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Section 8.1

Section 8.1. Statistics and Sampling Variability. Introduction. Suppose I want to know the average GPA of seniors at Glacier Peak. I could track down every senior at GP and ask them their GPA, this would give me the true mean of the population OR I could take a sample.

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Section 8.1

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  1. Section 8.1 Statistics and Sampling Variability

  2. Introduction • Suppose I want to know the average GPA of seniors at Glacier Peak. • I could track down every senior at GP and ask them their GPA, this would give me the true mean of the population • OR • I could take a sample

  3. The Foundation of Chapter 8 • Suppose that x is a discrete random variable that is equal to an individual’s score on the 2010 AP Statistics Exam. • As a first year AP stats teacher, I am interested in the mean score of all students who took the exam in 2010 so that I may set realistic expectations for myself and my students for this year. • What is the population that I am interested in?_____________________ • What statistic am I interested in? __________________

  4. The Foundation of Chapter 8 • Let let µ denote the true mean score of all students who took the AP Stats exam in 2010. • To learn something about µ, I might obtain a sample of 50 students and determine the mean score from the sample. This sample may produce a mean of 3.01. So, x = 3.01. • How close is this mean to the true population mean, µ? • If I selected another sample of 50 students and computed the mean score, would this second x be near 3.01 or would it be quite different? • If I repeated this process many, many times and plotted the resulting means, I would create a sampling distribution of x. This would give me an idea about the long run behavior of the sample mean and could help me determine the true population mean, µ.

  5. The Foundation of Chapter 8 • Questions regarding the repeated sampling can be addressed by studying what is called the sampling distribution of • Just as the distribution of a numerical variable describes its long-run behavior, the sampling distribution of provides information about the long-run behavior of when sample after sample is selected.

  6. I can obtain information about a population characteristic by selecting a sample. • Sample Mean: • Population Mean: • Often different from one another, and rarely actual values from the data set

  7. Basic Terms • Any quantity computed from values in a sample is called a statistic (x, s, p, etc.) • Any quantity computed from values in a population is called a populationcharacteristic or parameter (, , )

  8. Sampling Variability • The observed value of a statistic depends on the particular sample selected from the population. • Typically, the value of the statistic varies from sample to sample. • This variability is called sampling variability.

  9. Constructing a Sampling Distribution • Suppose I want to take a random sample of size n = 50 from the population of all students who took the 2009 AP Stats exam. • There are many, many different possible samples that might result. • We now define a hypothetical population, which consists of all the different possible samples of size n = 50. • This is called a population of samples. The population of samples is viewed as a population because it consists of every different sample; it is a complete collection of all possible samples.

  10. Constructing a Sampling Distribution • Just as a variable associates a value with every individual in the population and can be described by its distribution, a statistic associates a value with each individual sample in the population of samples. • Therefore, a statistic can also be described by a distribution. • The distribution of a statistic is called its sampling distribution.

  11. Ex: For 4 students, the number of siblings they each have is 0, 1, 3, 4. Select samples of size n=2 and find the average number of siblings in each sample. Obtain the sampling distribution of x.

  12. Ex: From the previous example, determine the x , the mean value of x .

  13. Ex: Consider a population consisting of the following five values, which represent the number of DVD rentals from the Red Box at Fred Meyer for a given month of five families. • The values are: 8, 14, 16, 10, 11. • Compute the mean of this population. • Notice that the 5 slips of paper you created for the warm up correspond to these values. • Turn your paper over and randomly select a sample of size n = 2 and compute the mean of your sample. • Come up and record your sample and mean on the following table. Copy the values in the table.

  14. Ex: Record your sample and mean in the table below.

  15. Ex: Construct a histogram for the sample mean data in the table.

  16. Ex: Consider the density histogram for the sample mean data in the previous slide. • Are most of the values of x near the population mean? • Do the x values differ a lot from sample to sample or do they tend to be similar?

  17. HomeworkPage 409 1,2,3,4a,7,10 Read pages 411-420

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