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 2/17/12

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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 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 ⎠ 2/17/12

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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

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⎛ 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: 2/17/12

⎛ 1.96(range / 4) ⎞ n = ⎜ ⎟ B ⎝ ⎠ Lecture 13

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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 Zcritical value we desire. 2/17/12

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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! 2/17/12

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Confidence interval for p •  To estimate the pop proportion p (or called π), we can use the sample proportion p ˆ –  Recall p is a number between 0 and 1

•  How to find a confidence interval for p? –  Need to know the mean, standard deviation ˆ and sampling distribution of p –  When the sampling distribution is known, we can use it to calculate the CI under certain confidence level 2/17/12

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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 ⎝ ⎠

ˆ , and get a •  We can then standardize p standard normal distribution z=

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pˆ − p ~ N ( 0,1) p(1 − p) n Lecture 13

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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 2/17/12

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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%). 2/17/12

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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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