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10.1 DAY 2: Confidence Intervals – The Basics

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How Confidence Intervals Behave We select the confidence interval, and the margin of error follows… We strive for HIGH confidence and a SMALL margin of error. HIGH confidence says that our method almost always gives correct answers. SMALL margin of error says that we have pinned down the parameter quite precisely.

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How Confidence Intervals Behave Consider margin of error… The margin of error gets smaller when… z gets smaller. To accept a smaller margin of error, you must be willing to accept lower confidence. σ gets smaller. The standard deviation σ measures the variation in the population. n gets larger. We must take four times as many observations in order to cut the margin of error in half.

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Ex 1: Video Screen Tension – Part 2 Suppose the manufacturer (from yesterday’s example) wants 99% confidence rather than 90%. The critical value for 99% confidence is z = 2.57. The 99% confidence interval for μ based on a SRS of 20 video terminals with mean x = 306.3 is: Demanding 99% confidence instead of 90% confidence has increased the margin of error from 15.8 to 24.7.

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Sample Size for a Desired Margin of Error To determine the sample size that will yield a confidence interval for a population mean with a specified margin of error, set the expression for the margin of error to be less than or equal to m and solve for n:

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Ex 2: How Many Monkeys? Researchers would like to estimate the mean cholesterol level μ of a particular variety of monkey that is often used in lab experiments. They would like their estimate to be within 1 mg/dl of the true value of μ at a 95% confidence level. A previous study indicated that σ = 5 mg/dl. Obtaining monkeys is time- consuming and expensive, so researchers want to know the minimum number of monkeys they will need to generate a satisfactory estimate. We must round up!!! We need 97 monkeys to estimate the cholesterol levels to our satisfaction.

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Ex 3: 2004 Election A poll taken immediately before the 2004 election showed that 51% of the sample intended to vote for John Kerry. The polling organization announced that they were 95% confident that the sample result was within + 2 points of the true percent of all voters who favored Kerry.

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Ex 3: 2004 Election Explain in plain language to someone who knows no statistics what “95% confident” means in this announcement. The method captures the unknown parameter 95% of the time. The poll showed Kerry leading. Yet the organization said the election was too close to call. Explain. Since the margin of error was 2%, the true value of p could be as low as 49%. Thus, the confidence interval contains some values of p, which suggests that Bush will win.

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Ex 3: 2004 Election On hearing the poll, a politician asked, “What is the probability that over half the voters prefer Kerry?” A statistician replied that this question can’t be answered from the poll results, and that it doesn’t even make sense to talk about such a probability. Explain. First, the proportion of voters who favor Kerry is not random – either a majority favors Kerry or they don’t. Discussing probabilities has little meaning: the “probability” the politician asked about is either 1 or 0.

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Some Cautions The size of the sample determines margin of error. The size of the population does not influence the sample size. The data must be a SRS from the population. Different methods are needed for different designs (other than a SRS). There is NO correct method for inference from data haphazardly collected with bias of unknown size.

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…More Cautions Outliers can distort results. The shape of the population distribution matters. When n>15, the confidence level is not greatly disturbed by non-Normal populations unless extreme outliers or quite strong skewness are present. So far, we have been given the standard deviation σ of the population. We will learn how to proceed with an unknown σ later.

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