large-sample confidence interval of a population proportion
TRANSCRIPT
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:
ns value)critical (z ±X
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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
s s 1.96 , 1.96n n
X X− +2/17/12 5 Lecture 13
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 formula by solving for n
• For 95% confidence, we have: • For any confidence level, we have
296.1⎟⎠
⎞⎜⎝
⎛=Bsn
2CritZ snB
⎛ ⎞= ⎜ ⎟⎝ ⎠2/17/12 6 Lecture 13
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?
, rounded up to 421.
2 21.96 1.96 5.23 420.30.5
snB
×⎛ ⎞ ⎛ ⎞= = =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
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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: 2)4/(96.1⎟⎠
⎞⎜⎝
⎛=Brangen
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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 Z-critical value we desire.
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One-sided Confidence Intervals (Confidence Bounds)
• Large-sample confidence bounds
• Upper:
• Lower:
ns value)critical (zX +<µ
ns value)critical (zX −>µ
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7.3 More Large Sample Confidence Intervals
• Be aware that most confidence intervals take a similar format
• 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!
estimateSEvaluecriticalestimate ⋅±
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Confidence interval for p
• To estimate the pop proportion p (or called π), we can use the sample proportion – 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 – When the sampling distribution is known, we
can use it to calculate the CI under certain confidence level
p̂
p̂
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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):
• We can then standardize , and get a
standard normal distribution
(1 )ˆ ~ , p pp N pn
⎛ ⎞−⎜ ⎟⎜ ⎟⎝ ⎠
( )ˆ
~ 0,1(1 )p pz Np pn
−=
−
p̂
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Confidence interval for π
• So, based on the previous formula, we can construct a confidence interval as such:
• So thankfully, when n is large (≥25), we have:
ˆ(| | ) confidence level
(1 )p pP Zcritp pn
−< =
−
ˆ ˆ(1 )ˆ p pp Zcritn−
±
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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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