Transcript t distribution and population proportions

Review
• Confidence Intervals
• Sample Size
Confidence Intervals
The Confidence Interval is expressed as:
xE
x  z 2 x  x  z
E is called the margin of error.
For samples of size > 30,
 s 
x  z 2 

 n

2
n
Sample Size
The sample size needed to estimate m so as
to be (1-)*100 % confident that the sample
mean does not differ from m more than E is:
 z 2
n  
 E
…round up



2
Small Samples
What happens if n is small (n < 30)?
Our formulas from the last section no longer
apply.
Small Samples
What happens if n is small (n < 30)?
Our formulas from the last section no longer
apply. There are two main issues that arise
for small samples:
1)  no longer can be approximated by s
2) The CLT no longer holds. That is the
distribution of the sampling means is not
necessarily normal.
t- distribution
If we have a small sample (n < 30) and wish
to construct a confidence interval for the
mean we can use a t-statistic, provided the
sample is drawn from a normally distributed
population.
t-distributions
 is unknown so we use s (the sample
standard deviation) as a point estimate of .
We convert the nonstandard t-distributed
problem to a standard t-distributed problem
through the use of the standard t-score
xm
t
s
n
t-distributions
• Mean 0
• Symmetric and bell-shaped
• Shape depends upon the degrees of
freedom, which is one less than the
sample size.
df = n-1
• Lower in center, higher tails than normal.
• See Table inside front cover in text
Example
In n=15 and after some calculation
/2=0.025, we use the table and
n-1 = 14 degrees of freedom to
deduce
t0.025 = 2.145
Confidence Interval for the mean
when  is unknown and n is small
The (1- )*100% confidence interval for
the population mean m is
x  tn 1, 
2
s
n
 m  x  tn1, 
2
s
n
The margin of error E, is in this case
E  tn 1, 
2
s
n
N.B. The sample is assumed to be drawn
from a normal population.
Confidence Intervals for a small
sample population mean
The Confidence Interval is expressed as:
xE
x  t
2
s
n
The degrees of freedom is n-1.
xm
t 
s/ n
Example
The following are the heat producing
capabilities of coal from a particular mine
(in millions of calories per ton)
8,500 8,330
8,480
7,960
8,030
Construct a 99% confidence interval for
the true mean heat capacity.
Solution:
sample mean is 8260.0
sample Std. Dev. is 251.9
degrees of freedom = 4
 = 0.01
7741.4  m  8778.6
Confidence intervals for a
population proportion
The objective of many surveys is to determine the
proportion, p, of the population that possess a
particular attribute..
Example: Determine the fraction of Canadians who
support gun control.
If the size of the population is N, and X people have
this attribute, then as we already know, p  X N is
the population proportion.
Confidence intervals for a
population proportion
If the size of the population is N, and X
people have this attribute, then as we already
know, p  X N is the population proportion.
The idea here is to take a sample of size n,
and count how many items in the sample
have this attribute, call it x. Calculate the
sample proportion, pˆ  x n . We would like
to use the sample proportion as an estimate
for the population proportion.
E  z 2
pˆ qˆ
n
The confidence interval for the population
proportion is :
pˆ  E  p  pˆ  E