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
(No Transcript)
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.