Download Chapter 11 Day 1

CHAPTER 11 DAY 1
Assumptions for Inference About a Mean


Our data are a simple random sample (SRS) of
size n from the population.
Observations from the population have a normal
distribution with mean μand standard deviation σ.
Both μand σ are unknown parameters.
 In
the previous chapter we made the unrealistic
assumption that we knew the value of σ, when in
practice σ is unknown.
Standard Error


Because we don’t know σ, we estimate it by the
sample standard deviation s.
When the standard deviation of a statistic is
estimated from the data, the result is called the
standard error of the statistic. The standard error of
the sample mean x is :
s
n
The One-Sample t Statistic
and the t Distributions


Draw an SRS of size n from a population that has
the normal distribution with mean μ and standard
deviation σ.
The one-sample t statistic
x
t
s
n
has the t distribution with n – 1 degrees of
freedom.
Facts About t Distributions



The density curves of the t distributions are similar in
shape to the standard normal curve. They are
symmetric about zero and are bell-shaped.
The spread of the t distributions is a bit greater
than that of the standard normal distribution. This
comes from using s instead of σ.
As the degrees of freedom increases, the density
curve approaches the standard normal curve.
t chart Examples

What critical values from Table C satisfies each of
the following conditions?
 A.
The t distribution with 8 degrees of freedom has
probability 0.025 to the right of t*
 B.
The t distribution with 17 degrees of freedom has
probability 0.20 to the left of t*
 C.
The one-sampled t statistics from a sample of 25
observations has probability 0.01 to the right of t*.
 D.
The one-sampled t statistics from an SRS of 30
observations has probability 0.95 to the left of t*.
Example

The one-sample t statistic for testing
 H0:
μ= 0
 Ha: μ> 0
From a sample of 10 observations has the value t = 3.12




A. What are the degrees of freedom for this statistic?
B. Give the two critical values of t* from the Table C from
bracket t.
C. Between what two values does the P-value of this test
fall?
D. Is the value t = 3.12 significant at the 5% level? Is it
significant at the 1% level?
Confidence Intervals

Confidence interval for t distribution
s
xt*
n
Example


Natalie placed an ad in the newspaper for her
beanbags. The following numbers are the beanbag
sales from 5 randomly chosen days:
37
41
35
36
31
Find a 99% confidence interval for the mean
number of beanbags sold.