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**Estimation of Means and Proportions**

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**Welcome to the interesting part of the course**

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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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**Estimation of a Population Mean**

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

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**Estimation of a Population Mean**

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). tα 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 tα

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Table: t Distribution

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**The Difference Between Two Means**

Suppose independent samples of n1 and n2 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**

The mean and standard deviation of is If the sampled populations are normally distributed, the sampling distribution of ( ) is exactly normally distributed regardless of n If the sampled populations are not normally distributed, the sampling distribution of ( ) is approximately normally distributed when n1 and n2 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 n1 and n2 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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Chapter 6 Sampling and Sampling Distributions

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