Estimating a Population Mean: σ Known

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Estimating a Population

Mean: σ Known

7-3, pg 355


Section 7.3: Estimation of a population mean µ

(σ is known)

In this section we cover methods for estimating a population mean. In addition to knowing the values of the sample data or statistics, we must also know the value of the population standard deviation, .


Point Estimate of the Population Mean

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


Confidence Interval for Estimating a Population Mean

(with Known)

= population mean

= population standard deviation

= sample mean

n = number of sample values

E = margin of error

z/2 = z score separating an area of a/2 in the right tail of the standard normal distribution

x


Requirements to check:

1. The value of the population standard deviation is known.

2. Either or both of these conditions is satisfied: The population is normally distributed or n > 30. (Just like in the Central Limit Theorem.)


Confidence Interval for Estimating a Population Mean

(with Known)

x E x E where E z 2

n

or x E

or x E,x E


Definition

The two values x – E and x + E are called confidence interval limits.


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 mean.

Round-Off Rule for Confidence Intervals Used to Estimate µ


• Press STAT and select TESTS

• Scroll down to ZInterval press ENTER

• choose Data or Stats. For Stats:

• Type in s: (known st. deviation)

• x: (sample mean)

• n: (sample size)

• C-Level: (confidence level)

• Press on Calculate

• Read the confidence interval (…..,..…)

Confidence Intervals by TI-83/84

_


Confidence Intervals by Excel


Finding a Sample Size for Estimating a Population Mean

(z/2) n =

E

2

σ = population standard deviation

E = desired margin of error

zα/2 = z score separating an area of /2 in the right tail of the standard normal distribution


Round-Off Rule for Sample Size n

If the computed sample size n is not a whole number, round the value of n up to the next larger whole number.


Example Using Excel:


Estimating a Population

Mean: σ Not Known

7-4, pg 355


Section 7.4: Estimation of a population mean µ(σ is not known)

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


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

Sample Mean

_


Construction of a confidence

intervals for mean µ(σ is not known)

With σ unknown, we use the Student t distribution instead of normal distribution.

It involves a new feature: number of degrees of freedom


degrees of freedom = n – 1

in this section.

Definition

The number of degrees of freedom for a collection of sample data is the number of sample values that can vary after certain restrictions have been imposed on all data values.

The degree of freedom is often abbreviated df.


Margin of Error E for Estimate of µ (With σ Not Known)

Formula 7-6

where tα/2 has n – 1 degrees of freedom.

nsE = tα/2

Table A-3 lists values for tα/2

t/2 = critical t value separating an area of /2 in the right tail of the t distribution


where E = tα/2 ns

x – E < µ < x + E

tα/2 found in Table A-3

Confidence Interval for the Estimate of μ (With σ Not Known)

df = n – 1


Important Properties of the Student t Distribution

1. The Student t distribution is different for different sample sizes (see the following slide, for the cases n = 3 and n = 12).

2. The Student t distribution has the same general symmetric bell shape as the standard normal distribution but it reflects the greater variability (with wider distributions) that is expected with small samples

3. The Student t distribution has a mean of t = 0 (just as the standard normal distribution has a mean of z = 0).

4. The standard deviation of the Student t distribution varies with the sample size and is greater than 1 (unlike the standard normal distribution, which has a σ = 1).

5. As the sample size n gets larger, the Student t distribution gets closer to the normal distribution.


Student t Distributions for n = 3 and n = 12

Figure 7-5


Choosing the Appropriate Distribution

Use the normal (z) distribution

known and normally distributed populationor known and n > 30

Use t distribution not known and normally distributed populationor not known and n > 30

Methods of Chapter 7 do not apply

Population is not normally distributed and n ≤ 30



• Scroll down to TInterval press ENTER

• choose Data or Stats. For Stats:

• Type in x: (sample mean)

• Sx: (sample st. deviation)

• n: (number of trials)





_



• Scroll down to TInterval press ENTER

• choose Data or Stats. For Data:

• Type in List: L1 (or L2 or L3)

• (specify the list containing your data)

• Freq: 1 (leave it)







Example 2, page 358, first calculate margin of error



Example 2, page 358, first calculate margin of error


Point estimate of µ:

x = (upper confidence limit) + (lower confidence limit)

2

Margin of Error:

E = (upper confidence limit) – (lower confidence limit)

2

Finding the Point Estimate and E from a Confidence Interval


Section 7-5 Estimating a Population

Variance

This section covers the estimation of a population variance 2 and standard deviation


Estimator of 2

The sample variance s2 is the best point estimate of the population variance 2.


Estimator of

The sample standard deviation s is a commonly used point estimate of .


Construction of confidence intervals for 2

We use the chi-square distribution, denoted by Greek character 2 (pronounced chi-square).


Properties of the Chi-Square Distribution

1. The chi-square distribution is not symmetric, unlike the normal and Student t distributions.

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

degrees of freedom = n – 1


2. The values of chi-square can be zero or positive, but they cannot be negative.

3. The chi-square distribution is different for each number of degrees of freedom, which is df = n – 1.

In Table A-4, each critical value of 2 corresponds to an area given in the top row of the table, and that area represents the cumulative area located to the right of the critical value.

Properties of the Chi-Square Distribution


Example

A sample of ten voltage levels is obtained.

Construction of a confidence interval for the population standard deviation requires the left and right critical values of 2 corresponding to a confidence level of 95% and a sample size of n = 10.

Find the critical value of 2 separating an area of 0.025 in the left tail, and find the critical value of 2 separating an area of 0.025 in the right tail.


ExampleCritical Values of the Chi-Square Distribution


Confidence Interval for Estimating a Population Variance

n 1 s2

R2

2 n 1 s2

L2


Confidence Interval for Estimating a Population Standard Deviation

n 1 s2

R2

n 1 s2

L2


Requirement:

The population must have normally distributed values (even if the sample is large)

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 Rule for Confidence Intervals Used to Estimate or 2


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.


Determining Sample Sizes


Example: We want to estimate the standard deviation . We want to be 95% confident that our estimate is within 20% of the true value of . How large should the sample be? Assume that the population is normally distributed.

From Table 7-2, 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.

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Estimating a Population Mean: σ Known