Title: Inference Concerning Populations Numeric Responses
1Chapter 7
- Inference Concerning Populations (Numeric
Responses)
2Inference for Population Mean
- Practical Problem Sample mean has sampling
distribution that is Normal with mean m and
standard deviation s / ?n (when the data are
normal, and approximately so for large samples).
s is unknown. - Have an estimate of s , s obtained from sample
data. Estimated standard error of the sample mean
is
When the sample is SRS from N(m , s) then the
t-statistic (same as z- with estimated standard
deviation) is distributed t with n-1 degrees of
freedom
3Family of t-distributions
- Symmetric, Mound-shaped, centered at 0 (like the
standard normal (z) distribution - Indexed by degrees of freedom (n ), the number of
independent observations (deviations) comprising
the estimated standard deviation. For one sample
problems n n-1 - Have heavier tails (more probability over extreme
ranges) than the z-distribution - Converge to the z-distribution as n gets large
- Tables of critical values for certain upper tail
probabilities are available (inside back cover of
text)
4Probability
Cri t ical Values
Degrees of Freedom
Critical Values
5(No Transcript)
6One-Sample Confidence Interval for m
- SRS from a population with mean m is obtained.
- Sample mean, sample standard deviation are
obtained - Degrees of freedom are n n-1, and confidence
level C are selected - Level C confidence interval of form
Procedure is theoretically derived based on
normally distributed data, but has been found to
work well regardless for large n
71-Sample t-test (2-tailed alternative)
- 2-sided Test H0 m m0 Ha m ? m0
- Decision Rule (t obtained such that P(t(n-1)?
t)a/2) - Conclude m gt m0 if Test Statistic (tobs) is
greater than t - Conclude m lt m0 if Test Statistic (tobs) is
less than -t - Do not conclude Conclude m ? m0 otherwise
- P-value 2P(t(n-1)? tobs)
- Test Statistic
8P-value (2-tailed test)
91-Sample t-test (1-tailed (upper) alternative)
- 1-sided Test H0 m m0 Ha m gt m0
- Decision Rule (t obtained such that P(t(n-1)?
t)a) - Conclude m gt m0 if Test Statistic (tobs) is
greater than t - Do not conclude m gt m0 otherwise
- P-value P(t(n-1)? tobs)
- Test Statistic
10P-value (Upper Tail Test)
111-Sample t-test (1-tailed (lower) alternative)
- 1-sided Test H0 m m0 Ha m lt m0
- Decision Rule (t obtained such that P(t(n-1)?
t)a) - Conclude m lt m0 if Test Statistic (tobs) is
less than -t - Do not conclude m lt m0 otherwise
- P-value P(t(n-1)? tobs)
- Test Statistic
12P-value (Lower Tail Test)
13Example Mean Flight Time ATL/Honolulu
- Scheduled flight time 580 minutes
- Sample n31 flights 10/2004 (treating as SRS
from all possible flights - Test whether population mean flight time differs
from scheduled time - H0 m 580 Ha m ? 580
- Critical value (2-sided test, a 0.05, n-130
df) t2.042 - Sample data, Test Statistic, P-value
14Paired t-test for Matched Pairs
- Goal Compare 2 Conditions on matched individuals
(based on similarities) or the same individual
under both conditions (e.g. before/after studies) - Obtain the difference for each pair/individual
- Obtain the mean and standard deviation of the
differences - Test whether the true population means differ
(e.g. H0mD 0) - Test treats the differences as if they were the
raw data
15Test Concerning mD
- Null Hypothesis H0mDD0 (almost always 0)
- Alternative Hypotheses
- 1-Sided HA mD gt D0
- 2-Sided HA mD ? D0
- Test Statistic
16Test Concerning mD
Decision Rule (Based on t-distribution with
nn-1 df) 1-sided alternative If tobs ? t
gt Conclude mD gt D0 If tobs lt t gt Do not
reject mD D0 2-sided alternative If tobs ? t
gt Conclude mD gt D0 If tobs ? -t gt
Conclude mD lt D0 If -t lt tobs lt t gt Do not
reject mD D0
Confidence Interval for mD
17Example Antiperspirant Formulations
- Subjects - 20 Volunteers armpits
- Treatments - Dry Powder vs Powder-in-Oil
- Measurements - Average Rating by Judges
- Higher scores imply more disagreeable odor
- Summary Statistics (Raw Data on next slide)
Source E. Jungermann (1974)
18Example Antiperspirant Formulations
19Example Antiperspirant Formulations
Evidence that scores are higher (more unpleasant)
for the dry powder (formulation 1)
20Comparing 2 Means - Independent Samples
- Goal Compare responses between 2 groups
(populations, treatments, conditions) - Observed individuals from the 2 groups are
samples from distinct populations (identified by
(m1,s1) and (m2,s2)) - Measurements across groups are independent
(different individuals in the 2 groups - Summary statistics obtained from the 2 groups
21Sampling Distribution of
- Underlying distributions normal ? sampling
distribution is normal - Underlying distributions nonnormal, but large
sample sizes ? sampling distribution
approximately normal - Mean, variance, standard deviation
22t-test when Variances are estimated
- Case 1 Population Variances not assumed to be
equal (s12?s22) - Approximate degrees of freedom
- Calculated from a function of sample variances
and sample sizes (see formula below) -
Satterthwaites approximation - Smaller of n1-1 and n2-1
- Estimated standard error and test statistic for
testing H0 m1m2
23t-test when Variances are estimated
- Case 2 Population Variances assumed to be equal
(s12s22) - Degrees of freedom n1n2-2
- Estimated standard error and test statistic for
testing H0 m1m2
24Example - Maze Learning (Adults/Children)
- Groups Adults (n114) / Children (n210)
- Outcome Average of Errors in Maze Learning
Task - Raw Data on next slide
- Conduct a 2-sided test of whether mean scores
differ - Construct a 95 Confidence Interval for true
difference
Source Gould and Perrin (1916)
25Example - Maze Learning (Adults/Children)
26Example - Maze LearningCase 1 - Unequal Variances
H0 m1-m2 0 HA m1-m2 ? 0 (a
0.05)
No significant difference between 2 age
groups Note Alternative would be to use 9 df
(10-1)
27Example - Maze LearningCase 2 - Equal Variances
H0 m1-m2 0 HA m1-m2 ? 0 (a
0.05)
No significant difference between 2 age groups
28SPSS Output
29C Confidence Interval for m1-m2