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Chapter 11- Confidence Intervals for Univariate Data Math 22 Introductory Statistics

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Introduction into Estimation n Point Estimate – the value of a sample statistic used to estimate the population parameter. n Interval Estimate – an interval bounded by two values calculated from the sample data, used to estimate a population parameter.

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Introduction into Estimation n Level of Confidence – The probability that the sample to be selected yields an interval that includes the parameter being estimated. n Confidence Interval – An interval estimate with a specified level of confidence. n Assumption – a condition that needs to exist in order to properly apply a statistical procedure to be valid.

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Confidence Interval n A confidence interval for a population parameter is an interval of possible values for the unknown parameter. n The interval is computed in such a way that we have a high degree of confidence that the interval contains the true value of a parameter.

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Confidence Level n The confidence, stated as a percent, is the confidence level. n In practice, estimates of unknown parameters are given in the form: estimate margin of error

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Developing a Confidence Interval Three determinations must be made to develop a Confidence Interval: n A good point estimator of the parameter. n The sampling dist. (or approximate sampling dist.) of the point estimator. n The desired confidence level, usually stated as a percentage.

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Standard Error of a Statistic n The standard deviation of its sampling dist. when all unknown population parameters have been estimated.

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Interpreting Confidence Intervals Q:What does a 99% C.I. really mean? A:A 99% C.I. means that of 100 different intervals obtained from 100 different samples, it is likely 99 of those intervals will contain the true parameter and one will not.

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Validity and Precision of Confidence Levels n Validity - Measured by the confidence level, which is the probability that the interval will contain the true value of the parameter. n Precision - measured by the length of the interval

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Confidence Interval for the Population Proportion

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Reducing the Margin of Error Two ways to reduce the margin of error: n Decrease z (Problem - Reduces Validity) n Increase n (No Problem)

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Calculating Sample Size for Proportions

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Estimation of the Mean When the Standard Deviation is Known When the population standard deviation is known, a (1- confidence interval for based on is given by the limits: When the population standard deviation is known, a (1- confidence interval for based on is given by the limits:

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Estimation of the Mean When the Standard Deviation is Unknown n We must make sure that the sampled population is normally distributed. n Normal Plots

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Student-t Distribution Many times we do not know what is. In these cases, we use s as the standard deviation. The standard error of the sample mean is now Many times we do not know what is. In these cases, we use s as the standard deviation. The standard error of the sample mean is now

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Characteristics of the Student- t Distribution n Bell shaped and symmetric, just like the normal distribution is bell shaped and symmetric. The t-distribution “looks” like the normal distribution but is not normal. n The t-distribution is a family of distributions, each member being uniquely identified by its degrees of freedom (df) which is simply n- 1 where n is the sample size.

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Characteristics of the Student- t Distribution n As the sample size increases the t- distribution becomes indistinguishable from the standard normal curve.

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The t-Interval

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Using the t-Interval For small sample sizes: If the sample size is less than 30, construct a normal plot of your data. If your data appears to be from a normal distribution, then use the t-distribution. If the data does not appear to be normal, then use a non-parametric technique that will be introduced later.

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Using the t-Interval For large sample sizes: If the sample size is 30 or more, use the t-distribution citing the Central Limit theorem as justification for having satisfied the required assumption of normality.

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Sample Size for Inference Concerning the Mean

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Confidence Interval for the Median Large Sample Confidence Interval for the Median: n Sample size must be 20 or more. We can construct a confidence interval for based on We can construct a confidence interval for based on We can then produce a confidence interval for with a sample proportion of.50 (this is used to represent the definition of the median, 50% below this mark, 50% above this mark.) We can then produce a confidence interval for with a sample proportion of.50 (this is used to represent the definition of the median, 50% below this mark, 50% above this mark.)

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Large Sample Confidence Interval for the Median Basic steps for conducting a large sample confidence interval for the median: n Construct a normal plot to see if the data is normal. If the normal assumption is violated, construct a (1- )100% for based on a sample proportion of.50. If the normal assumption is violated, construct a (1- )100% for based on a sample proportion of.50. n Multiply the upper and lower bound of the C.I. by n, the sample size. Round up the lower bound and round down the upper bound.

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Large Sample Confidence Interval for the Median n Sort the data and identify the data values in those positions identified by the previous step.

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Small Sample Confidence Interval for the Median n Sample size must be less than 20. n The method we will explore is based strictly on the binomial distribution.

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Small Sample Confidence Interval for the Median Basic steps for conducting a small sample confidence interval for the median: Create a table that contains the discrete cumulative probability distribution for 0 to n for a binomial distribution where =.50. Create a table that contains the discrete cumulative probability distribution for 0 to n for a binomial distribution where =.50. Identify the position for the lower bound with a cumulative probability as near /2 as possible. Identify the position for the lower bound with a cumulative probability as near /2 as possible.

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Small Sample Confidence Interval for the Median Identify the position for the upper bound with a cumulative probability as near 1- /2 as possible. Identify the position for the upper bound with a cumulative probability as near 1- /2 as possible. n Sort the data and identify the data values corresponding to the position located in the last two steps. n Report the actual confidence level by summing the tail probabilities associated with the positions chosen for the C.I. Bounds.

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