In practice, we rarely know the population standard deviation. In the past, when the
sample size was large, this did not present a problem to statisticians. They used the
sample standard deviation ss as an estimate for σσ and proceeded as before to calculate a
confidence interval with close enough results. However, statisticians ran into problems
when the sample size was small. A small sample size caused inaccuracies in the
confidence interval. William S. Gossett of the Guinness brewery in Dublin, Ireland ran
into this very problem. His experiments with hops and barley produced very few
samples. Just replacing σσ with ss did not produce accurate results when he tried to
calculate a confidence interval. He realized that he could not use a normal distribution
for the calculation. This problem led him to "discover" what is called the Student-t
distribution. The name comes from the fact that Gosset wrote under the pen name
"Student."
Up until the mid 1990s, statisticians used the normal distribution approximation for large
sample sizes and only used the Student-t distribution for sample sizes of at most 30.
With the common use of graphing calculators and computers, the practice is to use the
Student-t distribution whenever ss is used as an estimate for σσ.
If you draw a simple random sample of size nn from a population that has
approximately a normal distribution with mean μμ and unknown population
standard deviation σσ and calculate the t-score
t
=
x¯
-
μ
(
s
n
)
t=
x
-
μ
(
s
n
)
, then the t-scores follow a Student-t distribution with n-1n-1 degrees of freedom. The t-score has
the same interpretation as the z-score. It measures how far x¯x is from its mean μμ. For each sample size nn, there is a different Student-t distribution.
The degrees of freedom, n-1n-1, come from the sample standard deviation ss. In Chapter 2, we used nn deviations (x-x¯values)(x-xvalues)
to calculate ss. Because the
sum of the deviations is 0, we can find the last deviation once we know the
other n-1n-1 deviations. The other n-1n-1 deviations can change or vary freely.
We call the number n-1n-1 the degrees of freedom (df).
The following are some facts about the Student-t distribution:
- The graph for the Student-t distribution is similar to the normal curve.
- The Student-t distribution has more probability in its tails than the normal
because the spread is somewhat greater than the normal.
- The underlying population of observations is normal with unknown population
mean μμ and unknown population standard deviation σσ.
A Student-t table (there is one in the book - see the Table of Contents) gives t-scores
given the degrees of freedom and the right-tailed probability. The table is very limited.
Calculators and computers can easily calculate any Student-t probabilities.
The notation for the Student-t distribution is (using T as the random variable)
TT ~ tdftdf where df=n-1df=n-1.
If the population standard deviation is not known, then the error bound for a population mean formula is:
EBM
=
t
α
2
⋅
(
s
n
)
EBM=
t
α
2
⋅(
s
n
)
t
α
2
t
α
2
is the t-score with area to the right equal to
α
2
α
2
.
s
s
= the sample standard deviation
The mechanics for calculating the error bound and the confidence interval are the same
as when σσ is known.
Suppose you do a study of acupuncture to determine how
effective it is in relieving pain. You measure sensory rates for 15 subjects with
the results given below. Use the sample data to construct a 95% confidence
interval for the mean sensory rate for the population (assumed normal) from
which you took the data.
• 8.6; • 9.4; • 7.9; • 6.8; • 8.3; • 7.3; • 9.2; • 9.6; • 8.7; • 11.4; • 10.3; • 5.4; • 8.1; • 5.5; • 6.9- The first solution is step-by-step.
- The second solution uses the TI-83+ and TI-84 calculators.
To find the confidence interval, you need the sample mean, x¯x, and the EBM.
x¯
=
8.2267
s
=
1.6722
n
=
15
x
=8.2267s=1.6722n=15
CL
=
0.95
CL=0.95
so
α
=
1
-
CL
=
1
-
0.95
=
0.05
α=1-CL=1-0.95=0.05
EBM
=
t
α
2
⋅
(
s
n
)
EBM=
t
α
2
⋅(
s
n
)
α
2
=
0.025
t
α
2
=
t
.025
=
2.14
α
2
=0.025
t
α
2
=
t
.025
=2.14
(Student-t table with df=15-1=14df=15-1=14)
Therefore,
EBM
=
2.14
⋅
(
1.6722
15
)
=
0.924
EBM=2.14⋅(
1.6722
15
)=0.924
This gives
x¯
-
EBM
=
8.2267
-
0.9240
=
7.3
x
-EBM=8.2267-0.9240=7.3
and
x¯
+
EBM
=
8.2267
+
0.9240
=
9.15
x
+EBM=8.2267+0.9240=9.15
The 95% confidence interval is (7.30, 9.15).
You are 95% confident or sure that the true population average sensory rate is
between 7.30 and 9.15.
TI-83+ or TI-84: Use the function 8:TInterval in STAT TESTS. Once you
are in TESTS, press 8:TInterval and arrow to Data. Press ENTER. Arrow
down and enter the list name where you put the data for List, enter 1 for
Freq, and enter .95 for C-level. Arrow down to Calculate and press ENTER.
The confidence interval is (7.3006, 9.1527)
- Confidential Interval:
An interval estimate for unknown population parameter. This depends on:
- The desired confidence level.
- What is known for the distribution information (for ex., known variance).
- Gathering from the sampling information.
- Degrees of Freedom (df):
The number of objects in a sample that are free to vary.
- Error Bound for a Population Mean (EBM):
The margin of error. Depends on the confidence level, sample size, and known or estimated population standard deviation.
- Normal Distribution:
A continuous random variable (RV) with
pdf=1σ2πe−(x−μ)2/2σ2pdf=1σ2πe−(x−μ)2/2σ2 size 12{ ital "pdf"= { {1} over {σ sqrt {2π} } } e rSup { size 8{ - \( x - μ \) rSup { size 6{2} } /2σ rSup { size 6{2} } } } } {}, where μμ is the mean of the distribution and σσ is its standard deviation. Notation: XX ~ N
μ
σ
2
N
μ
σ
2
. If μ=0μ=0 and σ=1σ=1, the RV is called standard normal distribution, or z-score.
- Standard Deviation:
A number that is equal to the square root of the variance and measures how far data values are from their mean. Notations: ss for sample standard deviation and σσfor population standard deviation.
- Student-t Distribution:
Investigated and reported by William S. Gossett in 1908 and published under the pseudonym Student. The major characteristics of the random variable (RV) are:
- It is a continuous and assumes any real values.
- The pdf is symmetrical about its mean of zero. However, it is more spread out and flatter at the apex than the normal distribution.
- It approaches the standard normal distribution as n gets larger.
- There is a "family" of t distributions: every representative of family is completely defined by the number of degrees of freedom which is one less than the number of data.
- z-score:
Let’s consider the linear transformation of the form
z=x−msz=x−ms size 12{z= { {x-μ} over {σ} } } {}. If this transformation is applied to any normal distribution
X~N(μ,σ2)X~N(μ,σ2) size 12{X "~" N \( μ,σ rSup { size 8{2} } \) } {}, the result is the standard normal distribution
Z~N(0,1)Z~N(0,1) size 12{Z "~" N \( 0,1 \) } {}. If this transformation is applied to any specific value
xx size 12{x} {} of RV with mean
μμ size 12{μ} {} and standard deviation
σσ size 12{σ} {} , the result is called z-score of
xx size 12{x} {}. z-score allows to compare data that are normally distributed but scaled differently.
