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 (1876-1937) of the Guinness brewery in Dublin, Ireland ran
into this 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; he found that the actual distribution depends on the sample size. 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 calculation of 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 graph for the Student-t distribution is similar to the Standard Normal curve.
- The mean for the Student-t distribution is 0 and the distribution is symmetric about 0.
- The Student-t distribution has more probability in its tails than the Standard Normal distribution
because the spread of the t distribution is greater than the spread of the Standard Normal. So the graph of the Student-t distribution will be thicker in the tails and shorter in the center than the graph of the Standard Normal distribution.
- The exact shape of the Student-t distribution depends on the "degrees of freedom". As the degrees of freedom increases, the graph Student-t distribution becomes more like the graph of the Standard Normal distribution.
- The underlying population of individual observations is assumed to be normally distributed with unknown population
mean μμ and unknown population standard deviation σσ. In the real world, however, as long as the underlying population is large and bell-shaped, and the data are a simple random sample, practitioners often consider the assumptions met.
Calculators and computers can easily calculate any Student-t probabilities. The TI-83,83+,84+ have a tcdf function to find the probability for given values of t. The grammar for the tcdf command is tcdf(lower bound, upper bound, degrees of freedom). However for confidence intervals, we need to use inverse probability to find the value of t when we know the probability.
For the TI-84+ we will use the invT command on the DISTRibution menu.
The invT command works similarly to the invnorm.
The invT command requires two inputs: invT(area to the left, degrees of freedom) The output is the t-score that corresponds to the area we specified.
The TI-83 and TI-83+ do not have the invT command but you can download an invT program from your instructor that is easy to use. (The TI-89 has an inverse T command.)
A probability table for the Student-t distribution can also be used. The table gives t-scores that correspond to the confidence level (column) and degrees of freedom (row). (The TI-86 does not have an invT program or command, so if you are using that calculator, you need to use a probability table for the Student-t distribution.) When using t-table, note that some tables are formatted to show the confidence level in the column headings, while the column headings in some tables may show only corresponding area in one or both tails.
- TT ~ tdftdf where df=n-1df=n-1.
- For example, if we have a sample of size n=20 items, then we calculate the degrees of freedom as df=n−1=20−1=19 and we write the distribution as TT ~ t19t19
If the population standard deviation is not known, the error bound for a population mean 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
- use df=n-1df=n-1 degrees of freedom
-
s
s
= sample standard deviation
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 solution is shown step by step.
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
df=15-1=14df=15-1=14
CL
=
0.95
CL=0.95
so
α
=
1
-
CL
=
1
-
0.95
=
0.05
α=1-CL=1-0.95=0.05
α
2
=
0.025
t
α
2
=
t
.025
α
2
=0.025
t
α
2
=
t
.025
The area to the right of
t.025
t.025 is 0.025 and the area to the left of
t.025
t.025 is 1−0.025=0.975
t
α
2
=
t
.025
=
2.14
t
α
2
=
t
.025
=2.14 using invT(.975,14) on the TI-84+ calculator.
EBM
=
t
α
2
⋅
(
s
n
)
EBM=
t
α
2
⋅(
s
n
)
EBM
=
2.14
⋅
(
1.6722
15
)
=
0.924
EBM=2.14⋅(
1.6722
15
)=0.924
x¯
-
EBM
=
8.2267
-
0.9240
=
7.3
x
-EBM=8.2267-0.9240=7.3
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).
We estimate with 95% confidence that the true population average sensory rate is
between 7.30 and 9.15.
Note: When calculating the error bound, a probability table for the Student-t distribution can also be used to find the value of t. The table gives t-scores that correspond to the confidence level (column) and degrees of freedom (row); the t-score is found where the row and column intersect in the table.
- Confidence 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.
- Confidence Level:
The percent expression for the probability that the confidence interval contains the true population parameter. That is, for ex., if CL=90%, then in 90 out of 100 samples the interval estimate will enclose the true population parameter.
- 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.
