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isdstest3 - 8. 1 Population Mean: Known In order to develop...

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8. 1 Population Mean: σ Known In order to develop an interval estimate of a population mean, either the population standard deviation, σ , or the sample standard deviation, s , must be used to compute the margin of error In most applications σ is not known, and s is used to compute the margin of error In some applications, however, large amounts of relevant historical data are available and can be used to estimate the population standard deviation prior to sampling Also, in quality control applications where a process is assumed to be operating correctly, or “in control,” it is appropriate to treat the population standard deviation as known We refer to such cases as the σ known case Margin of Error and the Interval Estimate The sampling distribution of x can be used to compute the probability that x will be within a given distance of μ Therefore, σ = σ / (n) Because the sampling distribution shows how values of x are distributed around the population mean μ , the sampling distribution of x provides information about the possible differences between x and μ In the introduction to this chapter we said that the general form of an interval estimate of the population mean μ is x ± margin of error Any sample mean x that is within the darkly shaded region will provide an interval that contains the population mean μ Confidence interval – another name for an interval estimate Confidence coefficient – the confidence level expressed as a decimal value. For example: .95 is the confidence coefficient for a 95% confidence level Confidence level – the confidence associated with an interval estimate. For example: if an interval estimation procedure provides intervals such that 95% of the intervals formed using the procedure will include the population parameter, the interval estimate is said to be constructed at the 95% confidence level With the margin of error given by z ( σ / (n)), the general form of an interval estimate of a population mean for the σ known case follows: o x ± z ( σ / (n)) o where, (1- α ) is the confidence coefficient z is the z value providing an area of α /2 in the upper tail of the standard normal probability distribution Values of z for the Most Commonly Used Confidence Levels Confidence Level α α /2 z 90% .10 .05 1.645
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95% .05 .025 1.960 99% .01 .005 2.576 Comparing the results for the 90%, 95%, and 99% confidence levels, we see that in order to have a higher degree of confidence, the margin of error and thus the width of the confidence interval must be larger Practical Advice If the population follows a normal distribution, the confidence interval provided by the equation is exact If the population doesn’t follow a normal distribution, the confidence interval provided by the equation will be approximate In this case, the quality of the approximation depends on both the distribution fot he population and the sample size
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isdstest3 - 8. 1 Population Mean: Known In order to develop...

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