases it gets closer and closer to the x axis but never touches it. As df increases, the density curve becomes flatter and actually looks more normal.

Examples of Density Curvers for Chi-Square Distributions


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Critical Values


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Example: Visualization of 2

0.05 = 18.3070


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x

Example: Visualization of 2

0.99 = 1.2390.


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x


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Theorem Let S 2 be the sample variance of a

random sample of size n from a normal distribution with variance σ 2. The random variable

has a chi-square distribution with n 1 degrees of freedom.

2

2)1( SnX

CI for a Population Variance


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Example: Kiln-Fired Dishes

Earthenware dishes are made from clay

and are fired, or exposed to heat, in a large kiln. Large fluctuations in the kiln temperature can cause cracks, bumps, or other flaws (and increase cost).


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Example (cont.)

With the kiln set at 800C, a random sample of 19 temperature measurements (in C) was obtained.

The sample variance was s2 = 17.55.


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Example (cont.)

a) Find a 95% confidence interval for the true population variance in temperature of the kiln when it is set to 800C. Assume that the underlying distribution is normal.


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Example (cont.)

s2 = 17.55, n = 19, df = 19 1 =18

2 /2 = 20.025 = 31.5264

21- /2 = 2

0.975 = 8.2307


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x

x

x

x

Example (cont.):Left Endpoint


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0202.105264.31

)55.17)(18()1(2

2/

2

sn

Example (cont.):Right Endpoint


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3807.382307.8

)55.17)(18()1(2

2/1

2

sn

Example (cont.)

(10.0202, 38.3807) is a 95% confidence interval for the true population variance in temperature when the kiln is set to 800C.


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Example (cont.)

b) Quality control engineers have determined that the maximum variance in temperature during firing should be 16C. Using the confidence interval constructed in part a), is there any evidence to suggest that the true temperature variance is greater than 16C? Justify your answer.Copyright 2011 by W. H.


Example (cont.)

Claim: 2 = 16 Experiment: s2 = 17.55 Likelihood: The likelihood is

expressed as a 95% confidence interval, an interval of likely values for 2, (10.0202, 38.3807), from part a).


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Example (cont.)

Conclusion: The required kiln temperature variance, 16C, is included in this confidence interval. There is no evidence to suggest that 2 is different from 16C.


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Example (cont.)


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Remarks

1. This CI for σ 2 is valid only if the underlying population is normal.

2. Take the square root of each endpoint to obtain a 100(1 α)% CI for σ.

Copyright 2011 by W. H. Freeman and Company. All rights reserved.1 Introductory Statistics: A...

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Transcript of Copyright 2011 by W. H. Freeman and Company. All rights reserved.1 Introductory Statistics: A...