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Published byColin Owen Modified over 5 years ago

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+ “Statisticians use a confidence interval to describe the amount of uncertainty associated with a sample estimate of a population parameter.”confidence intervalparameter

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+ REFLECTION… In Chapter 7 they GAVE us either the population mean or the mean of the sampling distribution If we had the mean of the sampling distribution we could predict the population In real life, you seldom have this information So….how do you estimate your population mean?! Point Estimates vs. Interval Estimates

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+ We Don ’ t KNOW the sampling mean (which would be equal to our population mean) So wedon ’ t KNOW where our statistic is RELATIVE to the true population parameter.

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+ Making connections… In Chapter 2 we learned about the Empirical rule: 68% of values lie within (+1) σ of the mean, 95% of values lie within (+2) σ of the mean, 99% of values lie within (+3) σ of the mean. In Chapter 4, we learned if we randomly select the sample, we should be able to generalize our results to the population of interest. In Chapter 7 we learned that if we take multiple samples each may vary by a certain amount, but if we take all the possible combinations of samples, then, we can construct a sampling distribution.

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+ Confidence Intervals: The Basics Confidence Intervals: The Basics Definition: A point estimator is a statistic that provides an estimate of a population parameter. The value of that statistic from a sample is called a point estimate. Ideally, a point estimate is our “best guess” at the value of an unknown parameter.

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+ WHAT IS THE POINT ESTIMATOR? WHAT IS THE POINT ESTIMATE? CAN YOU IDENTIFY EACH IN AN APPLICATION? LET’S TRY THE FOLLOWING …

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+ What proportion p of U.S. high school students smoke? A 2007 Survey questioned a random sample of 14,041 students in grades 9-12. Of these 2,808 said they had smoked cigarettes a least one day in the past month

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+ The math department wants to know what proportion of students own a graphing calculator, so they take a random sample of 100 students and find that 28 own a graphing calculator.

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+ The makes of a new golf ball want to estimate the median distance the new balls will travel when hit by a mechanical driver. They select a SRS of 10 balls and measure the distance travelled after being hit by a mechanical driver. The following are the distances in yards: 285286284285282284287290288285

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+ B IG I DEA We are estimating the mean of a sampling distribution, and transitively the population parameter, by using a sample’s statistic [mean/ proportion]!

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+ INTERVAL ESTIMATE An interval estimate is defined by two numbers, between which a population parameter is said to lie. For example, (a, b) is an interval estimate that states the population mean is between a and b.

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+ CONFIDENCE INTERVALS: THE BASICS CONFIDENCE INTERVALS: THE BASICS The confidence level describes the uncertainty associated with a sampling method. Suppose we used the same sampling method to select different samples and to compute a different interval estimate for each sample. Some interval estimates would include the true population parameter and some would not. A 90% confidence level means that we would expect 90% of the interval estimates to include the population parameter. A 95% confidence level means that 95% of the intervals would include the parameter.

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+ Example: You give the SAT Test to a SRS of 500 high school seniors inCalifornia. The mean x for the math portion is 461. The standarddeviation of the population σ= 100. What is the sample Standard deviation? Would the sample mean vary if we took many samples of 500 seniors from the same population?

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+ *68-95-99.7 Rule says that in 95% of all samples, x of the samples will be within 2σ of the population mean μ. So the mean x of 500 SAT Math scores will be within 4.5*2 = 9 points of the unknown μ in 95% of all samples. − −

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+ So in 95% of all samples, the unknown μ lies between x-9 and x+9…. Or x + 9 Or (x-9, x+9) − −− − −

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+ CONFIDENCE LEVEL: To say that we are 95% confident is shorthand for “95% of all possible samples of a given size from this population will result in an interval that captures the unknown parameter.” CONFIDENCE INTERVAL: To interpret a C% confidence interval for an unknown parameter, say: “We are C% confident that the interval from _____ to _____ captures the actual value of the [population parameter in context].” I NTERPRETING C ONFIDENCE L EVEL AND C ONFIDENCE I NTERVALS

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+ C ONFIDENCE I NTERVALS

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+ Interpreting Confidence Levels and Confidence Intervals The confidence level tells us how likely it is that the method we are using will produce an interval that captures the populationparameter if we use it many times. Instead, the confidence interval gives us a set of plausible values for the parameter. We interpret confidence levels and confidence intervals in much the same way whether we are estimating a population mean, proportion, or some other parameter. The confidence level does not tell us the chance that a particular confidence interval captures the population parameter.

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+ CONFIDENCE INTERVAL DATA REQUIREMENTS To express a confidence interval, you need three pieces of information. Confidence level A confidence level C gives the probability that the interval will capture the true parameter value in repeated samples C% of the time. C is usually 90%, 95% or 99% but can be any %. Statistic x or p Margin of error Critical value (z score) * Standard deviation of the statistic − ^

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+ We usually choose a confidence level of 90% or higher because we want to be quite sure of our conclusions. The most common confidence level is 95%.

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+ Calculating a Confidence Interval Confidence Intervals: The Basics The confidence interval for estimating a population parameter has the form statistic ± (critical value) (standard deviation of statistic) where the statistic we use is the point estimator for the parameter. Calculating a Confidence Interval Properties of Confidence Intervals: The “margin of error” is the (critical value) (standard deviation of statistic) The user chooses the confidence level, and the margin of error follows from this choice. The critical value depends on the confidence level and the sampling distribution of the statistic. Greater confidence requires a larger critical value The standard deviation of the statistic depends on the sample size n The margin of error gets smaller when: The confidence level decreases The sample size n increases

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+ Conditions Before calculating a confidence interval for µ or p there are three important conditions 1)Random: The data should come from a well-designed random sample or randomized experiment. 2) Normal: The sampling distribution of the statistic is approximately Normal. For means: If the population distribution is Normal. If the population distribution is not Normal, then the CLT tells us the sampling distribution will be approximately Normal if n ≥ 30. For proportions: We can use Normal approximation to the sampling distribution as long as np ≥ 10 and n(1 – p) ≥ 10. 3) Independent: Individual observations are independent. When sampling without replacement, the sample size n should be no more than 10% of the population size N (the 10% condition) to use our formula for the standard deviation of the statistic.

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