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Unformatted text preview: ESTIMATING WITH CONFIDENCE Statistical inference provides methods for drawing conclusions about a population from sample data. SAT Math Scores in California Example 10.2 Suppose you want to estimate the mean SAT Math score for the more than 350,000 high school seniors in California. Only about 49% of California students take the SAT. These self-selected seniors are planning to attend college and so are not representative of all California seniors. You know better than to make inferences about the population based on any sample data. At considerable effort and expense, you give the test to a simple random sample (SRS) of 500 California high school seniors. The mean for your sample is x = 461. What can you say about the mean score in the population of all 350,000 seniors? The law of large numbers tells us that the sample mean x from a large SRS will be close to the unknown population mean . Because x = 461, we guess that is "somewhere around 461." To make "somewhere around 461" more precise, we ask: How would the sample mean x vary if we took many samples of 500 seniors from this same population? Recall the essential facts about the sampling distribution of x : The central limit theorem tells us that the mean x of 500 scores has a distribution that is close to normal. The mean of this normal sampling distribution is the same as the unknown mean of the entire population. The standard deviation of x for an SRS of 500 students is 500 where is the standard deviation of individual SAT Math scores among all California high school seniors. Let us suppose that we know that the standard deviation of SAT Math scores in the population of all California seniors is = 100. The standard deviation of x is then 5 . 4 500 100 = = n (It is usually not realistic to assume we know a. We will see in the next chapter how to proceed when u is not known. For now, we are more interested in statistical reasoning than in details of realistic methods.) If we choose many samples of size 500 and find the mean SAT Math score for each sample, we might get mean x = 461 from the first sample, x = 455 from the second, x = 463 from the third sample, and so on. If we collect all these sample means and display their distribution, we get the normal distribution with mean equal to the unknown and standard deviation 4.5. Inference about the unknown , starts from this sampling distribution. Figure 10.1 displays the distribution. The different values of x appear along the axis in the figure, and the normal curve shows how probable these values are. The sampling distribution of the mean score x of SRS of 500 California seniors on the SAT Math quantitative test....
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This note was uploaded on 12/09/2011 for the course STAT 101 taught by Professor O during the Fall '08 term at Lake Land.
- Fall '08