Transcript Chapter 11.1 Notes

Chapter 11
Inference for Distributions
AP Statistics
11.1 – Inference for the Mean of a
Population
σ is unknown
• Often the case in practice
• When σ is known
z
x

n
• When σ is unknown
• Whoa, way different!
Really just
x
x
t
s
n
Standard Error

n
Becomes
s
n
This is just Standard error of the Sampling
mean
x
t-distributions
• Cousins of the z-distribution (Normal)
• Conditions for inference about a mean

– Random? – to generalize about the population
– Normal? – Verify if the sampling distribution about the
mean
is approximately normal.
– N>=10n?
- Independent?
x
• t(k) distribution where k = n – 1 degrees of
freedom
– S has n-1 degrees of freedom
t(k) distributions
• Similar to Normal curve; symmetric, single
peaked, bell shaped
• Spread of t-dist. is greater than z-dist.
• As degrees of freedom increase, the t(k) density
curve approaches the normal curve more
closely.
– (s estimates
 more accurately as n increases)
• t* uses upper tail probabilities (look at table)
• Y1=normalpdf(x)
• Y2=tpdf(x,df)
Using the t* table
• What critical value t* would you use for a t
distribution with 18 degrees of freedom
having probability 0.9 to the left of t*?
Using the t* table
• What t* value would you use to construct a
95% confidence interval with mean 
and an SRS of n = 12?
Using the t* table
• What t* value would you use to construct a
80% confidence interval with mean 
and an SRS of n = 56?
t-CI’s & t-tests
• 1-sample t-interval VS. 1-sample t-test
Normal Probability Plot
T(9) distribution
Data Software Packages
Matched Pairs t-procedures
• Comparative Studies are more convincing
than single-sample investigations
• To compare the responses of the two
treatments in a matched pairs design,
apply the 1-sample t-procedures to the
Observed DIFFERENCES!
Statistical Software Packages
(con’t)
Robustness of t-procedures
• A CI or Significance Test is called robust
if the confidence level or P-value does not
change very much when assumptions of
the procedure or violated.
• Outliers? – Like x and s, the t-procedures
are strongly influenced by outliers.
Quite Robust when No Outliers
Sample size increases  CLT  more robust!
Using the t-procedure
• SRS is more important than normal (except in
the case of small samples)
• n < 15, use t-procedures if the data are close to
normal
• n ≥ 15, use t-procedures except in presence of
outliers or strong skewness
• Large samples (roughly n ≥ 40), t-procedures can be
used even for clearly skewed distributions
• p. 636-637 - histograms
The power of the t-test
• Power measures ability to detect
deviations from the null hypothesis Ho
• Higher power of a test is important!
• Usually assume a fixed level of
significance, α = 0.05
Here we go again. . . Power!
• Director hopes that n=20 teachers will detect an average
improvement of 2 pts in the mean listening score. Is this
realistic?
• Hypotheses?
• Test against the alternative  =2 when n=20.
• Impt: Must have a rough guess of the size of
to
compute power!
= 3 (from past samples)

