Transcript 2007

Topics 21 - 23
Topic 21
Comparing Two Proportion
Topic 22
Comparing Two Means
Topic 22 - Confidence Interval: Comparing two Mean
The purpose of confidence intervals is to use the sample statistic to construct an
interval of values that you can be reasonably confident contains the actual, though
unknown, parameter.
The estimated standard deviation of the sample statistic X-bar is called the standard error
2
s1
2

n1
n2
Confidence Interval for a population proportion :
2
estimate
 margin
of error
 X1  X
2
where n >= 30
t * is calculated based on level of confidence
When running for example 95% Confidence Interval:
95% is called Confidence Level and
we are allowing possible 5% for error, we call this alpha
(α )= 5% where α is the significant level
 t
*
s1
n1
2

s2
s2
n2
Topic 22 - Confidence Interval: Comparing two Mean
Use if the sample data is given, use the Stat,
Edit and enter data in the calculator before
running the Confidence Interval
L1 and L2 is where data is entered by you
C-Level: is the level you are running the
Confidence Interval
Use if the information about sample data is
given.
X-Bar mean of sample data
Sx is Standard deviation of the sample
n is sample size
C-Level: is the level you are running the
Confidence Interval
Topic 22 – Test of Significant: Comparing two Mean
The purpose of Test of Significant is when we do know the population Parameter but
we do not necessary agree with it or we have question about it. To do the test we
need to run a sample and we use the statistic to test its validity.
Step 1: Identify and define the parameter.
Step 2: we initiate hypothesis regarding the
question – we can not run test of significant
without establishing the hypothesis
H

H




0
:
1   2
a
:
1   2
or
1   2
or
1   2
Step 3: Decide what test we have to run, in case of proportion, we use t-test
t 
x1  x 2
2
s1
n1
2

s2
n2
Topic 22 – Test of Significant: Comparing two Mean
Step 4: Run the test from calculator
Step 5: From the calculator write
down the p-value T-test
Step 6: Compare your p-value with α – alpha – Significant Level
If p-value is smaller than α
we “reject” the null hypothesis, then it is statistically significant based on data.
If p-value is greater than the α
we “Fail to reject” the null hypothesis, then it is not statistically significant based on data.
Last step: we write conclusion based on step 6 at significant level α
•
•
•
•
•
p- value > 0.1: little or no evidence against H0
0.05 < p- value <= 0.10: some evidence against H0
0.01 < p- value <= 0.05: moderate evidence against H0
0.001 < p- value <= 0.01: strong evidence against H0
p- value <= 0.001: very strong evidence against H0
Topic 22 – Test of Significant: Comparing two Mean
Use if the sample data is given, use the Stat,
Edit and enter data in the calculator before
running the T-test
µ0 is mean–value in question
List: L1 where the raw data is entered by you
µ: is the alternative hypothesis
Use if the information about sample data is
given.
µ0 is mean–value in question
X-bar is sample mean
Sx is Sample Standard deviation
n is sample size
µ: is the alternative hypothesis
Activity 22-1 Close Friends
Recall from Activity 19- 17 that one of the questions asked of a random
sample of adult Americans in the 2004 General Social Survey was, “ From
time to time, most people discuss important matters with other people.
Looking back over the last six months— who are the people with whom
you discussed matters important to you? Just tell me their first names or
initials.” The interviewer then recorded how many names or initials the
respondent mentioned. Suppose you want to examine whether men and
women differ with regard to how many names they tend to mention. (
For convenience, we will refer to those named as “ close friends.”)
a. Is this an observational study or an experiment? Explain.
b. Identify the explanatory and response variable for this study. Also
classify each variable as categorical ( also binary) or quantitative.
Explanatory:
Type:
Response:
Type:
a. This is an observational study because the researcher simply observed the gender of each
subject—he/she did not randomly assign gender to subjects.
b. Explanatory: gender
Type: binary categorical
Response: number of “close friends”
Type: quantitative
Activity 22-1 Close Friends (Cont)
c. State the null and alternative hypotheses, in symbols and in words, for
testing whether the sample data provide evidence that men and women
differ with regard to the average number of close friends they tend to
mention in response to this question.
c. The null hypothesis is men and women tend to mention the same average number of
close friends in response to this question. In symbols, H 0 : µ m = µf . The alternative
hypothesis is men and women differ in the average number of close friends they tend to
mention in response to this question. In symbols, H a : µm ≠ µf .
Activity 22-1 Close Friends (Cont)
Before you learn a new test procedure for handling this situation, let’s begin
with a preliminary analysis of the data. Sample responses by gender are
tallied in the following table: (look at the book)
Some descriptive summaries follow: (book)
e. Are these values parameters or statistics? Explain.
f.
Produce box plots ( on the same scale) to compare the distributions of
the number of close friends between males and females.
g. Comment on any differences that you observe in the distributions of the
number of close friends between males and females.
h. Would it be possible to obtain sample means this far apart even if the
population means were equal? Explain.
e. These values are statistics because they describe samples, not
populations.
f. The following boxplots compare the distributions of the number of close
friends:
g. There are very few differences in the distributions of the number of close
friends mentioned between males and females. The males have a slightly
lower mean, median, and lower quartile, but the upper quartiles and
maximums are identical to those of females.
h. Yes, it would be possible to obtain sample means this far apart even if the
population means were equal.
Once again, because of sampling variability, you cannot conclude that
simply because these sample means differ, the means of the respective
populations must differ as well. As always, you can use a test of
significance to establish whether a sample result ( in this case, the
observed difference in sample mean number of close friends) is “
significant” in the sense of being unlikely to have occurred by chance (
from random sampling) alone.
Also, you can use a confidence interval to estimate the magnitude of the
difference in the population means.
Inference procedures for comparing the population means of two
different groups are similar to those for comparing population
proportions in that they take into account sample information from both
groups. These procedures are similar to those for a single population
mean in that they use the t- distribution, and the sample sizes, sample
means, and sample standard deviations are the relevant summary
statistics. The details for conducting confidence intervals and significance
tests concerning the difference between two population means, which
will be denoted by µ1 and µ2 , are presented here.
Notes
• As always, the symbols x-bar
and s represent a sample mean
and a sample standard
deviation, respectively.
The subscripts indicate the
population from which the
observational units are randomly
selected or the treatment group
to which they are randomly
assigned.
• The structure, reasoning, and
interpretation of this test and
interval procedure are the same
as for other tests and intervals
that you have studied.
Two- sample t- procedures
Two- sample t- procedures apply to scenarios involving random sampling
from two populations and/ or random assignment to two treatment
groups. As before, the calculations are identical, but the scope of
conclusions is very different for these two scenarios.
The degrees of freedom convention being used is a conservative
approximation, meaning the degrees of freedom is on the low side, so
the critical value will be slightly greater than it needs to be; thus the
interval will be slightly wider and, therefore, will succeed in capturing
µ1 - µ2 slightly more often than the confidence level indicates.
When using technology, a more exact critical value will be computed for
you. The p- value calculation can also differ a bit with technology, again
based on the degrees of freedom.
Activity 22-1 Close Friends (Cont)
i.
Use the summary statistics provided after part d to calculate the test statistic
for testing the hypotheses you stated in part c.
j.
Use Table III ( t- Distribution Critical Values) to find ( as accurately as possible)
the p- value of the test.
k.
Which of the following is a correct interpretation of the p- value?
• The p- value is the probability that males and females have the same mean
number of close friends in these samples.
• The p- value is the probability that males and females have the same mean
number of close friends in the populations.
• The p- value is the probability that males have a higher mean number of close
friends than females do.
• The p- value is the probability of getting sample data so extreme if, in fact, males
and females have the same mean number of close friends in the populations.
IJK
i. The test statistic is t = 2.45.
j. Using 500 degrees of freedom, p-value = XX
k. The correct interpretation of the p-value is
“The p-value is the probability of getting sample data so extreme if, in
fact, males and females have the same mean number of close friends in
the populations.”
Activity 22-1 Close Friends (Cont)
l.
Is this p- value small enough to reject the null hypothesis that these
population means are equal at the .05 significance level?
m. Is the observed difference in sample means statistically significant at the
.01 level?
n. State the technical conditions necessary for this procedure to be valid.
Does the strong skewness in the sample data provide any reason to
doubt the validity of this test?
LMN
l.
Yes, this p-value is small enough to reject the null hypothesis at the
alpha = .05 significance level (p-value = xxx).
m. No, the observed difference in sample means is not statistically
significant at the alpha = .01 significance level (p-value = XXX).
n. Technical conditions:
i. The data are a random sample broken into two distinct groups (see
page 418).
ii. Both sample size are large (greater than 30). Because both sample
sizes are large, you do not need to worry about the strong skewness in
the sample data. It does not provide any reason to doubt the validity of
this test.
Activity 22-1 Close Friends (Cont)
o. Determine and interpret a 95% confidence interval for the difference in
population means µf - µm .
Hint: Be sure to comment on whether the interval is entirely negative,
entirely positive, or contains zero. Also explain the importance of
whether the interval includes zero.
o.
For a 95% CI for µf - µm with 500 degrees of freedom, you calculate (0.045,
0.411).
This interval is entirely positive (and does not include zero), which means you
can be 95% confident that the mean number of close friends that women
have is between .045 and .411 greater than the mean number of close friends
that men have.
Watch Out
Remember that failing to reject a null hypothesis is not the same as
accepting it.
You do not have enough evidence to conclude that one route is faster
than the other on average for Alex, but you should not conclude that the
average commuting times are identical for the two routes.
Larger samples might produce a statistically significant difference.
Activity 22- 2: Hypothetical Commuting Times
Suppose Alex wants to determine which of two possible driving routes gets him
to school more quickly. Also suppose that over a period of 20 days, he randomly
decides which route to drive each day. He then records the commuting times (
in minutes) and displays them as follows:
Route 1 19.3 20.5 23.0 25.8
28.0
28.8 30.6 32.1 33.5 38.4
Route 2 23.7 24.5 27.7 30.0 31.9
32.5
32.6 35.5 38.7 42.9
a.
Does one route always get Alex to school more quickly than the other?
b. Do the data suggest that one route tends to get Alex to school more quickly
than the other? If so, which route appears to be quicker?
c.
Complete the table below
d.
hypothesis, t and p-value
Size
Alex Route 1
Alex Route 2
Mean
SD
P-Value
a. No, one route does not always get Alex to school more quickly than the
other.
b. Yes, the data suggest that Route 1 tends to get Alex to school more
quickly than Route 2.
c. Here is the completed table:
d. H 0 : µ 1 = µ2 H a : µ1 ≠ µ2
t = -1.49, the p-value is .1702
Activity 22- 2: Hypothetical Commuting Times (cont)
e. Are Alex’s data statistically significant at any of the commonly used
significance levels? Can Alex reasonably conclude that one route is
faster than the other route for getting to school? Explain.
f.
Use technology to calculate a 90% confidence interval for the difference
in Alex’s mean commuting times between route 1 and route 2. Record
the confidence interval.
g. Does this interval include the value zero? Explain the importance of this.
e. No, Alex’s data are not statistically significant at any of the commonly used significance
levels. Alex cannot reasonably conclude that one route is faster than the other route for
getting to school because his p-value is not small and will not allow you to reject the null
hypothesis that the average time required for both routes is the same.
f. (8.92, 0.92).
g. Yes, this interval includes the value zero. This means that zero is a plausible value for the
difference in the population means or that you cannot conclude there is a difference in
the population mean travel times (with 90% confidence).
k. For each commuter ( Barb, Carl, and Donna), use technology to conduct a
significance test of whether the difference in his or her sample mean
commuting times is statistically significant. Record the p- values of these
tests in the following table, along with the appropriate sample statistics:
Size
Barb: Route 1
Barb: Route 2
Carl: Route 1
Carl: Route 2
Donna: Route 1
Donna: Route 2
Mean
SD
P-Value
These comparisons should help you to see the roles of sample sizes, means,
and standard deviations in the two- sample t- test. All else being the
same, the test result becomes more statistically significant ( i. e., the pvalue becomes smaller) as
• The difference in sample means increases
• The sample sizes increase
• The sample standard deviations decrease
Note that the researcher has no control over the sample means but can
determine the sample sizes.
Larger samples are better, but they require more time and expense. The
researcher seems to have no control over the standard deviations, but
this third bullet reveals why statisticians like to reduce variability as much
as possible ( for example, using better measurement tools).
Watch Out (cont)
Always remember to check technical conditions before taking a test or
interval result seriously. Also notice that the second technical condition
here is an either/ or statement: the distributions do not have to be
normal if the sample sizes are large.
Remember the scope of conclusions, with regard to causation and
generalizability, depends on how the study is conducted.
Do not forget to relate your conclusions to the context of the study. Do
not simply say “ reject H0 ” and leave it at that. Such a conclusion would
not be very helpful
Topic 23
Analyzing Paired Data
Activity 23-1: Marriage Ages – page 498
Matched Pair Analysis – Page 503, 500
Topic 23 - Confidence Interval: Mean, σ is unknown
The purpose of confidence intervals is to use the sample statistic to construct
an interval of values that you can be reasonably confident contains the actual,
though unknown, parameter.
The estimated standard deviation of the sample statistic X-bar is called the
standard error
sd
n
Confidence Interval for a population proportion :
estimate
 margin
of error
 X
where n >= 30
t * is calculated based on level of confidence
When running for example 95% Confidence
Interval:
95% is called Confidence Level and
we are allowing possible 5% for error, we call this
alpha (α )= 5% where α is the significant level
d
 t
*
sd
n
Topic 23 - Confidence Interval: Mean, σ is unknown
Use if the sample data is given, use the
Stat, Edit and enter data in the calculator
before running the Confidence Interval
L1 is where data is entered by you
C-Level: is the level you are running the
Confidence Interval
Use if the information about sample
data is given.
X-Bar mean of sample data
Sx is Standard deviation of the sample
n is sample size
C-Level: is the level you are running
the Confidence Interval
Topic 23 – Test of Significant: Mean
The purpose of Test of Significant is when we do know the population
Parameter but we do not necessary agree with it or we have question about it.
To do the test we need to run a sample and we use the statistic to test its
validity.
Step 1: Identify and define the parameter.
Step 2: we initiate hypothesis regarding
the question – we can not run test of
significant without establishing the
hypothesis
H

H




0
:
  0
a
:
  0
or
  0
or
  0
Step 3: Decide what test we have to run, in case of proportion, we use t-test
t 
xd
sd
n
Topic 23 – Test of Significant: Mean
Step 4: Run the test from
calculator
Step 5: From the calculator
write down the p-value T-test
Step 6: Compare your p-value with α – alpha – Significant Level
If p-value is smaller than α
we “reject” the null hypothesis, then it is statistically significant based on data.
If p-value is greater than the α
we “Fail to reject” the null hypothesis, then it is not statistically significant based on
data.
Last step: we write conclusion based on step 6 at significant level α
•
•
•
•
•
p- value > 0.1: little or no evidence against H0
0.05 < p- value <= 0.10: some evidence against H0
0.01 < p- value <= 0.05: moderate evidence against H0
0.001 < p- value <= 0.01: strong evidence against H0
p- value <= 0.001: very strong evidence against H0
Exercise 23-6: Cow Milking – Page 508
Exercise 23-17: Mice Cooling – Page 511
Exercise 23-19: Exam Score Improvement - Page 512
Exercise 23-20: Exam Score Improvement - Page 513