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Chapter 9 Sampling Distributions 9.1 Sampling Distributions Key Term • A parameter is a number that describes the population. In statistical practice, the value of a parameter is not known. • A statistics is a number than can be computed from the sample data without making use of any unknown parameters. In practice, we often use a statistic to estimate an unknown parameter. • Refer to Example 9.1, page 564 Population and Sample Means • We write µ for the mean of a population. This is a fixed parameter that is unknown when we use a sample for inference • The mean of the sample is the sample is the familiar x-bar, the average of the observations in the sample. Key Concept / Term • Sampling Variability – For example 9.2 we found a p-hat of 160/515 = 0.31 However, the next time we take a random sample we choose different people and get a different value of p-hat. Key Term / Concept • The sampling distribution of a statistic is the distribution of values taken by the statistic in all possible samples of the same size from the same population • View Figures 9.4, 9.5, 9.6, 9.7, 9.8 • Larger samples sizes are less variable Two Digit Sample Mean Sampling Distribution of x-bar for Samples of size n = 2 Example 9.5: Proportions of samples who watched Survivor : Guatemala in Sample size of n = 100, 1000 samples Figure 9.6: Sampling distribution of the sample proportion p-hat from SRSs of size 1000 drawn from a population with population proportion p = 0.37, 100 samples Same Sample different scale illustrated the normal distribution Comparison of the two sample sizes Key Term • A statistic used to estimate a parameter is unbiased if the mean of its sampling distribution is equal to the true value of the parameter being estimated. See Figure 9.9, page 576 A statistic used to estimate a parameter is unbiased if the mean of its sampling distribution is equal to the true value of the parameter being estimated. Homework • Read 9.2 • Complete Exercises 1,2,4, 9 -12, 15 - 17 Chapter 9 Sampling Distributions 9.2 Sampling Proportions Key Concept • The variability of a statistic is described by the spread of its sampling distribution. This spread is determined by the sampling design and the size of the sample. Larger samples give smaller spread. • As long as the population is much larger than the sample (at least 10 times as large), the spread of the sample distribution is approximately the same for any population size. Key Term / Concept • Retailers would like to know what proportion of all adults find clothes shopping frustrating and time-consuming. This unknown population proportion is a parameter p. A random sample of 2500 people found 1650 frustrated by clothes shopping. The sample proportion p-hat = 1650/2500 = .66 is a statistic that we use to gain information about the parameter p. Sampling Distribution of a Sample Proportion Choose an SRS of size n from a large population with population proportion p having some characteristic of interest. Let p-hat be the proportion of the sample having that characteristic. Then: • The sampling distribution of p-hat is approximately normal and is closer to a normal distribution when the sample size n is large. • The mean of the sampling distribution is exactly p. • The standard deviation of the sampling distribution is p(1 p) n Rule of Thumb 1 • Use the recipe for the standard deviation of p-hat only when the population is at least 10 times as large as the sample • Example 9.7, page 584 Rule of Thumb 2 • We will use the normal approximation to sample distribution of p-hat values of n and p that satisfy: np > 10 and n(1-p) > 10 • Example 9.5, page 476 Homework • Read 9.3 • Complete Exercises #19, 20, 22, 24, 25, 28, 29, 30 Chapter 9 Sampling Distributions 9.3 Sample Means Rate of Return (Individual Stocks) Rate of Returns (Portfolios) Key Terms • The mean and standard deviation of a population are parameters. We use Greek letters to write these parameters: μ for the mean and σ for the standard deviation. • The mean and standard deviation calculated from sample data are statistics. We write the sample mean as x-bar and the sample standard deviation as s. Mean and Standard Deviation of a Sample Mean Suppose that x-bar is the mean of an SRS of size n drawn from a large population with mean μ and standard deviation σ. Then the mean of the sampling distribution of x-bar is μ and its standard deviation is n Sampling Distribution of a Sample Mean • Draw an SRS of size n from a population that has the normal distribution with mean μ and the standard deviation σ. Then the sample mean x-bar has the normal distribution N ( , ) n with the mean μ and standard deviation n Central Limit Theorem • Draw an SRS of size n from any population whatsoever with mean μ and finite standard deviation σ. When n is large, the sampling distribution of the sample mean x-bar is close to the normal distribution N (, n ) with mean μ and standard deviation n Law of Large Numbers Draw observations at random from any population with finite mean μ. As the number of observations drawn increases, the mean x-bar of the observed values get closer and closer to μ. Homework • Complete Exercises #35-46