Download 9.2 Confidence Interval of mean when SD is unknown

Confidence Interval Estimation of
Population Mean, μ, when σ is
Unknown
Chapter 9
Section 2
The BIG Idea!
Most of the time, the value for the populations
standard deviation is NOT known. For example,
what’s the likelihood anyone know the average
number of kids in each family in this school or the
standard deviation for that population?
How do we develop a confidence interval when the
standard deviation for the population is
unknown????
Confidence Interval Estimation of Population
Mean, μ, when σ is Unknown
• If the population standard deviation σ is
unknown, we can substitute the sample
standard deviation, S
• This introduces extra uncertainty, since S
varies from sample to sample
• So we use the student’s t distribution instead
of the normal Z distribution
t-distribution
The t – distribution is actually a family of curves
based on the concept of degrees of freedom,
which relate to sample size.
As the sample increases, the t-distribution
approaches the standard normal distribution.
• Student’s t distribution
• Note: t
Z as n increases
Standard
Normal
t (df = 13)
t-distributions are bellshaped and symmetric, but
have ‘fatter’ tails than the
normal
t (df = 5)
0
t
Degrees of Freedom
The degrees of freedom (denoted d.f.) are the number of
values that are free to vary after a sample statistic has
been computed, and tell the researcher which specific
curve to use when a distribution consists of a family of
curves.
For example: If the mean of 5 values is 10, then 4 of the 5
values are free to vary. But once 4 values are selected,
the fifth value must be a specific number to get the
sum of 50, since 50/5 = 10.
Hence, the degrees of freedom are 5 – 1 = 4, and this tells
the researcher which curve to use.
d.f. Formula
• The formula for finding the degrees of
freedom for the confidence interval of a mean
is simply
d.f. = n – 1
• Confidence Interval Estimate Use Student’s
t Distribution :
S
  X  t n-1
n
(where t is the critical value of the t distribution with n-1 d.f. and an
area of α/2 in each tail)
• t distribution is symmetrical around its mean of zero, like Z dist.
• Compare to Z dist., a larger portion of the probability areas are in the
tails.
• As n increases, the t dist. approached the Z dist.
• t values depends on the degree of freedom.
Example #1
A sample of 14 commuters in Chicago showed
the average of the commuting time was 33.2
minutes. If the standard deviation of the
sample was 8.3 minutes, fine the 95%
confidence interval of the true mean.
Example #2
Poll Everywhere poll on sleep last night.
Link to Poll
What is the mean and standard deviation for our sample?
What is the degrees of freedom for our sample?
Find a 90% confidence interval of the mean time.
Assume the situation is normally distributed.