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January 11, 2010

Statistics Notes — Confidence Intervals for the Mean (Large Samples)

Filed under: Statistics, Statistics Notes — Tags: — bowman @ 8:30 am

There are two types of estimation. They can be first understood through the following example: Let’s say that you take your car to your local car dealer’s service department and you ask the service manager how much it will cost to repair your car. If the manager says it will cost you $500 then he is providing a point estimate. If the manager says it will cost somewhere between $400 and $600 then he is providing an interval estimate.

Point estimate is a single value estimate for a population parameter. The most unbiased point estimate of the population mean μ is the sample mean x .

The sample mean is not the same thing as the population mean. It could be an underestimate or an overestimate. But it is equally likely to be either. As long as the sample size is large enough, you can use the sample mean as the population mean.

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Example: Here are the IQ test scores of 31 seventh-grade girls in a school district:

114, 100, 104, 89, 102, 91, 114, 114, 103, 105, 108, 130, 120, 132, 111,

128, 118, 119, 86, 72, 111, 103, 74, 112, 107, 103, 98, 96, 112, 112, 93

Treat the 31 girls as a simple random sample of all seventh-grade girls in the school district. Find a point estimate of the population mean µ.

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. So, the point estimate for the mean IQ test score of all seventh-grade girls in this schools district is 105.8387.

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Interval estimate is an interval or range of values used to estimate a population parameter. Interval estimates are often desirable because the estimate of the mean varies from sample to sample. Instead of a single estimate for the mean, a confidence interval generates a lower and upper limit for the mean. The interval estimate gives an indication of how much uncertainty there is in the estimate of the true mean. The narrower the interval, the more precise is the estimate.

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The beauty of confidence intervals is that we know over the long run the chances of them including the true population parameter are very likely. You can’t do this with a point estimate – a point estimate is either it or its not.

Level of confidence (c) is the probability that the interval estimate contains the population parameter.

In this course, 90%, 95%, and 99% levels of confidence will be used most often.

The following z-scores correspond to certain levels of confidence.

70% level of confidence zc = 1.04
75% level of confidence zc = 1.15
80% level of confidence zc = 1.28
85% level of confidence zc = 1.44
90% level of confidence zc = 1.645
95% level of confidence zc = 1.96
98% level of confidence zc = 2.33
99% level of confidence zc = 2.575

More precise…

70% level of confidence zc = 1.0364

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90% level of confidence zc = 1.6449

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Error of estimate = | xμ | . Since in most cases, μ is unknown and x varies from sample to sample, it would be impossible to find the exact error of estimate. But what we can find is a maximum value of error if we know the level of confidence we are trying to achieve and the sampling distribution.

Maximum error of estimate (E), sometimes called the margin of error or error tolerance, is the greatest possible distance between the point estimate and the value of the parameter it is estimating. It is not the exact error of estimate, it is the greatest possible value. So not exact.

When n 30, s may be used instead of σ.

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Example: Here are the IQ test scores of 31 seventh-grade girls in a school district:

114, 100, 104, 89, 102, 91, 114, 114, 103, 105, 108, 130, 120, 132, 111,

128, 118, 119, 86, 72, 111, 103, 74, 112, 107, 103, 98, 96, 112, 112, 93

Treat the 31 girls as a simple random sample of all seventh-grade girls in the school district. Use a 95% confidence level to find the maximum error of estimate for the mean IQ test score of the population mean µ.

You don’t know the population standard deviation σ. But since n ≥ 30, you can use s in place of σ.

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s = 14.2714

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Using a point estimate and a maximum error of estimate, we can construct an interval estimate of a population parameter. This interval estimate is called a confidence interval.

Confidence interval:

The probability that the confidence interval contains µ is c.

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Guidelines: Finding a confidence interval for a population mean

( n ≥ 30 or σ known with a normally distributed population)

1. Find the sample mean

2. If you have σ , great! Otherwise, if n ≥ 30 , find the sample standard deviation s and use it as an estimate for σ .

3. Determine the critical value zc that corresponds to the desired level of confidence.

4. Find the maximum error of estimate E.

5. Find the left and right endpoints and form the confidence interval.

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Example: Here are the IQ test scores of 31 seventh-grade girls in a school district:

114, 100, 104, 89, 102, 91, 114, 114, 103, 105, 108, 130, 120, 132, 111,

128, 118, 119, 86, 72, 111, 103, 74, 112, 107, 103, 98, 96, 112, 112, 93

Treat the 31 girls as a simple random sample of all seventh-grade girls in the school district. Construct a 95% confidence interval for the mean IQ score for all seventh-grade girls in the school district.

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So with 95% confidence, we can say that the interval from 100.8148 and 110.8626 contains the population mean IQ score for all seventh-grade girls in this school district.

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So with 95% confidence, we can say that the interval from 100.81 and 110.86 contains the population mean IQ score for all seventh-grade girls in this school district. I like using the fancy graphing utility. Its pretty.

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When we form a confidence interval for µ, we usually express our confidence in the interval. Unlike point estimators, confidence intervals have some measure of reliability, the confidence coefficient, associated with them. For that reason they are generally preferred to point estimators.

So why not just use 99% confidence intervals rather than 95% intervals, since you will make fewer mistakes? The answer is that for a given sample size, the 99% confidence interval will be wider, therefore less precise than a 95% confidence interval.

As the level of confidence increases, if the sample stays the same, the confidence interval widens. When this happens, the precision of the estimate decreases. One way to improve the precision of an estimate without decreasing the level of confidence is to increase the sample size.

70% confidence interval (103.18, 108.5)

80% confidence interval (102.55, 109.12)

90% confidence interval (101.62, 110.05)

95% confidence interval (100.81, 110.86)

99% confidence interval (99.236, 112.44)

Notice how as the level of confidence increases, the interval widens.

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The size of the sample needed for a desired level of confidence

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Example: How large of a sample of seventh-grade girls from the previous examples would be needed to estimate the mean IQ score within ± 5 points with 99% confidence.

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. Round up to obtain a whole number. So, we should include at least 55 seventh-grade girls from the district in our sample.

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