Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 8-1 Chapter 8.

Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall8-1

Chapter 8

Confidence Intervals


Population Mean

σ Unknown

ConfidenceIntervals

PopulationProportion

σ Known

Section 8.2

Section 8.4

Section 8.3


• A confidence interval for the mean

– interval (or range) around the sample mean ( ) within which true (population) mean (µ) is expected to be

– µ = 362.3 ± 1.96 * 15 =

• A confidence level (dependent on Z(α))

– probability that the interval estimate will include the population parameter of interest (1-α)

– Here α = .05 => 95% confidence

8.2 Calculating Confidence Intervals for the Mean when the Standard Deviation (σ) of a Population is Known

2α / xx z σ

x

Confidence Intervals for the Mean, σ Known

• Assumptions for section 8.2:– The sample size is at least 30 (n ≥ 30)– The population standard deviation (σ) is known

• Recall from Chapter 7:– Formula for the Standard Error of the Mean


n

σσ x

whereσ = Population standard deviationn = Sample size

• Formulas for the Confidence Interval for the Mean (σ Known)



xα/x

xα/x

σzxLCL

σzxUCL

2

2

mean the of error standard The

score- critical The

mean sample The

where

x

α/

σ

zz

x

2

• is called the critical z-score

• The variable is known as the significance level

• Example: if = .10, then = 1.645 is the value that encloses 90% of the area under the normal distribution and leaves 5% in each tail– The total area to the left of the

right-hand boundary is

0.90 + 0.05 = 0.95– The total area to the left

of the left-hand

boundary is 0.05


2/z

05.02/ zz

α/2 = 0.050.90

z0-1.645 1.645


α/2 = 0.05

0.95

0.05

Calculating the Margin of Error

• The Margin of Error ME is the amount added and subtracted to the point estimate to form the confidence interval


Point EstimateLower Confidence Limit

UpperConfidence Limit

Width of confidence interval

x

x

xα/xx

MEx

σzxLCLUCL

2 , xα/x σzME 2

Margin of Error Margin of Error

Calculating the Margin of Error

• Increasing the sample size while keeping the confidence level constant will reduce the margin of error, resulting in a narrower (more precise) confidence interval


xx MEσn

Interpreting a Confidence Interval

• We are 90% confident that the true mean is between 137.10 and 153.90

– Population mean (µ) may/may not be in this interval 90% of the sample means drawn from samples

of this population will produce confidence intervals that include that population’s mean

• An incorrect interpretation is that there is 90% probability that this interval contains the true population mean

– (This interval either does or does not contain the true mean, there is no probability for a single interval)



xμμ


Each interval extends from

to

But varies from sample to sample

For 90% confidence, 90% of intervals constructed contain μ ; 10% do not

xα/ σzx 2x1

x2

/2 /21

xα/ σzx 2

x

Interpreting a Confidence Interval

x

Changing Confidence Levels

• The significance level, ,

– Probability any given confidence interval will not contain the true population mean

• The confidence level of an interval is the complement to the significance level, 1 ─ – i.e., a 100(1 – )% confidence interval has a

significance level equal to

• The confidence interval gets wider if the confidence level increases (as Z(α) increases)


• encloses 95% of the area under the

curve, with 2.5% in each tail


• Consider a 95% confidence interval:

0.951

0

1.96z


0.025/2 0.025/2

-1.96/2 z -1.96/2 z0.975

Confidence Level:

Significance Level:

Critical z-score:

80%

90%

95%

98%

99%

20%

10%

5%

2%

1%

z0.10 = ±1.28

z0.05 = ±1.645

z0.025 = ±1.96

z0.01 = ±2.33

z0.005 = ±2.575

• z-scores for the most commonly used confidence levels are shown in this table:


)%100(1 )%(100 2/z


Using Excel to Determine Confidence Intervals for the Mean (σ Known)

• Excel’s CONFIDENCE function calculates the margin of error for confidence intervals

• The CONFIDENCE function has the following characteristics:

=CONFIDENCE (alpha, standard_dev, size)

where: alpha = The significance level of the confidence interval

standard_dev = The standard deviation of the population

size = Sample size


Confidence Intervals for the Mean with Small Samples when σ is Known

• When the sample size is less than 30 and sigma is known, the population must be normally distributed to calculate a confidence interval

– With n < 30 the Central Limit Theorem cannot be applied, so we can’t say the sampling distribution will be approximately normal…

– …but the sampling distribution is always normal (regardless of sample size) if the population is normally distributed



• Population standard deviation is unknown, – Use s, the sample standard deviation, in its place

• calculate the standard error (of the mean)

• Formula for the Sample Standard Deviation (recall from Chapter 3):

8.3 Calculating Confidence Intervals when the Population Standard Deviation (σ) is Unknown

11

2

n

)x(xs

n

ii

where: x = The sample mean n = The sample size (number of data values) (xi – x ) = The difference between each data value and the sample mean



Population Mean

σ Unknown

ConfidenceIntervals


σ Known

Section 8.2

Section 8.4

Section 8.3

Using the Student’s t-distribution

• Formula for the Approximate Standard Error of the Mean

• The Student’s t-distribution – used instead of normal probability distribution when

the sample standard deviation, s, is used in place of the population standard deviation, σ


n

sσ x ˆ


• Formulas for the Confidence Interval for the Mean (σ Unknown)


xα/x

xα/x

σtxLCL

σtxUCL

ˆ

ˆ

2

2

where:

= The sample mean

= The critical t-score

= The approximate standard error of the meanx

α/

σ

t

x

ˆ2



t

t (df = 5)

t (df = 13)

t-distributions are bell-shaped and symmetric, but have ‘fatter’ tails than the normal

Normal distribution


• The t-distribution is a continuous probability distribution with the following properties:

– It is bell-shaped and symmetrical around the mean

– The shape of the curve depends on the degrees of freedom (df), df = n – 1

– The area under the curve is equal to 1.0

– The t-distribution is flatter and wider than the normal distribution

– The critical score for the t-distribution is greater than the critical z-score for the same confidence level


• Comparing t-scores and z-scores:


Confidence t t t z Level (10 df ) (20 df ) (30 df )

.80 1.372 1.325 1.310 1.28

.90 1.812 1.725 1.697 1.645

.95 2.228 2.086 2.042 1.96

.99 3.169 2.845 2.750 2.575

Note: t z as n increases



• The t-distribution is actually a family of distributions. As the number of degrees of freedom increases, the shape of the t-distribution becomes similar to the normal distribution

– With more than 100 degrees of freedom (a sample size of more than 100), the two distributions are practically identical


Using Excel and PHStat2 to Determine Confidence Intervals for the Mean (σ Unknown)

• The critical t-score can be found with Excel’s TINV function, which has the following characteristics:

=TINV (alpha, degrees of freedom)


where: alpha ( ) = The significance level of the confidence interval degrees of freedom = n - 1 n = Sample size



Population Mean

σ Unknown

ConfidenceIntervals


σ Known

Section 8.2

Section 8.4

Section 8.3


• Proportion data follow the binomial distribution, which can be approximated by the normal distribution under the following conditions:

nπ ≥ 5 and n(1 – π) ≥ 5

where:

π = The probability of a success in the population

n = The sample size

8.4 Calculating ConfidenceIntervals for Proportions

Calculating Confidence Intervals for Proportions

• The confidence interval for the proportion is an interval estimate around a sample proportion that provides us with a range of where the true population proportion lies


n

xp

Formula for the Standard Error of the Proportion:

Formula for the Sample Proportion:

nσ p

)1(

where: π = The population proportion x = The number of observations of interest in the sample (successes) n = The sample size


• Since the population proportion π is unknown, it is estimated using the sample proportion, p

• Formula for the Approximate Standard Error of the Proportion:


n

ppσ p

)1(ˆ

where: p = The sample proportion n = The sample size


• Formulas for the Confidence Interval for a Proportion:


where: p = The sample proportion = The critical z-score

= The approximate standard error of the proportion

p

p

σzpLCL

σzpUCL

α/p

α/p

ˆ

ˆ

2

2

pσ

zα/ˆ2


• Formula for the Margin of Error for a Confidence Interval for the Proportion


p

p

MEp

σzpLCLUCL α/pp

ˆ, 2

pα/p σzME ˆ2

Example: From a random sample of U.S. citizens, 22 of 100 people are found to have blue eyes.

Calculate a 98% confidence interval for the population proportion of blue eyes for U.S. citizens

1. Calculate the sample proportion and the approximate standard error

of the proportion:

2. Find for 98%:

3. Calculate the interval endpoints: (next slide)


2/z 2.33 01.02/ zz


0.041100

0.22)0.22(1

n

ppσ p

)1(ˆ0.22

100

22

n

xp


• Increasing the sample size, holding all else constant, reduces the margin of error and provides a narrower confidence interval

• The sample size needed to achieve a specific margin of error can be calculated, given the following information:

– The confidence level

– The population standard deviation

8.5 Determining the Sample Size

Calculating the Sample Size to Estimate a Population Mean

• Solving for n:


x

α/

α/x

xα/x

ME

σzn

n

σzME

σzME

2

2

2

so

2

222

)(

)(

x

α/

ME

σzn

Formula for the Sample Size Needed to Estimate a Population Mean:

Calculating the Sample Size Needed to Estimate a Population Proportion

• Solving for n:


p

α/

α/p

pα/p

ME

ppzn

n

ppzME

σzME

)1(

)1(

ˆ

2

2

2

so

Formula for the Sample Size Needed to Estimate a Population Proportion:

2

22

)(

)1()(

p

α/

ME

ppzn


• In order to calculate the required sample size to estimate π, the population proportion, we need to know p, the sample proportion

– Select a pilot sample and use the sample proportion, p

– If it is not possible to select a pilot sample, set p = 0.5

• Setting it equal to 0.50 provides the most conservative estimate for a sample size (a sample size that is at least large enough to satisfy the margin-of-error requirement)


Example: What sample size is needed to estimate with 95% confidence the population proportion of U.S. citizens with blue eyes within a margin of error of ± 5%? Assume a pilot sample of 100 people found 22 with blue eyes.

1.Find for 95%:

2.Calculate the required sample size:

so use a sample of size n = 264 (always round up)


2/z 1.96 025.02/ zz


263.69(0.05)

0.22)(0.22)(1(1.96)2

2

2

22

)(

)1()(

p

α/

ME

ppzn

Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 8-1 Chapter 8.

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