IS 310 Business Statistics CSU Long Beach

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IS 310 – Business StatisticsIS 310 – Business Statistics

IS 310

Business Statistic

sCSU

Long Beach

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Interval EstimationInterval Estimation

In chapter 7, we studied how to estimate a population In chapter 7, we studied how to estimate a population parameter with a sample statistic. We used a point estimate parameter with a sample statistic. We used a point estimate as a single value. as a single value.

To increase the level of confidence in estimation, we use a To increase the level of confidence in estimation, we use a range of values (rather than a single value) as the estimate range of values (rather than a single value) as the estimate of a population parameter.of a population parameter.

Let’s take a real-life example. Suppose, someone asks you how Let’s take a real-life example. Suppose, someone asks you how long it takes to go from CSULB to LAX. What would be a long it takes to go from CSULB to LAX. What would be a more reliable estimate – 30 minutes or between 25 and 40 more reliable estimate – 30 minutes or between 25 and 40 minutes? minutes?

If you use 30 minutes as estimate, you are using a point If you use 30 minutes as estimate, you are using a point estimate. On the other hand, if you use between 25 and 40 estimate. On the other hand, if you use between 25 and 40 minutes as estimate, you are using an interval estimationminutes as estimate, you are using an interval estimation..

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Interval estimation uses a range of values.Interval estimation uses a range of values.

The width of the range indicates the level of The width of the range indicates the level of confidence. The narrower the range, lower the confidence. The narrower the range, lower the confidence. Wider the range, higher the confidence. Wider the range, higher the confidence.confidence.

Once a confidence level is specified, the interval Once a confidence level is specified, the interval estimate can be calculated using the formula, 8.1, estimate can be calculated using the formula, 8.1, in the book. If one wants 95% confidence, the in the book. If one wants 95% confidence, the interval estimate is known as 95% Confidence interval estimate is known as 95% Confidence Interval. For a 90% confidence, the interval is Interval. For a 90% confidence, the interval is known as 90% Confidence Interval.known as 90% Confidence Interval.

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In this chapter, we will cover the following:In this chapter, we will cover the following:

Interval Estimate for a Population Mean Interval Estimate for a Population Mean (known (known σσ))

Interval Estimate for a Population Mean Interval Estimate for a Population Mean (unknown (unknown σσ))

Determining Size of a SampleDetermining Size of a Sample

Interval Estimate for a Population Proportion Interval Estimate for a Population Proportion

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The general form of an interval estimate of aThe general form of an interval estimate of a population mean ispopulation mean is

The general form of an interval estimate of aThe general form of an interval estimate of a population mean ispopulation mean is

Margin of Errorx Margin of Errorx

Margin of Error and the Interval EstimateMargin of Error and the Interval Estimate

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Margin of ErrorMargin of Error

The Margin of Error implies bothThe Margin of Error implies both

Confidence level andConfidence level and

Amount of error Amount of error

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Interval Estimate ofInterval Estimate of

Interval Estimate of a Population Mean:Interval Estimate of a Population Mean: Known Known

x zn

/2x zn

/2

where: is the sample meanwhere: is the sample mean 1 -1 - is the confidence coefficient is the confidence coefficient zz/2 /2 is the is the zz value providing an area of value providing an area of /2 in the upper tail of the standard /2 in the upper tail of the standard

normal probability distributionnormal probability distribution is the population standard deviationis the population standard deviation nn is the sample size is the sample size

xx

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Interval Estimate for Population Mean Interval Estimate for Population Mean (Known (Known σσ))

Example:Example: Lloyd’s Department Store wants to determine Lloyd’s Department Store wants to determine

a 95% confidence interval on the average a 95% confidence interval on the average amount spent by all of its customers amount spent by all of its customers (population mean).(population mean).

Lloyd took a sample of 100 customers and Lloyd took a sample of 100 customers and found that the average amount spent is found that the average amount spent is $82 (this is sample mean or x‾ ). Lloyd $82 (this is sample mean or x‾ ). Lloyd assumes that the population standard assumes that the population standard deviation ( deviation ( σσ ) is $20. ) is $20.

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Interval Estimation for Population MeanInterval Estimation for Population Mean(Known (Known σσ ) )

Using Formula 8.1, the 95% confidence interval is:Using Formula 8.1, the 95% confidence interval is: __ x ± z (x ± z (σσ/√ n) = 0.05/√ n) = 0.05 0.0250.025

82 ± 1.96 (20/√100)82 ± 1.96 (20/√100)

82 ± 3.9282 ± 3.92

78.08 to 85.9278.08 to 85.92

We are 95% sure that the average amount spent by all We are 95% sure that the average amount spent by all Lloyd customers is between $78.08 and $85.92.Lloyd customers is between $78.08 and $85.92.

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Sample ProblemSample Problem

Problem # 5 (10-Page 306; 11-Page 315)Problem # 5 (10-Page 306; 11-Page 315)

__

Given: x = 24.80 n = 49 Given: x = 24.80 n = 49 σσ = 5 confidence level = 95% = = 5 confidence level = 95% = 0.05 or 5%0.05 or 5%

a.a. Margin of Error = z x (Margin of Error = z x (σσ / √n) / √n)

/2/2

= 1.96 (5/√49) = 1.4= 1.96 (5/√49) = 1.4

b.b. 95% Confidence Interval is:95% Confidence Interval is:

_ _

x ± Margin of Error 24.8 ± 1.4 23.4, 26.2x ± Margin of Error 24.8 ± 1.4 23.4, 26.2

It means that we are 95% confident that the average amount It means that we are 95% confident that the average amount spent by all customers for dinner at the Atlanta restaurant is spent by all customers for dinner at the Atlanta restaurant is between $23.40 and $26.20between $23.40 and $26.20

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Interval Estimation of a Population Mean:Interval Estimation of a Population Mean: Unknown Unknown

If an estimate of the population standard If an estimate of the population standard deviation deviation cannot be developed prior to cannot be developed prior to sampling, we use the sample standard sampling, we use the sample standard deviation deviation ss to estimate to estimate . . This is the This is the unknown unknown case. case.

In this case, the interval estimate for In this case, the interval estimate for is based is based on the on the tt distribution. distribution.

(We’ll assume for now that the population is (We’ll assume for now that the population is normally distributed.)normally distributed.)

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The The t t distribution distribution is a family of similar probability is a family of similar probability distributions.distributions. The The t t distribution distribution is a family of similar probability is a family of similar probability distributions.distributions.

tt Distribution Distribution

A specific A specific tt distribution depends on a parameter distribution depends on a parameter known as the known as the degrees of freedomdegrees of freedom.. A specific A specific tt distribution depends on a parameter distribution depends on a parameter known as the known as the degrees of freedomdegrees of freedom..

Degrees of freedom refer to the sample size minus 1.Degrees of freedom refer to the sample size minus 1. Degrees of freedom refer to the sample size minus 1.Degrees of freedom refer to the sample size minus 1.

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A A tt distribution with more degrees of freedom has distribution with more degrees of freedom has less dispersion.less dispersion. A A tt distribution with more degrees of freedom has distribution with more degrees of freedom has less dispersion.less dispersion.

As the number of degrees of freedom increases, theAs the number of degrees of freedom increases, the difference between the difference between the tt distribution and the distribution and the standard normal probability distribution becomesstandard normal probability distribution becomes smaller and smaller.smaller and smaller.

As the number of degrees of freedom increases, theAs the number of degrees of freedom increases, the difference between the difference between the tt distribution and the distribution and the standard normal probability distribution becomesstandard normal probability distribution becomes smaller and smaller.smaller and smaller.

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StandardStandardnormalnormal

distributiondistribution

tt distributiondistribution(20 degrees(20 degreesof freedom)of freedom)

tt distributiondistribution(10 degrees(10 degrees

of of freedom)freedom)

00zz, , tt

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Degrees Area in Upper Tail

of Freedom .20 .10 .05 .025 .01 .005

. . . . . . .

50 .849 1.299 1.676 2.009 2.403 2.678

60 .848 1.296 1.671 2.000 2.390 2.660

80 .846 1.292 1.664 1.990 2.374 2.639

100 .845 1.290 1.660 1.984 2.364 2.626

.842 1.282 1.645 1.960 2.326 2.576

Standard Standard normalnormalzz values values

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Interval EstimateInterval Estimate

x tsn

/2x tsn

/2

where: 1 -where: 1 - = the confidence coefficient = the confidence coefficient

tt/2 /2 == the the tt value providing an area of value providing an area of /2/2 in the upper tail of a in the upper tail of a t t distribution distribution with with nn - 1 degrees of freedom - 1 degrees of freedom ss = the sample standard deviation = the sample standard deviation

Interval Estimation of a Population Mean:Interval Estimation of a Population Mean: Unknown Unknown

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Interval Estimate for Population Mean Interval Estimate for Population Mean (Unknown (Unknown σσ))

Example:Example: We want to estimate the mean credit card debt We want to estimate the mean credit card debt

of all customers in the U.S. with a 95% of all customers in the U.S. with a 95% confidence.confidence.

A sample of 70 households provided the credit A sample of 70 households provided the credit card balances shown in Table 8.3. The sample card balances shown in Table 8.3. The sample standard deviation (s) is $4007.standard deviation (s) is $4007.

Using formula 8.2,Using formula 8.2, __ x ± t (s/√n) = 0.05x ± t (s/√n) = 0.05 0.0250.025 9312 ± 1.995 (4007/√70)9312 ± 1.995 (4007/√70)

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Interval Estimate (Continued)Interval Estimate (Continued)

9312 ± 9559312 ± 955 8357 to 10,2678357 to 10,267

We are 95% confident that the average credit We are 95% confident that the average credit card balance of all customers in the U.S. are card balance of all customers in the U.S. are between $8,357 and $10,267. between $8,357 and $10,267.

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Summary of Interval Estimation Summary of Interval Estimation ProceduresProcedures

for a Population Meanfor a Population Mean

Can theCan thepopulation standardpopulation standard

deviation deviation be assumed be assumed known ?known ?

Use the sampleUse the samplestandard deviationstandard deviation

ss to estimate to estimate ss

UseUse

YesYes NoNo

/ 2

sx t

n / 2

sx t

nUseUse

/ 2x zn

/ 2x z

n

KnownKnownCaseCase

UnknownUnknownCaseCase

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Problem # 16 (10-Page 315; 11-Page 324)Problem # 16 (10-Page 315; 11-Page 324)

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Given: x = 49 n = 100 s = 8.5 Confidence Level = 90%Given: x = 49 n = 100 s = 8.5 Confidence Level = 90%

a.a. At 90% confidence, the Margin of Error is:At 90% confidence, the Margin of Error is:

t . (s/√n) 1.660 x (8.5/10) = 1.411 t . (s/√n) 1.660 x (8.5/10) = 1.411

/2 /2

b.b. The 90% Confidence Interval is:The 90% Confidence Interval is:

_ _

x ± 1.411 49 ± 1.411 47.59, 50.41x ± 1.411 49 ± 1.411 47.59, 50.41

It means that we are 90% confident that the average hours It means that we are 90% confident that the average hours of flying for Continental pilots is between 47.59 and 50.41of flying for Continental pilots is between 47.59 and 50.41

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Let Let EE = the desired margin of error. = the desired margin of error. Let Let EE = the desired margin of error. = the desired margin of error.

EE is the amount added to and subtracted from the is the amount added to and subtracted from the point estimate to obtain an interval estimate.point estimate to obtain an interval estimate. EE is the amount added to and subtracted from the is the amount added to and subtracted from the point estimate to obtain an interval estimate.point estimate to obtain an interval estimate.

Sample Size for an Interval EstimateSample Size for an Interval Estimateof a Population Meanof a Population Mean

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Sample Size for an Interval EstimateSample Size for an Interval Estimateof a Population Meanof a Population Mean

E zn

/2E zn

/2

nz

E

( )/ 22 2

2n

z

E

( )/ 22 2

2

Margin of ErrorMargin of Error

Necessary Sample SizeNecessary Sample Size

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Determination of Sample SizeDetermination of Sample Size

Example Problem (10-Page 317; 11-Page 326):Example Problem (10-Page 317; 11-Page 326): We want to know the average daily rental rate of all We want to know the average daily rental rate of all

midsize automobiles in the U.S. We want our estimate midsize automobiles in the U.S. We want our estimate with a margin of error of $2 and a 95% level of confidence.with a margin of error of $2 and a 95% level of confidence.

What should be the size of our sample?What should be the size of our sample? Given for this problem: E =2, = 0.05, Given for this problem: E =2, = 0.05, σσ = 9.65 = 9.65

Using Formula 8.3,Using Formula 8.3, 2 2 22 2 2 n = (z ) (n = (z ) (σσ ) / E ) / E 0.0250.025 = 89.43= 89.43 The sample size needs to be at least 90.The sample size needs to be at least 90.

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Problem #26 (10-Page 318; 11-Page 327)Problem #26 (10-Page 318; 11-Page 327)

Given: µ = 2.41 Given: µ = 2.41 σσ = 0.15 Margin of Error = 0.07 = 0.15 Margin of Error = 0.07

Confidence Level = 95%Confidence Level = 95%

Margin of Error = z . (Margin of Error = z . (σσ /√n) /√n)

/2 2 2 2/2 2 2 2

0.07 = 1.96 (0.15/√n) n = (1.96) (0.15) / 0.07)0.07 = 1.96 (0.15/√n) n = (1.96) (0.15) / 0.07)

= 17.63 or 18= 17.63 or 18

A sample size of 18 is recommended for The Cincinnati A sample size of 18 is recommended for The Cincinnati Enquirer for its studyEnquirer for its study

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The general form of an interval estimate of aThe general form of an interval estimate of a population proportion ispopulation proportion is

The general form of an interval estimate of aThe general form of an interval estimate of a population proportion ispopulation proportion is

Margin of Errorp Margin of Errorp

Interval EstimationInterval Estimationof a Population Proportionof a Population Proportion

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The sampling distribution of plays a key role inThe sampling distribution of plays a key role in computing the margin of error for this intervalcomputing the margin of error for this interval estimate.estimate.

The sampling distribution of plays a key role inThe sampling distribution of plays a key role in computing the margin of error for this intervalcomputing the margin of error for this interval estimate.estimate.

pp

The sampling distribution ofThe sampling distribution of can be approximated can be approximated by a normal distribution whenever by a normal distribution whenever npnp >> 5 and 5 and nn(1 – (1 – pp) ) >> 5. 5.

The sampling distribution ofThe sampling distribution of can be approximated can be approximated by a normal distribution whenever by a normal distribution whenever npnp >> 5 and 5 and nn(1 – (1 – pp) ) >> 5. 5.

pp

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/2/2 /2/2


Normal Approximation of Sampling Distribution Normal Approximation of Sampling Distribution of of

pp

Samplingdistribution of

Samplingdistribution of pp

(1 )p

p p

n

(1 )p

p p

n

pppp

/ 2 pz / 2 pz / 2 pz / 2 pz

1 - of all values1 - of all valuespp

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Interval EstimateInterval Estimate


p zp pn

/

( )2

1p z

p pn

/

( )2

1

where: 1 -where: 1 - is the confidence coefficient is the confidence coefficient

zz/2 /2 is the is the zz value providing an area of value providing an area of

/2 in the upper tail of the standard/2 in the upper tail of the standard

normal probability distributionnormal probability distribution

is the sample proportionis the sample proportionpp

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Interval Estimate for a Population Interval Estimate for a Population ProportionProportion

Example (10-Page 320; 11-Page 329):Example (10-Page 320; 11-Page 329): We want to estimate the proportion of all women golfers in the We want to estimate the proportion of all women golfers in the

U.S. who are satisfied with the availability of tee times with a U.S. who are satisfied with the availability of tee times with a 95% confidence level.95% confidence level.

We take a sample of 900 women golfers and find that 396 are We take a sample of 900 women golfers and find that 396 are satisfied with the tee times.satisfied with the tee times.

__ Given: p = 396/900 = 0.44 = 0.05Given: p = 396/900 = 0.44 = 0.05 The 95% confidence interval is:The 95% confidence interval is: _ _ __ _ _ p ± z √ [p (1 – p)]/n = 0.44 ± 1.96√ [0.44 (1 – 0.44)]/900p ± z √ [p (1 – p)]/n = 0.44 ± 1.96√ [0.44 (1 – 0.44)]/900 0.44 ± 0.0324 0.4076 to 0.4724 or 40.76% to 47.24%0.44 ± 0.0324 0.4076 to 0.4724 or 40.76% to 47.24% We are 95% confident that the proportion of all women golfers We are 95% confident that the proportion of all women golfers

who are satisfied with tee times is between 40.76% and who are satisfied with tee times is between 40.76% and 47.24%.47.24%.

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#38 (10-Page 323; 11-Page 332))#38 (10-Page 323; 11-Page 332))

a.a. Point estimate of the proportion of companies that Point estimate of the proportion of companies that

fell short of estimates = 29/162 = 0.179fell short of estimates = 29/162 = 0.179

_ _

b.b. Given : p = 104/162 = 0.642 _ _Given : p = 104/162 = 0.642 _ _

Margin of error = z √ [(p (1 – p ) / n)]Margin of error = z √ [(p (1 – p ) / n)]

0.025 0.025

= 1.96 √ [(0.642) (1 – 0.642) / = 1.96 √ [(0.642) (1 – 0.642) / 162]162]

= 0.0738= 0.0738

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Sample Problem (contd)Sample Problem (contd)

95% confidence interval:95% confidence interval: 0.642 ± 0.07380.642 ± 0.0738 0.5682 and 0.71580.5682 and 0.7158

We are 95% confident that the proportion of companies We are 95% confident that the proportion of companies that beat estimates of their profits is between 56.82% that beat estimates of their profits is between 56.82% and 71.58%. and 71.58%.

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c. Required sample size, n = [(1.96) (0.642) c. Required sample size, n = [(1.96) (0.642) (0.358)]/(0.05) (0.358)]/(0.05)

= 353.18 use 354= 353.18 use 354

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End of Chapter 8End of Chapter 8

IS 310 Business Statistics CSU Long Beach

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