Download Large-Sample Confidence Interval of a Population Mean

Large-Sample C.I.s for a
Population Mean;
Large-Sample C.I. for a
Population Proportion
Chapter 7: Estimation and
Statistical Intervals
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Confidence Intervals (CIs):
•  Typically: estimate ± margin of error
•  Always use an interval of the form (a, b)
•  Confidence level (C) gives the probability that
such interval(s) will cover the true value of
the parameter.
–  It does not give us the probability that our
parameter is inside the interval.
–  In Example 1: C = 0.95, what Z gives us the
middle 95%? (Look up on table)
Z-Critical for middle 95% = 1.96
–  What about for other confidence levels?
•  90%? 99%?
•  1.645 and 2.575, respectively.
Lecture 11
A large-sample Confidence Interval:
•  Data: SRS of n observations (large sample)
•  Assumption: population distribution is N
(µ,σ) with unknown µ and σ
•  General formula:
s
X ± (z critical value)
n
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Interpreting CI
•  Given a 95% Confidence Level, the
Confidence Interval of a population mean
should be interpreted as:
–  We are 95% confident that the population
mean falls in the interval (lower limit, upper
limit)
•  For the example we just saw, we say
–  We are 95% confident that the mean corn
yield is between X − 1.96 s , X + 1.96 s
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Lecture 13
n
n
5
Choosing a sample size:
•  The margin of error or half-width of the
interval is sometimes called the bound on
the error of estimation
•  Before collecting data, we can determine the
sample size for a specific bound, B.
•  We just rearrange the margin of error
2
formula by solving for n
⎛ 1.96 s ⎞
•  For 95% confidence, we have: n = ⎜ B ⎟
⎝
⎠
•  For any confidence level, we have
2
⎛ Z Crit s ⎞
n = ⎜
⎟
⎝ B ⎠
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Example 2 (cont.)
•  Suppose we wanted to estimate the mean
breakdown voltage in our previous example but we
wanted a bound, B, of no more than 0.5kV with
95% confidence.
•  What is the required sample size to achieve this
bound?
2
2
⎛ 1.96 s ⎞ ⎛ 1.96 × 5.23 ⎞
n = ⎜
, ⎟ = 420.3
⎟ = ⎜
0.5
⎝ B ⎠ ⎝
⎠
rounded up to 421.
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If s is unknown?
•  If you don’t have a sample standard
deviation, you may use a “best guess” from
a previous study of what it might be.
•  OR, as long as the population is not too
skewed, dividing the range by 4 often gives
a rough idea of what s might be.
•  For 95% confidence:
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⎛ 1.96(range / 4) ⎞
n = ⎜
⎟
B
⎝
⎠
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8
One-sided Confidence Intervals
(Confidence Bounds)
•  There are circumstances where we are only
interested in a bound or limit on some
measurement
–  Examples? Cutoff score for the top 10% students
in a Science Competition.
•  To do this we simply put all the area on one
side, maintaining the confidence and Zcritical value we desire.
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One-sided Confidence Intervals
(Confidence Bounds)
•  Large-sample confidence bounds
•  Upper:
s
µ < X + (z critical value)
n
•  Lower:
s
µ > X − (z critical value)
n
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7.3
More Large Sample Confidence Intervals
•  Be aware that most confidence intervals take
a similar format
estimate ± critical value ⋅ SEestimate
•  Understanding the sampling distribution of
the estimate is the critical part that gives us
the pieces above
•  We’ll come back to this in a few minutes!
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Confidence interval for p
•  To estimate the pop proportion p (or
called π), we can use the sample
proportion p̂
–  Recall p is a number between 0 and 1
•  How to find a confidence interval for p?
–  Need to know the mean, standard deviation
and sampling distribution of p̂
–  When the sampling distribution is known, we
can use it to calculate the CI under certain
confidence level
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Sampling Distribution of p
•  As we’ve seen in chapter 5, from the CLT
we have (when n is sufficiently large):
⎛
p(1 − p) ⎞
pˆ ~ N ⎜⎜ p,
⎟⎟
n
⎝
⎠
•  We can then standardize p̂, and get a
standard normal distribution
z=
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pˆ − p
~ N ( 0,1)
p(1 − p)
n
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Confidence interval for π
•  So, based on the previous formula, we can
construct a confidence interval as such:
P(|
pˆ − p
|< Zcrit ) = confidence level
p(1 − p)
n
•  So thankfully, when n is large (≥25), we have:
pˆ (1 − pˆ )
pˆ ± Zcrit
n
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Example 3: Parking problem?!
•  To estimate the proportion of Purdue
Students who think parking is a problem, we
sample 100 students and find that 67 of them
agree that parking is indeed a problem.
•  Give a 95% confidence interval for the true
proportion of students that think parking is a
problem.
–  Make sure you can interpret the interval.
Answer: (58%,76%).
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After Class…
•  Review Sections 7.1 through 7.3
•  Read sections 7.4 (till Pg 316) and 7.5
•  Exam#1, next Tuesday evening.
•  Lab#3, next Wed.
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