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Section 7-4. Estimating a Population Mean: σ Not Known. ASSUMPTIONS: σ NOT KNOWN. 1.The sample is a simple random sample. 2.Either the sample is from a normally distributed population or n > 30. When σ is not known we will use the Student t Distribution.

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Section 7-4

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Section 7 4

Section 7-4

Estimating a Population Mean:

σ Not Known


Assumptions not known

ASSUMPTIONS: σ NOT KNOWN

1.The sample is a simple random sample.

2.Either the sample is from a normally distributed population or n > 30.

When σ is not known we will use the Student t Distribution.


The student t distribution

THE STUDENT t DISTRIBUTION

If the distribution of a population is essentially normal, then the distribution of

is essentially a Student t distributionfor all samples of size n, and is used to find critical values denoted by tα/2. The Student t distribution is often referred to as the t distribution.


Degrees of freedom

DEGREES OF FREEDOM

Degrees of Freedom (df) corresponds to the number of sample values that can vary after certain restrictions have been imposed on all data values.

In this section,

degrees of freedom = n− 1


Margin of error estimate of with not known

MARGIN OF ERROR ESTIMATE OF µ (WITH σ NOT KNOWN)

where (1 −α) is the confidence level andtα/2 has n − 1 degrees of freedom.

NOTE: The values for tα/2 are found in Table A­3 which is found on page 606, inside the back cover, and on the Formulas and Tables card.


Confidence interval estimate of the population mean with not known

CONFIDENCE INTERVAL ESTIMATE OF THE POPULATION MEAN μ (WITH σ NOT KNOWN)

where


Constructing a confidence interval for not known

CONSTRUCTING A CONFIDENCE INTERVAL FOR μ (σ NOT KNOWN)

  • 1.Verify that the required assumptions are met.

  • Using n− 1 degrees of freedom, refer to Table A-3 and find the critical value tα/2 that corresponds to the desired confidence interval. (For the confidenc level, refer to “Area in Two Tails.”)

  • 3.Evaluate the margin of error

  • 4.Find the values of x – E and x + E. Substitute these in the general format of the confidence interval:

  • x – E < μ < x + E.

  • 5.Round the result using the same round-off rule from the last section.


Finding a confidence interval for with ti 83 84

FINDING A CONFIDENCE INTERVAL FOR µ WITH TI-83/84

  • Select STAT.

  • Arrow right to TESTS.

  • Select 8:TInterval….

  • Select input (Inpt) type: Data or Stats. (Most of the time we will use Stats.)

  • Enter the sample mean, x.

  • Enter the sample standard deviation, Sx.

  • Enter the size of the sample, n.

  • Enter the confidence level (C-Level).

  • Arrow down to Calculate and pressENTER.


Properties of the student t distribution

PROPERTIES OF THESTUDENT t DISTRIBUTION

  • The Student t distribution is different for different sample sizes (see Figure below for the cases n = 3 and n = 12).


Properties of the student t distribution continued

PROPERTIES OF THESTUDENT t DISTRIBUTION (CONTINUED)

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

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

  • 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).

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


Choosing the appropriate distribution

CHOOSING THE APPROPRIATE DISTRIBUTION


Choosing between z and t

CHOOSING BETWEEN z AND t


Finding a point estimate and e from a confidence interval

FINDING A POINT ESTIMATE AND E FROM A CONFIDENCE INTERVAL

Point estimate of µ:

Margin of error:


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