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.

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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

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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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