Transcript Chapter 11: Inference for Distributions The Practice of Statistics (Yates) 11.1 Inference for the Mean of a Population • Confidence intervals and tests of significance for.

Chapter 11: Inference for
Distributions
The Practice of Statistics
(Yates)
11.1 Inference for the Mean of a
Population
• Confidence intervals and tests of significance
for the mean of a normal population
– Mean    is based on sample mean  x 
– Population standard deviation   is unknown so
estimate using sample data
• Standard Error of the sample mean:
s
n
Assumptions for Inference about a
Mean
• Data are a simple random sample (SRS) on size
n from the population
• Observations from the population have a
normal distribution with mean and standard
deviation  both of which are unknown
One-Sample t Statistic and
t-Distribution
• The one-sample t statistic
x
t
s/ n
has the t distribution with n – 1 degrees of
freedom
Facts about the t Distributions
• Density curves – similar in shape to standard
normal curve
– Symmetric about zero
– Bell-shaped
• Spread – somewhat greater than the standard
normal distribution
– More probability in tails and less in the center
– Due to substituting the estimate for population
standard deviation
• As the degrees of freedom k increase the t(k)
density curve approaches the N(0, 1) curve
One-Sample t Procedures
• To test the hypothesis H0 :   u0
– Compute the one-sample t statistic
– Against one of the following
H a :   0 (right tail)
H a :   0 (left tail)
H a :   0 (both tails)
Matched Pairs t Procedures
Matched pairs design
– Subjects are matched in pairs
– Each treatment given to one subject in each pair
– Also used in before-and-after observations on the
same subjects
Robustness of t Procedures
• A confidence interval or significance test is
called robust if the confidence level or P-value
does not change very much when the
assumptions of the procedure are violated
– t procedures are strongly influenced by outliers
– Skewness affects the t procedures
– Make a plot to check for outliers and skewness
– When population is not normal, larger sample
sizes improve accuracy (central limit theorem)
Using t Procedures
• SRS assumption is more important than
assumption of Normal distribution
• Sample size less than 15: use t procedures
only if data is nearly normal, without outliers
• Sample size at least 15: use t procedures
except if there are outliers or strong skewness
• Large samples, n  40 : t procedures can be
used even in clearly skewed distributions
The Power of the t Test
• Power: measures the ability of test to detect
deviations from the null hypothesis
• Power of one-sample t test: probability that
test will reject null hypothesis when the mean
has alternative value
• To calculate
– assume fixed level of significance
– Usually   .05
11.2 Comparing Two Means
Two –Sample Problems
– Goal of inference is
• to compare the responses to two treatments
• to compare the characteristics of two populations
– Have a separate sample from each treatment or
each population
Assumptions for Comparing Two
Means
• Two SRSs from two distinct populations
• The samples are independent
– One sample has no influence on the other
– Measure the same variable for both samples
• Both populations are normally distributed
– Means and standard deviations of the populations are
unknown
Two-Sample z Statistic
• Normal distribution of the statistic x1  x2
x  x      

z
1
2

1
2
1
n1


2
2
n2
2
Two-Sample t Procedures
• The population standard deviations are
unknown
x  x      

t
1
2
1
2
1
2
2
s
s

n1 n2
2
Two-Sample Problems
• Two-Sample t statistic is used with t critical
values in inference
– Option 1: use procedures based on the statistic t
with critical values from a t distribution with
degrees of freedom computed from the data
– Option 2: use procedures based on the statistic t
with critical values from a t distribution with
degrees of freedom equal to the smaller of
n1  1 and n2  1