Estimation - Estimation Sample Proportions and Point...

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Estimation
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Sample Proportions and Point Estimation Sample Proportions Let p be the proportion of successes of a sample from a population whose total proportion of successes is π and let μ p be the mean of p and σ p be its standard deviation. Then The Central Limit Theorem For Proportions 1. μ p = π 2. 3. For n large, p is approximately normal. Example Consider the next census. Suppose we are interested in the proportion of Americans that are below the poverty level. Instead of attempting to find all Americans, Congress has proposed to perform statistical sampling. We can concentrate on 10,000 randomly selected people from 1000 locations. We can determine the proportion of people below the poverty level in each of these regions. Suppose this proportion is .08 . Then the mean for the sampling distribution is μ p = 0.8 and the standard deviation is Point Estimations A Point Estimate is a statistic that gives a plausible estimate for the value in question. Example x is a point estimate for μ s is a point estimate for σ
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A point estimate is unbiased if its mean represents the value that it is estimating. Confidence Intervals for Means (Both Large and Small Samples) Confidence Intervals for a Mean Point Estimations Usually, we do not know the population mean and standard deviation. Our goal is to estimate these numbers. The standard way to accomplish this is to use the sample mean and standard deviation as a best guess for the true population mean and standard deviation. We call this "best guess" a point estimate . A Point Estimate is a statistic that gives a plausible estimate for the value in question. Example: x is a point estimate for μ s is a point estimate for σ A point estimate is unbiased if its mean represents the value that it is estimating. Confidence Intervals We are not only interested in finding the point estimate for the mean, but also determining how accurate the point estimate is. The Central Limit Theorem plays a key role here. We assume that the sample standard deviation is close to the population standard deviation (which will almost always be true for large samples). Then the Central Limit Theorem tells us that the standard deviation of the sampling distribution is We will be interested in finding an interval around x such that there is a large probability that the actual mean falls inside of this interval. This interval is called a confidence interval and the large probability is called the confidence level .
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Example Suppose that we check for clarity in 50 locations in Lake Tahoe and discover that the average depth of clarity of the lake is 14 feet. Suppose that we know that the standard deviation for the entire lake's depth is 2 feet. What can we conclude about the average clarity of the lake with a 95% confidence level? Solution
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Estimation - Estimation Sample Proportions and Point...

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