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Title: Introduction to Inference


1
Chapter 13
  • Introduction to Inference

2
Statistical Inference
  • Provides methods for drawing conclusions about a
    population from sample data
  • Confidence Intervals
  • What is the population mean?
  • Tests of Significance
  • Is the population mean larger than 66.5?

3
Inference about a MeanSimple Conditions
  1. SRS from the population of interest
  2. Variable has a Normal distribution N(m, s) in the
    population
  3. Although the value of m is unknown, the value of
    the population standard deviation s is known

4
Confidence Interval
  • A level C confidence interval has two parts
  • An interval calculated from the data, usually of
    the form estimate margin of error
  • The confidence level C, which is the probability
    that the interval will capture the true parameter
    value in repeated samples that is, C is the
    success rate for the method.

5
Case Study
NAEP Quantitative Scores (National Assessment of
Educational Progress)
Rivera-Batiz, F. L., Quantitative literacy and
the likelihood of employment among young adults,
Journal of Human Resources, 27 (1992), pp.
313-328.
What is the average score for all young adult
males?
6
Case Study
NAEP Quantitative Scores
The NAEP survey includes a short test of
quantitative skills, covering mainly basic
arithmetic and the ability to apply it to
realistic problems. Scores on the test range
from 0 to 500, with higher scores indicating
greater numerical abilities. It is known that
NAEP scores have standard deviation s 60.
7
Case Study
NAEP Quantitative Scores
In a recent year, 840 men 21 to 25 years of age
were in the NAEP sample. Their mean quantitative
score was 272. On the basis of this sample,
estimate the mean score m in the population of
all 9.5 million young men of these ages.
8
Case Study
NAEP Quantitative Scores
  1. To estimate the unknown population mean m, use
    the sample mean 272.
  2. The law of large numbers suggests that will be
    close to m, but there will be some error in the
    estimate.
  3. The sampling distribution of has the Normal
    distribution with mean m and standard deviation

9
Case Study
NAEP Quantitative Scores
10
Case Study
NAEP Quantitative Scores
11
Case Study
NAEP Quantitative Scores
12
Confidence IntervalMean of a Normal Population
  • Take an SRS of size n from a Normal population
    with unknown mean m and known standard deviation
    s. A level C confidence interval for m is
  • z is called the critical value, and z and z
    mark off the Central area C under a standard
    normal curve (next slide) values of z for many
    choices of C can be found at the bottom of Table
    C in the back of the textbook, and the most
    common values are on the next slide.

13
Confidence IntervalMean of a Normal Population
14
Case Study
NAEP Quantitative Scores
Using the 68-95-99.7 rule gave an approximate
95 confidence interval. A more precise 95
confidence interval can be found using the
appropriate value of z (1.960) with the previous
formula.
We are 95 confident that the average NAEP
quantitative score for all adult males is between
267.884 and 276.116.
15
Careful Interpretation of a Confidence Interval
  • We are 95 confident that the mean NAEP score
    for the population of all adult males is between
    267.884 and 276.116.
  • (We feel that plausible values for the
    population of males mean NAEP score are between
    267.884 and 276.116.)
  • This does not mean that 95 of all males will
    have NAEP scores between 267.884 and 276.116.
  • Statistically 95 of all samples of size 840
    from the population of males should yield a
    sample mean within two standard errors of the
    population mean i.e., in repeated samples, 95
    of the C.I.s should contain the true population
    mean.

16
Reasoning of Tests of Significance
  • What would happen if we repeated the sample or
    experiment many times?
  • How likely would it be to see the results we saw
    if the claim of the test were true?
  • Do the data give evidence against the claim?

17
Stating HypothesesNull Hypothesis, H0
  • The statement being tested in a statistical test
    is called the null hypothesis.
  • The test is designed to assess the strength of
    evidence against the null hypothesis.
  • Usually the null hypothesis is a statement of no
    effect or no difference, or it is a statement
    of equality.
  • When performing a hypothesis test, we assume that
    the null hypothesis is true until we have
    sufficient evidence against it.

18
Stating HypothesesAlternative Hypothesis, Ha
  • The statement we are trying to find evidence for
    is called the alternative hypothesis.
  • Usually the alternative hypothesis is a statement
    of there is an effect or there is a
    difference, or it is a statement of inequality.
  • The alternative hypothesis should express the
    hopes or suspicions we bring to the data. It is
    cheating to first look at the data and then frame
    Ha to fit what the data show.

19
Case Study I
Sweetening Colas
Diet colas use artificial sweeteners to avoid
sugar. These sweeteners gradually lose their
sweetness over time. Trained testers sip the
cola and assign a sweetness score of 1 to 10.
The cola is then retested after some time and the
two scores are compared to determine the
difference in sweetness after storage. Bigger
differences indicate bigger loss of sweetness.
20
Case Study I
Sweetening Colas
Suppose we know that for any cola, the sweetness
loss scores vary from taster to taster according
to a Normal distribution with standard deviation
s 1. The mean m for all tasters measures loss
of sweetness. The sweetness losses for a new
cola, as measured by 10 trained testers, yields
an average sweetness loss of 1.02. Do the
data provide sufficient evidence that the new
cola lost sweetness in storage?
21
Case Study I
Sweetening Colas
  • If the claim that m 0 is true (no loss of
    sweetness, on average), the sampling distribution
    of from 10 tasters is Normal with m 0 and
    standard deviation
  • The data yielded 1.02, which is more than
    three standard deviations from m 0. This is
    strong evidence that the new cola lost sweetness
    in storage.
  • If the data yielded 0.3, which is less than
    one standard deviations from m 0, there would
    be no evidence that the new cola lost sweetness
    in storage.

22
Case Study I
Sweetening Colas
23
The Hypotheses for Means
  • Null H0 m m0
  • One sided alternatives
  • Ha m gt m0
  • Ha m lt m0
  • Two sided alternative
  • Ha m ¹ m0

24
Case Study I
Sweetening Colas
The null hypothesis is no average sweetness loss
occurs, while the alternative hypothesis (that
which we want to show is likely to be true) is
that an average sweetness loss does occur. H0 m
0 Ha m gt 0 This is considered a one-sided
test because we are interested only in
determining if the cola lost sweetness (gaining
sweetness is of no consequence in this study).
25
Case Study II
Studying Job Satisfaction
Does the job satisfaction of assembly workers
differ when their work is machine-paced rather
than self-paced? A matched pairs study was
performed on a sample of workers, and each
workers satisfaction was assessed after working
in each setting. The response variable is the
difference in satisfaction scores, self-paced
minus machine-paced.
26
Case Study II
Studying Job Satisfaction
The null hypothesis is no average difference in
scores in the population of assembly workers,
while the alternative hypothesis (that which we
want to show is likely to be true) is there is an
average difference in scores in the population of
assembly workers. H0 m 0 Ha m ? 0 This is
considered a two-sided test because we are
interested determining if a difference exists
(the direction of the difference is not of
interest in this study).
27
Test StatisticTesting the Mean of a Normal
Population
  • Take an SRS of size n from a Normal population
    with unknown mean m and known standard deviation
    s. The test statistic for hypotheses about the
    mean (H0 m m0) of a Normal distribution is the
    standardized version of

28
Case Study I
Sweetening Colas
If the null hypothesis of no average sweetness
loss is true, the test statistic would
be Because the sample result is more than
3 standard deviations above the hypothesized mean
0, it gives strong evidence that the mean
sweetness loss is not 0, but positive.
29
P-value
  • Assuming that the null hypothesis is true, the
    probability that the test statistic would take a
    value as extreme or more extreme than the value
    actually observed is called the P-value of the
    test.
  • The smaller the P-value, the stronger the
    evidence the data provide against the null
    hypothesis. That is, a small P-value indicates a
    small likelihood of observing the sampled results
    if the null hypothesis were true.

30
P-value for Testing Means
  • Ha m gt m0
  • P-value is the probability of getting a value as
    large or larger than the observed test statistic
    (z) value.
  • Ha m lt m0
  • P-value is the probability of getting a value as
    small or smaller than the observed test statistic
    (z) value.
  • Ha m ¹ m0
  • P-value is two times the probability of getting a
    value as large or larger than the absolute value
    of the observed test statistic (z) value.

31
Case Study I
Sweetening Colas
For test statistic z 3.23 and alternative
hypothesisHa m gt 0, the P-value would
be P-value P(Z gt 3.23) 1 0.9994
0.0006 If H0 is true, there is only a 0.0006
(0.06) chance that we would see results at least
as extreme as those in the sample thus, since we
saw results that are unlikely if H0 is true, we
therefore have evidence against H0 and in favor
of Ha.
32
Case Study I
Sweetening Colas
33
Case Study II
Studying Job Satisfaction
Suppose job satisfaction scores follow a Normal
distribution with standard deviation s 60.
Data from 18 workers gave a sample mean score of
17. If the null hypothesis of no average
difference in job satisfaction is true, the test
statistic would be
34
Case Study II
Studying Job Satisfaction
For test statistic z 1.20 and alternative
hypothesisHa m ? 0, the P-value would
be P-value P(Z lt -1.20 or Z gt 1.20) 2 P(Z
lt -1.20) 2 P(Z gt 1.20) (2)(0.1151) 0.2302
If H0 is true, there is a 0.2302 (23.02)
chance that we would see results at least as
extreme as those in the sample thus, since we
saw results that are likely if H0 is true, we
therefore do not have good evidence against H0
and in favor of Ha.
35
Case Study II
Studying Job Satisfaction
36
Statistical Significance
  • If the P-value is as small as or smaller than the
    significance level a (i.e., P-value a), then we
    say that the data give results that are
    statistically significant at level a.
  • If we choose a 0.05, we are requiring that the
    data give evidence against H0 so strong that it
    would occur no more than 5 of the time when H0
    is true.
  • If we choose a 0.01, we are insisting on
    stronger evidence against H0, evidence so strong
    that it would occur only 1 of the time when H0
    is true.

37
Tests for a Population Mean
  • The four steps in carrying out a significance
    test
  • State the null and alternative hypotheses.
  • Calculate the test statistic.
  • Find the P-value.
  • State your conclusion in the context of the
    specific setting of the test.
  • The procedure for Steps 2 and 3 is on the next
    page.

38
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39
Case Study I
Sweetening Colas
  • Hypotheses H0 m 0 Ha m gt 0
  • Test Statistic
  • P-value P-value P(Z gt 3.23) 1 0.9994
    0.0006
  • Conclusion
  • Since the P-value is smaller than a 0.01,
    there is very strong evidence that the new cola
    loses sweetness on average during storage at room
    temperature.

40
Case Study II
Studying Job Satisfaction
  • Hypotheses H0 m 0 Ha m ? 0
  • Test Statistic
  • P-value P-value 2P(Z gt 1.20) (2)(1 0.8849)
    0.2302
  • Conclusion
  • Since the P-value is larger than a 0.10, there
    is not sufficient evidence that mean job
    satisfaction of assembly workers differs when
    their work is machine-paced rather than
    self-paced.

41
Confidence Intervals Two-Sided Tests
A level a two-sided significance test rejects
the null hypothesis H0 m m0 exactly when the
value m0 falls outside a level (1 a) confidence
interval for m.
42
Case Study II
Studying Job Satisfaction
A 90 confidence interval for m is
Since m0 0 is in this confidence
interval, it is plausible that the true value of
m is 0 thus, there is not sufficient
evidence(at ? 0.10) that the mean job
satisfaction of assembly workers differs when
their work is machine-paced rather than
self-paced.
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