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Section 7-4

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Section 7-4

Estimating a Population Mean:

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.

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 (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

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.

where

- 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.

- 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.

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

- 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.

Point estimate of µ:

Margin of error: