Section 7.4 Estimation of a Population Mean (s is unknown )

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Section 7.4Estimation of a Population Mean

is unknown

This section presents methods for estimating a population mean when the population standard deviation is not known.


The sample mean x is still the best point estimate of the population mean.

Best Point Estimate

_


When σ is unknown, we must use the Student t distribution instead of the normal distribution.

Requires new parameter df = Degrees of Freedom

Student t Distribution( t-dist )


The number of degrees of freedom (df) for a collection of sample data is defined as:

“The number of sample values that can vary after certain restrictions have been imposed on all data values.”

In this section: df = n – 1

Basically, since σ is unknown, a data point has to be “sacrificed” to make s. So all further calculations use n – 1 data points instead of n.

Definition


Using the Student t Distribution

The t-score is similar to the z-score but applies for the t-dist instead of the z-dist. The same is true for probabilities and critical values.

P(t < -1) tα (Area under curve) (Critical value)

NOTE: The values depend on df

-1 0 0

α (area)


Important Properties of the Student t Distribution

1. Has a symmetric bell shape similar to the z-dist

2. Has a wider distribution than that the z-dist

3. Mean μ = 0

4. S.D. σ > 1 (Note: σ varies with df)

5. As df gets larger, the t-dist approaches the z-dist


Student t Distributions for n = 3 and n = 12


z-Distribution and t-Distribution

Wider Spread

df = 2 df = 100

As df increases, the t-dist approaches the z-dist

Almost the same


df = 2 df = 3 df = 4

df = 5

df = 6 df = 7 df = 8

df = 20 df = 50 df = 100

Progression of t-dist with df


Choosing the Appropriate Distribution

Use the normal (Z) distribution

known and normally distributed populationor known and n > 30

Use t distribution

Methods of Ch. 7do not apply

Population is not normally distributed and n ≤ 30

not known and normally distributed populationor not known and n > 30


Calculating values from t-dist

Stat → Calculators → T



Enter Degrees of Freedom (DF) and t-score



P(t<-1) = 0.1646 when df = 20



tα = 1.697 when α = 0.05 df = 20


Margin of Error E for Estimate of (σ unknown)

Formula 7-6

where t2 has n – 1 degrees of freedom.

t/2 = The t-value separating the right tail so it has an area of /2


C.I. for the Estimate of μ (With σ Not Known)


Point estimate of µ:

Margin of Error:

Finding the Point Estimate and E from a C.I.


Find the 90%confidence interval for the population mean using a sample of size 42, mean 38.4, and standard deviation 10.0

Example:

s

Note: Same parameters as example used in Section 7-3 7-3: Etimating a population mean: σ known

Using σ = 10 ( instead of s = 10.0 ) we found the 90% confidence interval:

C.I. = (35.9, 40.9)



Example:

sDirect Computation:

T Calculator (df = 41)

.0



Example:

s .0

Using StatCrunch

Stat → T statistics → One Sample → with Summary



Example:

s .0

Using StatCrunch

Enter Parameters, click Next



Example:

s .0

Using StatCrunch

Select Confidence Interval and enter Confidence Level, then click Calculate



Example:

s .0

Using StatCrunch

From the output, we find the Confidence interval isCI = (35.8, 41.0)

Lower LimitUpper Limit

Standard Error


Example:

s

If σ known Used σ = 10 to obtain 90% CI:

If σ unknown Used s = 10.0 to obtain 90% CI:

Notice: σ known yields a smaller CI (i.e. less uncertainty)


Results

(35.8, 41.0)

(35.9, 40.9)


Section 7.5Estimation of a Population

Variance

This section presents methods for estimating a population variance

and standard deviation .


The sample variance s2 is the best point estimate of

the population variance

Best Point Estimate of


The sample standard deviation s is the best point estimate of the

population standard deviation

Best Point Estimate of


Pronounced “Chi-squared”

Also dependent on the number degrees of freedom df.

The 2 Distribution( 2-dist )


Properties of the 2 Distribution

Chi-Square Distribution

Use StatCrunch to Calculate values (similar to z-dist and t-dist)

Chi-Square Distribution for df = 10 and df = 20

1. The chi-square distribution is not symmetric, unlike the z-dist and t-dist.

2. The values can be zero or positive, they are nonnegative.3. Dependent on the Degrees of Freedom: df = n – 1


Calculating values from 2-dist

Stat → Calculators → Chi-Squared


Calculating values from 2-dist

Enter Degrees of Freedom DF and parameters( same procedure as with t-dist )

P(2 < 10)= 0.5595 when df = 10


Find the 90% left and right critical values (2

L and 2R) of the 2-dist when df = 20

Example:

Need to calculate values when the left/right areas are 0.05 ( i.e. α/2 )

2L = 10.851 2

R = 31.410


The 2-distribution is used for calculating the Confidence Interval of the Variance σ2

Take the square-root of the values to get the Confidence Interval of the Standard Deviation σ

( This is why we call it 2 instead of )

Important Note!!


Confidence Interval for Estimating a Population Variance

Note: Left and Right Critical values on opposite sides


Confidence Interval for Estimating a Population Standard Deviation

Note: Left and Right Critical values on opposite sides


Requirement for Application

The population MUST be normally distributed to hold(even when using large samples)

This requirement is very strict!


1. When using the original set of data, round the confidence interval limits to one more decimal place than used in original set of data.

2. When the original set of data is unknown and only the summary statistics (n, x, s) are used, round the confidence interval limits to the same number of decimal places used for the sample standard deviation.

Round-Off Rules for Confidence Intervals Used to Estimate or 2


Example

Direct Computation:

Chi-Squared Calculator (df = 39)

Suppose the scores a test follow a normal distribution. Given a sample of size 40 with mean 72.8 and standard deviation 4.92, find the 95% C.I. of the population standard deviation.


Using StatCrunch

Stat → Variance → One Sample → with Summary

ExampleSuppose the scores a test follow a normal distribution. Given a sample of size 40 with mean 72.8 and standard deviation 4.92, find the 95% C.I. of the population standard deviation.


Using StatCrunch

Enter parameters, then click NextBe sure to enter the sample variance s2 (not s)

Sample Variance



Using StatCrunch

Select Confidence Interval, enter Confidence Level, then click Calculate



Using StatCrunch Remember:The result is the C.I for the Variance σ2

Take the square root for Standard Deviation σ

Variance Upper Limit: ULσ2

Variance Lower Limit: LLσ2

CI = ( LLσ2, ULσ2 ) = (16.2, 39.9)

CI = ( LLσ2, ULσ2 ) = (4.03, 6.32) σσ2



Determining Sample Sizes

The procedure for finding the sample size necessary to estimate 2 is based on Table 7-2

You just read the required sample size from an appropriate line of the table.


Table 7-2


ExampleWe want to estimate the standard deviation . We want to be 95% confident that our estimate is within 20% of the true value of .

Assume that the population is normally distributed.

How large should the sample be?

For 95% confident and within 20%

From Table 7-2 (see next slide), we can see that 95% confidence and an error of 20% for correspond to a sample of size 48.

We should obtain a sample of 48 values.


For 95% confident and within 20%

Section 7.4 Estimation of a Population Mean (s is unknown )

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