Confidence intervals

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Lecture 20
Outline

• Confidence intervals
• CIs for the mean based on CLT
• Small sample CIs
• the t-distribution
• Other CIs

Next Example and bootstrap confidence intervals
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Confidence intervals

• When estimating a mean, what is
• A confidence interval (CI) is a range of possible
  guesses for a parameter.
• decide on a confidence level, usually 95,then
  the CI is the statistics LL and UL s.t.

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Example CI for the mean
Pivotal quantity
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CI for the mean

• Now solve for parameter,

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Example confidence interval

• Consider the aiconditioner example from lecture
  19. Recall these are times between successive
  failures in a Boeing 720 aircraft
• We decided these followed an exponential
  distribution with s59.6

aircraft6_c(23,261,87,7,120,14,62,47, 225,71,246,2
1,42,20,5,12,120,11,3,14,71,11, 14,11,16,90,1,16,5
2,95)
use CLT CI for s
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Example confidence intervals

• Confidence interval based on CLT

gt xbar lt- mean(aircraft6) gt s2 lt-
var(aircraft6) gt n lt- length(aircraft6) gt xbar
1.96 sqrt(s2/n) 1 85.32364 gt xbar - 1.96
sqrt(s2/n) 1 33.87636
Or in the menus statistics
one sample
compare samples
T-test
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Notes about confidence intervals
See figure 7.5 on page 280 of the textbook.
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Notes about confidence intervals

• One sided confidence intervals are used if we
  want to make a one sided statement about a
  parameter. Same procedure, see page 230.
• These confidence intervals are approximate for
  two reasons

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Small sample confidence intervals

• With small samples, we cannot rely on the CLT,
  however if we have normally distributed data we
  can still produce confidence intervals based on
  the t distribution
• If the data is normally distributed, then
• Where is a t distribution with n-1
  degrees of freedom

t-based intervals
Where t is the appropriate T-critical value
from t-tables (page 540).
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Notes about the t-distribution

• Symmetric
• Quickly approaches the normal distribution in
  shape as n-1 increases
• Probabilities can be found in the t-tables (page
  540)

• Although often used in practice, if the data are
  not normally distributed
  and we do not know what to do. However in
  either case the t has longer tails than normal so
  supplies conservative inference.

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Summary of CI for the mean

• -- depends on CLT

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Other confidence intervals

• In general confidence intervals have the form
• Examples are CI for
• a proportion
• the difference between two means

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z from the normal- tables.
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