Confidence%20intervals:%20the%20basics

1
Chapter 13

• Confidence intervals the basics

2
Statistical Inference

• Two general types of statistical inference
• Confidence Intervals (introduced this chapter)
• Tests of Significance (introduced next chapter)

3
Starting Conditions

1. SRS from population
2. Normal distribution XN(m, s) in the population
3. Although the value of m is unknown, the value of
   the population standard deviation s is known

4
Case Study
NAEP Quantitative Scores (National Assessment of
Educational Progress)
Rivera-Batiz, F. L. (1992). Quantitative literacy
and the likelihood of employment among young
adults. Journal of Human Resources, 27, 313-328.
The NAEP survey includes a short test of
quantitative skills, covering mainly basic
arithmetic and the ability to apply it to
realistic problems. Young people have a better
chance of good jobs and wages if they are good
with numbers.
5
Case Study
NAEP Quantitative Scores

• Given
• Scores on the test range from 0 to 500
• Higher scores indicate greater numerical ability
• It is known NAEP scores have standard deviation s
  60.
• In a recent year, 840 men 21 to 25 years of age
  were in the NAEP sample
• Their mean quantitative score was 272 (x-bar).
• On the basis of this sample, estimate the mean
  score µ in the population of 9.5 million young
  men in this age range

6
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 a Normal
   distribution with unknown mean m and standard
   deviation

7
Case Study
NAEP Quantitative Scores
8
Case Study
NAEP Quantitative Scores
9
Case Study
NAEP Quantitative Scores
10
NAEP Illustration (cont.)

• The confidence interval has the formestimate
  margin of error
• estimate (x-bar in this case) is our guess for
  unknown µ
• margin of error ( 4.2 in this case) shows
  accuracy of estimate

11
Level of Confidence (C)

• Probability that interval will capture the true
  parameter in repeated samples the success rate
  for the method
• You can choose any level of confidence, but the
  most common levels are
• 90
• 95
• 99
• e.g., If we use 95 confidence, we are saying we
  got this interval by a method that gives correct
  results 95 of the time (next slide)

12
Fig 13.4

• Twenty-five samples from the same population gave
  25 95 confidence intervals
• In the long run, 95 of samples give an interval
  that capture the true population mean µ

13
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

Confidence level C 90 95 99
Critical value z 1.645 1.960 2.576
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Confidence IntervalMean of a Normal Population
15
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.
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How Confidence Intervals Behave

• The margin of error is
• The margin of error gets smaller, resulting in
  more accurate inference,
• when n gets larger
• when z gets smaller (confidence level gets
  smaller)
• when s gets smaller (less variation)

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Case Study
NAEP Quantitative Scores
The 90 CI is narrower than the 95 CI.
18
Choosing the Sample Size
The confidence interval for the mean of a Normal
population will have a specified margin of error
m when the sample size is
19
Case Study
NAEP Quantitative Scores
Suppose that we want to estimate the population
mean NAEP scores using a 90 confidence interval,
and we are instructed to do so such that the
margin of error does not exceed 3 points.
What sample size will be required to enable us
to create such an interval?
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Case Study
NAEP Quantitative Scores
Thus, we will need to sample at least 1082.41
men aged 21 to 25 years to ensure a margin of
error not to exceed 3 points. Note that since
we cant sample a fraction of an individual and
using 1082 men will yield a margin of error
slightly more than 3 points, our sample size
should be n 1083 men.