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Statistics for Business and Economics

Chapter 6 Inferences Based on a Single Sample

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Content1. Identifying and Estimating the Target Parameter2. Confidence Interval for a Population Mean:

Normal (z) Statistic3. Confidence Interval for a Population Mean:

Student’s t-Statistic4. Large-Sample Confidence Interval for a

Population Proportion5. Determining the Sample Size6. Finite Population Correction for Simple Random

Sampling7. Confidence Interval for a Population Variance

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Learning Objectives

1. Estimate a population parameter (means, proportion, or variance) based on a large sample selected from the population

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

3. Show how to select the proper sample size for estimating a population parameter

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Thinking ChallengeSuppose you’re interested in the average amount of money that students in this class (the population) have on them. How would you find out?

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Statistical Methods

StatisticalMethods

Estimation HypothesisTesting

InferentialStatistics

DescriptiveStatistics

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6.1

Identifying and Estimatingthe Target Parameter

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Estimation Methods

Estimation

IntervalEstimation

PointEstimation

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Target Parameter

The unknown population parameter (e.g., mean or proportion) that we are interested in estimating is called the target parameter.

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Target Parameter

Determining the Target Parameter

Parameter Key Words of Phrase Type of Data

µ Mean; average Quantitative

p Proportion; percentagefraction; rate Qualitative

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Point Estimator

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 as an estimate of the target parameter.

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Point Estimation1. Provides a single value

• Based on observations from one sample

2. Gives no information about how close the value is to the unknown population parameter

3. Example: Sample mean x = 3 is the point estimate of the unknown population mean

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Interval Estimator

An interval estimator (or confidence interval) is a formula that tells us how to use the sample data to calculate an interval that estimates the target parameter.

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Interval Estimation1. Provides a range of values

• Based on observations from one sample

2. Gives information about closeness to unknown population parameter• Stated in terms of probability

– Knowing exact closeness requires knowing unknown population parameter

3. Example: Unknown population mean lies between 50 and 70 with 95% confidence

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6.2

Confidence Interval for a Population Mean:

Normal (z) Statistic

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Estimation Process

Mean, , is unknown

Population

Sample

Random Sample

I am 95% confident that is between 40 &

60.

Meanx = 50

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Key Elements of Interval Estimation

Sample statistic (point estimate)

Confidence interval

Confidence limit (lower)

Confidence limit (upper)

A confidence interval provides a range of plausible values for the population parameter.

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Confidence IntervalAccording to the Central Limit Theorem, the sampling distribution of the sample mean is approximately normal for large samples. Let us calculate the interval estimator:

x 1.96 x x 1.96

nThat is, we form an interval from 1.96 standard deviations below the sample mean to 1.96 standard deviations above the mean. Prior to drawing the sample, what are the chances that this interval will enclose µ, the population mean?

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Confidence IntervalIf sample measurements yield a value of that falls between the two lines on either side of µ, then the interval will contain µ.

The area under the normal curve between these two boundaries is exactly .95. Thus, the probability that a randomly selected interval will contain µ is equal to .95.

x

x 1.96 x

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Confidence CoefficientThe confidence coefficient is the probability that a randomly selected confidence interval encloses the population parameter - that is, the relative frequency with which similarly constructed intervals enclose the population parameter when the estimator is used repeatedly a very large number of times. The confidence level is the confidence coefficient expressed as a percentage.

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95% Confidence LevelIf our confidence level is 95%, then in the long run, 95% of our confidence intervals will contain µ and 5% will not.For a confidence coefficient of 95%, the area in the two tails is .05. To choose a different confidence coefficient we increase or decrease the area (call it ) assigned to the tails. If we place /2 in each tailand z/2 is the z-value, the confidence interval with coefficient (1 – ) is

x z 2 x .

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Conditions Required for a Valid Large-Sample

Confidence Interval for µ

1. A random sample is selected from the target population.

2. The sample size n is large (i.e., n ≥ 30). Due to the Central Limit Theorem, this condition guarantees that the sampling distribution of is approximately normal. Also, for large n, s will be a good estimator of .

x

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

where z/2 is the z-value with an area /2 to its right and in the standard normal distribution. The parameter is the standard deviation of the sampled population, and n is the sample size.Note: When is unknown and n is large (n ≥ 30), the confidence interval is approximately equal to

where s is the sample standard deviation.

x z 2 x x z 2n

x z 2sn

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Thinking ChallengeYou’re a Q/C inspector for Gallo. The for 2-liter bottles is .05 liters. A random sample of 100 bottles showed x = 1.99 liters. What is the 90% confidence interval estimate of the true mean amount in 2-liter bottles?

2 liter

© 1984-1994 T/Maker Co.

2 liter

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Confidence Interval Solution*

x z /2

n x z /2

n

1.99 1.645.05

100 1.99 1.645

.05

100

1.982 1.998

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6.3

Confidence Interval for a Population Mean:

Student’s t-Statistic

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Small Sample Unknown

Instead of using the standard normal statistic

use the t–statistic

z x µ x

x µ n

t x µs n

in which the sample standard deviation, s, replaces the population standard deviation, .

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Student’s t-StatisticThe t-statistic has a sampling distribution very much like that of the z-statistic: mound-shaped, symmetric, with mean 0.

The primary difference between the sampling distributions of t and z is that the t-statistic is more variable than the z-statistic.

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Degrees of Freedom

The actual amount of variability in the sampling distribution of t depends on the sample size n. A convenient way of expressing this dependence is to say that the t-statistic has (n – 1) degrees of freedom (df).

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zt

Student’s t Distribution

0

t (df = 5)

Standard Normal

t (df = 13)Bell-Shaped

Symmetric

‘Fatter’ Tails

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t - Table

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t-valueIf we want the t-value with an area of .025 to its right and 4 df, we look in the table under the column t.025 for the entry in the row corresponding to 4 df. This entry is t.025 = 2.776. The corresponding standard normal z-score is z.025 = 1.96.

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Small-SampleConfidence Interval for µ

where ta/2 is based on (n – 1) degrees of freedom.

x t 2sn

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Conditions Required for a Valid Small-Sample

Confidence Interval for µ

1. A random sample is selected from the target population.

2. The population has a relative frequency distribution that is approximately normal.

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Estimation Example Mean ( Unknown)

x t /2 sn

x t /2 sn

50 2.064 825

50 2.064 825

46.70 53.30

A random sample of n = 25 has = 50 and s = 8. Set up a 95% confidence interval estimate for .

x

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Thinking ChallengeYou’re a time study analyst in manufacturing. You’ve recorded the following task times (min.): 3.6, 4.2, 4.0, 3.5, 3.8, 3.1. What is the 90% confidence interval estimate of the population mean task time?

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Confidence Interval Solution*• x = 3.7

• s = 3.8987

• n = 6, df = n – 1 = 6 – 1 = 5

• t.05 = 2.015

3.7 2.015.38987

6 3.7 2.015

.38987

6.492 6.908

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6.4

Large-Sample Confidence Interval for a Population

Proportion

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Sampling Distribution of

1. The mean of the sampling distribution of is p; that is, is an unbiased estimator of p.

pp

3. For large samples, the sampling distribution of is approximately normal. A sample size is considered large if both np 15 and nq 15.

p

2. The standard deviation of the sampling distribution of is ; that is, where q = 1–p.

pq np p pq n

p

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Large-Sample Confidence Interval for

where

p z 2 p p z 2 pqn

p z 2 pqn

p xn

and q 1 p.

Note: When n is large, can approximate the value of p in the formula for .

p p

p

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2. The sample size n is large. (This condition will be satisfied if both . Note that and are simply the number of successes and number of failures, respectively, in the sample.).

np 15 and nq 15 npnq

Conditions Required for a Valid Large-Sample

Confidence Interval for p1. A random sample is selected from the target population.

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Estimation Example Proportion

A random sample of 400 graduates showed 32 went to graduate school. Set up a 95% confidence interval estimate for p.

/2 /2

ˆ ˆ ˆ ˆ 32ˆ ˆ ˆ 0.08400

.08 .92 .08 .92.08 1.96 .08 1.96

400 400

.053 .107

pq pqp Z p p Z pn n

p

p

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Thinking ChallengeYou’re a production manager for a newspaper. You want to find the % defective. Of 200 newspapers, 35 had defects. What is the 90% confidence interval estimate of the population proportion defective?

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Confidence Interval Solution*

/2 /2

ˆ ˆ ˆ ˆˆ ˆ

.175(.825) .175(.825).175 1.645 .175 1.645200 200

.1308 .2192

p q p qp z p p zn n

p

p

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Adjusted (1 – )100% Confidence Interval for a Population Proportion, p

where is the adjusted sample proportion of observations with the characteristic of interest, x is the number of successes in the sample, and n is the sample size.

2

14

p p

p zn

p

x 2n 4

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6.5

Determining the Sample Size

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Sampling Error

In general, we express the reliability associated with a confidence interval for the population mean µ by specifying the sampling error within which we want to estimate µ with 100(1 – )% confidence. The sampling error (denoted SE), then, is equal to the half-width of the confidence interval.

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Sample Size Determination for 100(1 – ) %

Confidence Interval for µIn order to estimate µ with a sampling error (SE) and with 100(1 – )% confidence, the required sample size is found as follows:

z 2n

SE

The solution for n is given by the equation

n z 2 2 2

SE 2

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Sample Size ExampleWhat sample size is needed to be 90% confident the mean is within 5? A pilot study suggested that the standard deviation is 45.

n (z 2 )2 2

(SE) 2 1.645 2

45 2

5 2 219.2 220

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Sample Size Determination for 100(1 – ) %

Confidence Interval for pIn order to estimate p with a sampling error SE and with 100(1 – )% confidence, the required sample size is found by solving the following equation for n:

z 2pqn

SE

The solution for n can be written as follows:

n z 2 2 pq

SE 2

Note: Always round n up to the nearest integer value.

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Sample Size ExampleWhat sample size is needed to estimate p within .03 with 90% confidence?

.03 .0152 2

widthSE

n (Z 2 )2 pq

(SE) 2 1.645 2 .5.5

.015 2 3006.69 3007

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Thinking ChallengeYou work in Human Resources at Merrill Lynch. You plan to survey employees to find their average medical expenses. You want to be 95% confident that the sample mean is within ± $50. A pilot study showed that was about $400. What sample size do you use?

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Sample Size Solution*

n (z 2 )2 2

(SE)2

1.96 2

400 2

50 2

245.86 246

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6.6

Finite Population Correction for Simple Random Sample

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Finite Population Correction Factor

In some sampling situations, the sample size n may represent 5% or perhaps 10% of the total number N of sampling units in the population. When the sample size is large relative to the number of measurements in the population (see the next slide), the standard errors of the estimators of µ and p should be multiplied by a finite population correction factor.

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Rule of Thumb for Finite Population Correction Factor

Use the finite population correction factor when n/N > .05.

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Simple Random Sampling with Finite Population of Size N

Estimation of the Population Mean

Estimated standard error:

Approximate 95% confidence interval:

x sn

N nN

ˆ2 xx

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Simple Random Sampling with Finite Population of Size N

Estimation of the Population Proportion

Estimated standard error:

Approximate 95% confidence interval:p 2 p

p p(1 p)

nN n

N

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Finite Population Correction Factor Example

You want to estimate a population mean, μ, wherex =115, s =18, N =700, and n = 60. Find an approximate 95% confidence interval for μ.

is greater than .05 use the finite correction factor

086.70060 N

nSince

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Finite Population Correction Factor Example

You want to estimate a population mean, μ, wherex =115, s =18, N =700, and n = 60. Find an approximate 95% confidence interval for μ.

x 2sn

N nN

115 2 1860

700 60700

115 4.4

110.6, 119.4

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6.7

Confidence Interval for a Population Variance

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Confidence Interval for a Population Variance

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Conditions Required for a Valid Confidence Interval for 2

1. A random sample is selected from the target population.

2. The population of interest has a relative frequency distribution that is approximately normal.

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Thinking ChallengeYou’re a marketing manager for a 5K race. You take a random sample of the times of 292 runners from the last race, with mean of 28.5 minutes and standard deviation of 8.3 minutes. What is the 95% confidence interval estimate of the population variance?

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Confidence Interval Solution*

2

2 22

2 21 2

2 22

2

1 1

292 1 8.3 292 1 8.3349.874 253.912

57.30 78.95

n s n s

df = 292 1 = 291 (use 300 df) .0252

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Key IdeasPopulation Parameters, Estimators, and Standard Errors

Parameter Estimator Standard Error of

Estimator

Estimated Std Error

Mean, µ

Proportion, p pq n

pq np

s n nx

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Key Ideas

Population Parameters, Estimators, and Standard ErrorsConfidence Interval: An interval that encloses an unknown population parameter with a certain level of confidence (1 – )

Confidence Coefficient: The probability (1 – ) that a randomly selected confidence interval encloses the true value of the population parameter.

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Key IdeasKey Words for Identifying the Target Parameter

µ – Mean, Average

p – Proportion, Fraction, Percentage, Rate, Probability

2 - Variance

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Key Ideas

Commonly Used z-Values for a Large-Sample Confidence Interval

90% CI: (1 – ) = .10 z.05 = 1.645

95% CI: (1 – ) = .05 z.025 = 1.96

98% CI: (1 – ) = .02 z.005 = 2.326

99% CI: (1 – ) = .01 z.005 = 2.575

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Key Ideas

Determining the Sample Size n

2 22

2 MEn zEstimating µ:

Estimating p: 2 2

2 MEn z pq

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Key Ideas

Finite Population Correction Factor

Required when n/N > .05

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Key IdeasConfidence Interval for Population Variance

Uses chi-square (2) distribution

Need to know and df.2

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Key IdeasIllustrating the Notion of “95% Confidence”

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Key IdeasIllustrating the Notion of “95% Confidence”