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Transcript
Section 7-4
Estimating a Population Mean:
Οƒ Not Known
ASSUMPTIONS: Οƒ NOT KNOWN
1. The sample is a simple random sample.
2. Either the sample is from a normally
distributed population or n > 30.
When Οƒ is not known we will use the
Student t Distribution.
THE STUDENT t DISTRIBUTION
If the distribution of a population is essentially
normal, then the distribution of
π‘₯βˆ’πœ‡
𝑑= 𝑠
𝑛
is essentially a Student t distribution for all
samples of size n, and is used to find critical
values denoted by tΞ±/2. The Student t distribution
is often referred to as the t distribution.
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 βˆ’ 1
MARGIN OF ERROR ESTIMATE
OF µ (WITH Οƒ NOT KNOWN)
𝑠
𝐸 = 𝑑𝛼/2 βˆ™
𝑛
where (1 βˆ’ Ξ±) is the confidence level and tΞ±/2
has n βˆ’ 1 degrees of freedom.
NOTE: The values for tΞ±/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 ΞΌ (WITH Οƒ NOT KNOWN)
π‘₯βˆ’πΈ <πœ‡ <π‘₯+𝐸
where
𝑠
𝐸 = 𝑑𝛼/2 βˆ™
𝑛
CONSTRUCTING A CONFIDENCE
INTERVAL FOR ΞΌ (Οƒ NOT KNOWN)
1.
2.
Verify that the required assumptions are met.
Using n βˆ’ 1 degrees of freedom, refer to Table A-3
and find the critical value tΞ±/2 that corresponds to the
desired confidence interval. (For the confidenc level,
refer to β€œArea in Two Tails.”)
𝑠
𝑛
3.
Evaluate the margin of error 𝐸 = 𝑑𝛼/2 βˆ™
4.
Find the values of π‘₯ βˆ’ 𝐸 and π‘₯ + 𝐸. Substitute these
in the general format of the confidence interval: π‘₯ βˆ’
𝐸 < πœ‡ < π‘₯ + 𝐸.
Round the result using the same round-off rule from
the last section.
5.
FINDING A CONFIDENCE
INTERVAL FOR µ WITH TI-83/84
1.
2.
3.
4.
5.
6.
7.
8.
9.
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 press ENTER.
PROPERTIES OF THE
STUDENT t DISTRIBUTION
1. 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)
2.
3.
4.
5.
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 Οƒ = 1).
As the sample size n gets larger, the Student t
distribution gets closer to the normal distribution.
CHOOSING THE APPROPRIATE
DISTRIBUTION
CHOOSING BETWEEN z AND t
Method
Conditions
Use normal (z)
distribution
Οƒ known and normally distributed population
or
Οƒ known and n > 30
Use t distribution
Οƒ not known and normally distributed
population
or
Οƒ not known and n > 30
Population is not normally distributed and n ≀
30.
Use a
nonparametric
method or
bootstrapping
FINDING A POINT ESTIMATE AND E
FROM A CONFIDENCE INTERVAL
Point estimate of µ:
upper confidence limit + lower confidence limit
π‘₯=
2
Margin of error:
upper confidence limit βˆ’ lower confidence limit
𝐸=
2