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Estimation of Means and Proportions

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Concepts Estimator: a rule that tells us how to estimate a value for a population parameter using sample data Estimate: a specific value of an estimator for particular sample data

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Concepts A point estimator is a rule that tells us how to calculate a particular number from sample data to estimate a population parameter An interval estimator is a rule that tells us how to calculate two numbers based on sample data, forming a confidence interval within which the parameter is expected to lie

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Properties of a Good Estimator Unbiasedness: mean of the sampling distribution of the estimator equals the true value of the parameter Efficiency: The most efficient estimator among a group of unbiased estimators is the one with the smallest variance

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Properties of a Good Estimator

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Estimation of a Population Mean The CLT suggests that the sample mean may be a good estimator for the population mean. The CLT says that: –Sampling distribution of sample mean will be approximately normally distributed regardless of the distribution of the sampled population if n is large –The sample mean is an unbiased estimator –The standard error of the sample mean is

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A point estimator of the population mean is: An interval estimator of the population mean is a confidence interval, meaning that the true population parameter lies within the interval of the time, where is the z value corresponding to an area in the upper tail of a standard normal distribution Estimation of a Population Mean

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Usually σ (the population standard deviation) is unknown. –If n is large enough (n ≥ 30) then we can approximate it with the sample standard deviation s.

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One Sided Confidence Intervals In some cases we may be interested in the probability the population parameter falls above or below a certain value Lower One Sided Confidence Interval (LCL): –LCL= (point estimate) – Upper One Sided Confidence Interval (UCL): –UCL = (point estimate) +

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Small Sample Estimation of a Population Mean If n is large, we can use sample standard deviation s as reliable estimator of population standard deviation –No matter what distribution the population has, sampling distribution of sample mean is normally distributed As the sample size n decreases, the sample standard deviation s becomes a less reliable estimator of the population standard deviation (because we are using less information from the underlying distribution to compute s) How do we deal with this issue?

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t Distribution Assume (1) The underlying population is normally distributed (2) Sample is small and σ is unknown Using the sample standard deviation s to replace σ, the t statistic follows the t – distribution

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Properties of the t Distribution mound-shaped perfectly symmetric about t=0 more variable than z (the standard normal distribution) affected by the sample size n (as n increases s becomes a better approximation for σ) n-1 is the degrees of freedom (d.f.) associated with the t statistic

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More on the t Distribution Remember the t-distribution is based on the assumption that the sampled population possesses a normal probability distribution. –This is a very restrictive assumption. Fortunately, it can be shown that for non-normal but mound-shaped distributions, the distribution of the t statistic is nearly the same shape as the theoretical t- distribution for a normal distribution. Therefore the t distribution is still useful for small sample estimation of a population mean even if the underlying distribution of x is not known to be normal

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How to use the t-distribution table The t-distribution table is in the book (Appendix II, Table 4, pp611). t α is the value of t such that an area α lies to its right. To use the table: Determine the degrees of freedom Determine the appropriate value of α Lookup the value for t α

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Table: t Distribution

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The Difference Between Two Means Suppose independent samples of n 1 and n 2 observations have been selected from populations with means, and variances, The Sampling Distribution of the difference in means ( ) will have the following properties

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The Difference Between Two Means 1.The mean and standard deviation of is 2.If the sampled populations are normally distributed, the sampling distribution of ( ) is exactly normally distributed regardless of n 3.If the sampled populations are not normally distributed, the sampling distribution of ( ) is approximately normally distributed when n 1 and n 2 are large

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Point Estimation of the Difference Between Two Means Point Estimator: A confidence interval for ( ) is

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Difference Between Two Means (small sample) If n 1 and n 2 are small then the t statistic is distributed according to the t distribution if the following assumptions are satisfied: 1. Both samples are drawn from populations with a normal distribution 2. Both populations have equal variances

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Difference Between Two Means (small sample) In practice, the t statistic is still appropriate even if the underlying distributions are not exactly normally distributed. To compute s, we can pool the information from both samples: or

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Difference Between Two Means (small sample) Point Estimate: Interval Estimate: a confidence interval for is Where s is computed using the pooled estimate described earlier

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Sampling Distribution of Sample Proportions Recall from Chapter 6: –If a random sample of n objects is selected from the population and if x of these possess a chararacteristic of interest, the sample proportion is –The sampling distribution of will have a mean and standard deviation

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Estimators for p Assuming n is sufficiently large and the interval lies in the interval from 0 to 1, the: Point Estimator for p: Interval Estimator for p: A confidence interval for p is

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Estimating the Difference Between Two Binomial Proportions Point estimate Confidence interval for the difference

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Choosing Sample Size How many measurements should be included in the sample? –Increasing n increases the precision of the estimate, but increasing n is costly Answer depends on: –What level of confidence do you want to have (i.e., the value of 100(1- α )? –What is the maximum difference (B) you want to permit between the estimate of the population parameter and the true population parameter

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Choosing Sample Size Once you have chosen B and α, you can solve the following equation for sample size n: If the resulting value of n is less than 30 and an estimate

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Choosing Sample Size

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