CHAPTER 7 – MORE
CONFIDENCE INTERVALS AND
HYPOTHESIS TESTS
STAT 200
The Big Picture of Chapter 7
2
Up to now we’ve seen the Ztest and the confidence
intervals for means.
Both methods required three conditions:
Independent observations
Normally distributed data (Or large sample size)
Standard deviation is known
In this chapter we look to overcome the third
assumption and to generalize to two sample
situations.
One Sample Procedures
PART 1
3
tConfidence Interval
4
In situations where we seek to estimate the mean
,
we almost never know the standard deviation
.
In order to overcome this issue in confidence
intervals, we may be tempted to simply replace
by
S
– the sample standard deviation used to
estimate
.
This is a correct step, but it is not the only change
that we must bring to the interval.
What changes when
σ is unknown?
Notice that we are now
dividing by an estimated
standard deviation. This
isn’t a Zscore! As a result
the distribution of the
middle term is not
Normal
5
Not so Normal Anymore
6
Since we use
S
, a random variable in the
denominator instead of
a fixed value, we add
uncertainty to the middle term.
This translates into higher variance which is
accounted for by the new distribution we use, called
the tdistribution.
So where does the distribution come into play in all
this?
It’s from the distribution that we get the confidence
coefficients.
t 
Distribution
If the population is normally distributed and
σ
is
unknown, the appropriate interval is based on the
t

distribution
The
t
Distribution are a family of similar distributions
defined by a single parameter called its
degrees of
freedom (d.f.)
. They are
symmetric, bellshaped
curves.
Basically they are like the standard Normal but with a
larger spread (centered at zero with
thicker tails
).
The larger the degrees of freedom, the closer to the
standard Normal it gets and hence it gets less
dispersion.
7
8
TTable (Table A.4)
9
Calcium
Using specimens obtained from 10 individuals,
determinations of percent calcium content of sound
teeth gave the following results:
x
1
+x
2
+…x
10
: 356.58
s=0.7137
Find 95 and 99 percent CIs for the mean percentage
of calcium in the population.
10
Validity of CI’s
If the population distribution is
Normal
and
σ is known: the CI based on the normal
distribu7on is exact
σ is unknown: the CI based on the t‐distribu7on is
exact.
If the popula7on distribu7on is not Normal, but n is
suﬃciently large, the CI based on the
corresponding t‐distribu7on is approximately
correct (due to a theorem similar to the CLT).
11
continued
In practice, if n is large, the CI based on the normal
distribution with
s
substituted for
σ
also is
approximately correct since the tdistribution with
large df is almost identical to the standard Normal
distribution.