Survey

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Psychometrics wikipedia, lookup

Bootstrapping (statistics) wikipedia, lookup

Taylor's law wikipedia, lookup

German tank problem wikipedia, lookup

Resampling (statistics) wikipedia, lookup

Student's t-test wikipedia, lookup

Transcript
```TWO SAMPLE
STATISTICAL INFERENCES
1
How can we compare two means?
• Suppose that we have two populations.
Furthermore, suppose that
the first population has a mean 1 and the second population a
mean 2.
• We would like to use inferential statistics to compare 1 and 2 by
using either a confidence interval and/or a significance test.
2
Independent Samples
• Independent Samples occur when observations are gathered
independently from two different populations.
•
For example, suppose we want to compare starting salaries for males
and females at a certain bank. We collect a random sample of males and
a random sample of females and measure their starting salaries.
3
Paired Samples
• Paired Samples occur when observations are
gathered in pairs, where each pair consists of an
observation from population 1 and an observation
from population 2, and these two observations are
related.
• Although there are other types of paired samples, a
common type is a before-after sample.
4
Paired Samples
• For example, suppose we want to determine the effectiveness
of a new drug in reducing blood pressure.
• For each individual in the sample a measurement is taken prior to the
medication and another measurement is taken after using the
medication for a fixed period of time.
• These two observations (for each individual) are related. This constitutes
a paired sample.
5
Outline
• Two sample comparisons:
• Paired Samples (dependent samples)
• Independent Samples – Population Standard Deviations
Not Assumed to be Equal (σ1≠ σ2)
• Independent Samples – Population Standard Deviations
Assumed to be Equal (σ1= σ2)
• All methods assume random samples from normal
populations with unknown means and unknown
standard deviations.
• For small sample sizes, it is critical that our observations
are normal or approximately normal.
• However, for sufficiently large sample sizes (from the
C.L.T.) the assumptions of normality can be relaxed.
6
PAIRED SAMPLES
7
Paired (dependent) Samples
8
Paired (dependent) Samples
9
Paired (dependent) Samples
t-score = di invttail(n-1,α/2)
10
Paired (dependent) Samples
11
INDEPENDENT TWO
SAMPLES
12
Independent Two Samples
• We gather independent and random samples
from each population.
13
Independent Two Samples
14
Independent Two Samples
15
Independent Two Samples: Not Assuming Equal
Standard Deviations (σ1≠ σ2)
~ tdf=v,
Note: Unless we obtain our degrees of freedom using a computer, we will
round v down to the nearest integer.
16
Independent Two Samples: Not Assuming Equal
Standard Deviations (σ1≠ σ2)
• A (1-α)×100% CI is
( x1  x2 )  t
*
df  v , / 2
2
1
2
2
s
s

n1 n2
*
where tdf v, / 2 a critical value that makes the right tail
probability equal to α/2
Stata: di invttail(df, α/2)
17
Independent Two Samples: Not Assuming Equal
Standard Deviations (σ1≠ σ2)
18
Exercise: Does alcohol affect males and
females differently?
A study involving males and females with similar physical
characteristics was conducted. In a controlled setting, each
individual was asked to consume 4 ounces of alcohol. One
hour after consumption each participant took a Breathalyzer to
measure his or her blood alcohol level. The following results
were obtained:
•Is
there a real difference in blood alcohol levels between males and
females? Carry out an appropriate test to answer this question.
•Calculate a 99% confidence interval for the difference in blood alcohol
levels between males and females.
19
Exercise
• Stata command and outputs:
•
ttesti n1 u1 s1 n2 u2 s2, unequal level(#)
20
Independent Samples: Assuming Equal Standard
Deviations (σ1= σ2)
21
Independent Samples: Assuming Equal Standard
Deviations (σ1= σ2)
22
Independent Samples: Assuming Equal Standard
Deviations (σ1= σ2)
23
Independent Samples: Assuming Equal Standard
Deviations (σ1= σ2)
24
To be pooled, or not to be pooled? Equal S.D. or
not equal?
• We
could base our decision on an informal rule (usually
works and is much less complicated).
• If
no sample standard deviation is twice the other, (i.e 0.5 < s1/s2 <
2), then the assumption of equal standard deviations should be ok.
• We
could perform a graphical analysis and look at the
box-plots for the samples to informally assess the equal
standard deviations assumption.
• There are formal tests to assess the evidence against
equal population standard deviations
• Variance Ratio Test
• Levene’s Test
25
Variance Ratio Test
• Ho: σ12 = σ22
vs. Ha: σ12 ≠ σ22
• Test statistic: F= s12 / s22 ~ Fdf1=n1-1, df2=n2-1
• p-value:
26
Variance Ratio Test
• Stata command:
sdtesti n1 . sd1 n2 . sd2
• Example: Test σ1= σ2 when observed n1=75,
sd1=6.5, n2=65, and sd2= 7.5
• sdtesti 75 . 6.5
65 . 7.5
27
Exercise
• Two
machines are used to fill plastic bottles with
dishwashing detergent. Two random samples are taken
and the following results are obtained:
• Is
there a real difference between their average fills?
Carry out an appropriate test to answer this question.
• Calculate a 95% confidence interval for the difference in
the mean fill between the two machines
28
Exercise
• s1=√0.113=0.336; s2= √0.125 =0.354
• Testing for Equal Variances
29
Exercise
• Stata command for two-sample t test with equal variances
• ttesti N1 u1 s1 N2 u2 S2, level(#)
Compared with machine 2, machine 1 had a statistically significantly lower
mean fill. The difference in mean fill was -0.41 (95%CI: -0.59 to -0.23) for
machine 2 relative to machine 1 (p<0.001).
30
```
Related documents