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Chapter 11 Problems of Estimation
11.1 Estimation of means 11.2 Estimation of means (unknown variance) 11.3 Skip 11.4 Estimation of proportions
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11.1 The Estimation of Means
How to estimate the population mean μ, and standard deviation σfrom sample data x1, x2, …, xn? We usually use sample mean to estimate μ and sample standard deviation s to estimate σ. and s are called point estimates.
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Point estimate of the mean
For a certain sample, sample mean, which is the point estimate of the population mean, is a single number. Since sample means fluctuate from sample to sample, we must expect an error . A point estimate along does not tell us about the possible size of the error.
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Interval Estimate—Confidence intervals
An interval estimate consists of an interval which will contain the quantity it is supposed to estimate with a specified probability (or degree of confidence). Recall that for large random samples from infinite populations, the sampling distribution of the mean is approximately a normal distribution with So we will utilize some properties of normal distribution to explain a confidence interval.
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For a standard normal curve
-za/2 za/2 Standard normal Define Za/2 to be such that P(Z > Za/2)=a/2. Hence the area under the standard normal curve between -Za/2 and Za/2 is equal to 1-a. 1-a a / Za/
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For X normal with mean m and standard deviation s,
Distribution of m With probability 1-a, deviates from m by no more than This is called maximum error of estimate with probability 1-a.
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For X normal with mean m and standard deviation s,
.95 .05 Distribution of m The probability is 0.95 that will differ from m by at most or approximately to be “off” either way by at most 1.96 standard errors of the mean.
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Maximum error E with probability 1-a
With probability 0.95, deviates from μ by no more than (approximately 2 standard error away from the true value) Probability Maximum error E 0.80 0.90 0.95 0.99
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Maximum error E with probability
The maximum error depends on both the confidence level and sample size! You can determine the sample size according to the confidence level and the maximum error.
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Sample size for estimating m
How large must our sample to keep our error no more than E with probability 1-a? As s2 increases, n increases. As E decreases, n increases. As our error probability a decreases, n increases.
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Confidence Interval for Means
After computing sample mean , find a range of values such that 95% of the time the resulting range includes the true value m.
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Degree of Confidence The degree of confidence states the probability that the interval will give a correct answer. If you use 95% confidence interval often, in the long run 95% of your intervals will contain the true parameter value. When the method is applied once, you do not know if your interval gave a correct value (95% of the time) or not (5% of the time).
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Example 11.1 Suppose we measure specific gravity of a metal, and σ=0.025. Send each of you into the lab to take n=25 measurements:
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Example 11.1 95% CI for the mean:
If the true value is 2, then about 95% of students will find this is true:
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Confidence Intervals 100(1-a)% CI: 80% 90% 95% 99%
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Example 11.2 X=breaking strength of a fish line.
σ= In a random sample of size n=10, Find a 95% confidence interval for μ, the true average breaking strength.
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Solution: Standard error of the mean:
Critical value=1.96; maximum error is CI: from to 10.36
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Example 11.2 (continued) How large a sample size is needed in order to get a maximum error no more than 0.01with 95% probability if the sample mean is used to estimate the true mean? Solution n=385, always round up!
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11.2 Estimation of Means (unknown variance)
A sample of size n: x1, x2, …, xn from a normal population with mean μ, and standard deviation, σ. If σis known, with probability
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If σis unknown Estimate σby sample standard deviation s
The estimated standard error of the mean will be Using the estimated standard error we have a confidence interval of The multiplier needs to be bigger than Za/2 (e.g., 1.96). The confidence interval needs to be wider to take into account the added uncertainty in using s to estimate s. The correct multipliers were figured out by a Guinness Brewery worker.
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What is the correct multiplier? “t”
100(1-a)% confidence interval when s is unknown 95% CI =100(1-0.05)% confidence interval when s is unknown
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Properties of t distribution
The value of ta/2 depends on how much information we have about s. The amount of information we have about s depends on the sample size. The information is “degrees of freedom” and for a sample from one normal population this will be: df=n-1.
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t curve and z curve Both the standard normal curve N(0,1) (the z distribution), and all t(k) distributions are density curves, symmetric about a mean of 0, but t distributions have more probability in the tails. You can verify this for yourself by comparing values from Table B with those on the n=infinity line of Table C. As the sample size increases, this decreases and the t distribution more closely approximates the z distribution. By n = 1000 they are virtually indistinguishable from one another.
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Critical values of t distribution
t table is given in the book (p. 497) It depends on the degrees of freedom as well Df alpha t
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Areas under the curve The area between and is
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Confidence interval for the mean when s is unknown
With probability Maximum error
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Example (ex. 11.16, p 273) Noise level, n=12
Point estimate for the average noise level of vacuum cleaners; 95% Confidence interval
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Solution n=12, Critical value with df=11 95% CI:
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11.4 The Estimation of Proportions
Notation: 1. μ, σ mean and variance p proportion=probability of a success Consider count data: n=# of trials, p=probability of a success
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Estimate of p Xi=0, or 1 with probability 1-p or p
Mean of Xi =p: population mean X=sum of Xi Sample proportion (mean) X/n p
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Example 11.4 Toss a coin 100 times and you get 45 heads
Estimate p=probability of getting a head Solution: Is the coin balanced one?
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Estimate of p If np≥5 and n(1-p)≥5, then is approximately normal.
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Maximum error We have (1-a)100% confidence that the error in our estimate is at most (worst case is p=1/2.)
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CI An approximate 100(1-a)% confidence interval for p is
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Sample Size The sample size required to have probability 1-a that our error is no more than E is Since p is unknown, you have to estimate it in the formula.
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Maximize p(1-p) to get the sample size
If you don’t have any prior information about p, then Maximum p(1-p)=1/4
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If you know p is somewhere …
If then maximum p(1-p)=0.3(1-0.3)=0.21 maximum p(1-p)=0.4(1-0.4)=0.24
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How to estimate the maximum
Estimate p(1-p) by substitute p with the value closest to 0.5 (0, ), p=0.1 (0.3, 0.4), p=0.4 (0.6, 1.0), p=0.6
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Example 11.4 (continued) 95% CI for p 0.3525<p< with 95% probability
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Example 11.5 (example 11.13 in text)
A state highway dept wants to estimate what proportion of all trucks operating between two cities carry too heavy a load 95% probability to assert that the error is no more than 0.04 Sample size needed if p between 0.10 to 0.25 no idea what p is
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Solution E=0.04, p=0.25 Round up to get n=451 E=0.04, p(1-p)=1/4 n=601
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