Estimation This is our introduction to the field of inferential statistics. We already know why we want to study samples instead of entire populations, (e.g. limited resources, destructive sampling etc.). By studying the sample and its statistics, can we make inferences about the population and its parameters.

Estimator and Point Estimate An estimator is a “sample statistic” (such as the sample mean, or sample standard deviation) used to approximate a population parameter. A Point Estimate is a single value or point used to approximate a population parameter.

“The sample mean is the best point estimate of the population mean  ”

Age of STFX Students There are approximately 4000 students at STFX. Our goal is to determine the mean age of all STFX students, (the parameter to be estimated is mean age,  ). Can we use a statistic of the sample to estimate a parameter of a population? We take a sample of 50 STFX students and calculate the mean age of the sample, we find that the sample mean is 21.4 years. Therefore the best point estimate of the population mean  of the ages of all STFX students is the sample mean = 21.4 We see that 21.4 is the best point estimate of the age of all STFX students, but we have no indication of just how good that estimate really is.

The Concept of Sampling Distributions Parameter – numerical measure of a population. It is usually unknown Sample Statistic - numerical descriptive measure of a sample. It is usually known. Taking all the possible sample statistics, we get a sample distribution.

Sampling Distribution of Sample Means The sampling distribution of Sample Means is the distribution of the sample means obtained when we repeatedly draw samples of the same size from the same population. For sufficiently large samples, this distribution is almost always Bell Shaped, that is, it is almost always Normal.

Copyright © 2013 Pearson Education, Inc.. All rights reserved. Figure 6.4 Uniform distribution for thickness of steel sheets

Copyright © 2013 Pearson Education, Inc.. All rights reserved. Table 6.3

Copyright © 2013 Pearson Education, Inc.. All rights reserved. Figure 6.5 MINITAB histograms for sample mean and sample median, Example 6.2

Notes Both the sample mean and the sample median are “centered” at the true value of the population mean. But it is clear that the values of the sample means are less spread out, i.e., have smaller variance. This is called the minimum variance property of the sample mean as an estimator of the population mean. The fact that the sample means are centered at the true value of the population mean is called the unbiased property.

Central Limit Theorem Take ANY random variable X and compute  and  for this variable. If samples of size n are randomly selected from the population, then: 1) For large n, the distribution of the sample means, will be approximately a normal distribution, 2) The mean of the sample means will be the population mean  and 3) The standard deviation of the sample means will be

Let samples of size n be selected from a population with mean  and standard deviation , The mean of the sample means is denoted as Therefore the CLT says that Notation

Let samples of size n be selected from a population with mean  and standard deviation , The standard deviation of the sample means is denoted as According to the CLT, we have that for large populations. Notation

Application of the CLT Therefore, the distribution of the sample mean x of a random sample drawn from practically any population with mean  and standard deviation  can be approximated by a normal distribution with mean , and standard deviation provided the population is large.

Finding the probability that a sample mean is between a and b

Example: In human engineering and product design, it is often important to consider the weights of peoples so that airplanes or elevators are not overloaded, chairs don’t break and other unpleasant happenings don’t occur. Assume that the population of men and women has normally distributed weights with a mean of 173 pounds and a standard deviation of 30 pounds. –A) If 1 person is randomly selected, find the probability that their weight is greater than 180 pounds. –B) If 36 different people are randomly selected from this population find the probability that their mean weight is greater than 180 pounds. Solutions: A).4090 B).0808

Example To avoid false advertisement suits, a beverage bottler must make reasonably certain that 500 ml bottles actually contain 500 ml. To infer whether a bottling machine is working satisfactorily, the bottler randomly samples 10 bottles per hour and measures the amount of beverage in each bottle. The mean of the 10 measurements is used to determine whether to readjust the amount of beverage delivered to each bottle. Records show that the amount of fill per bottle is normally distributed with a standard deviation of 1.5 ml and a mean fill of 501 ml. What is the probability that the sample mean of 10 test bottles is less than 500 ml.

Practice Problems #6.34 page 310 #6.41 page 311

Overview Central Limit Theorem.

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