Aem 201 1st Edition Lecture 18Outline of Last LectureI. The Sampling Distribution of the Sample MeanII. The Sampling Distribution of the Sample ProportionCurrent LectureI. Interval EstimationII. Interval Estimation of the Population Mean-Population Standard Deviation KnownIII. Interval Estimation of the Population Mean-Population Standard Deviation UnknownINTERVAL ESTIMATION- A few important definition (including some reminders)- The only way to know is to take a census- x bar is a good estimate of -on average the sample mean equals the population mean- The sample mean or proportion will never be exactly equal to the population’s- Point Estimate: a single numerical value used as an estimate of a parameter- Point Estimator: the sample statistic that provides the point estimate of the parameter- Precision: the exactness of an estimator- Accuracy: the correctness of an estimator - Remember that precision and accuracy are inversely related- Point estimators are:o Perfectly preciseo Almost certainly inaccurate- Sampling Error: the absolute difference between a parameter and its point estimator- How do we address the existence of sampling error? - A single value is precise-but almost certainly not accurate- We can make an estimate more accurate by estimating a range- Gaining accuracy gives you a better chance of being correct- Confidence Interval: range of values, used as an estimate of a parameter, that will contain the true value of the parameter a given proportion of times over many independent, identical repeated trials-also referred to as an interval estimate- Confidence Level: the proportion of times a confidence interval can be expected to contain the true value of the parameter over many independent, identical repeated trials. The correctness ofan estimator. OR the probability that the interval estimation procedure will generate an interval that does contain the true value of the parameter. Also referred to as the confidence coefficientThese notes represent a detailed interpretation of the professor’s lecture. GradeBuddy is best used as a supplement to your own notes, not as a substitute.o Note that: (1-confidence level) is often referred to as  or the significance level. So confidence level=1-INTERVAL ESTIMATION OF THE POPULATION MEAN-POPULATIONSTANDARD DEVIATION KNOWN- Recall that we can find a symmetric interval that satisfies the formula:o Tolerance level is 1-- If a=0.10, then we would expect (if we took 10 separate identical samples of size n) 9 intervals would contain the true mean- Problem: we are trying to estimate -but the tolerance intervals are constructed around o How can we estimate - If the parent population’s standard deviation is known and either i.) the parent population is normal or ii.) the sample is sufficiently large (n is greater than or equal to 30)o This is often referred to as interval estimate or a confidence intervalo (1-)*100% is referred to as the confidence level or confidence coefficient- A correct interpretation of a 90% confidence interval:o If we select several independent samples of the same size from the sample population using the same sampling procedure and calculate a 90% confidence interval for each sample, we will expect 90% of these intervals to include the true value of the populationmean- Of course in reality, we cannot take several samples of size n from the population and calculate aconfidence interval of each sample-realistically we must work with a single random sample and the confidence interval that sample generates- The probability of a confidence interval containing the real population mean is either 1 or 0. We can say that 90% of the time it will be 1 or 10% of the time it will be 0 for a 90% confidence interval- When we go to the table we lose precision but gain accuracy- Less confidence, smaller interval and vice versa- Before we collect our sample data, the confidence interval is unknown (it is still random, since we are taking to estimate the confidence intervalInterval estimation of the population mean-population standard deviation unknown- If the parent population’s standard deviation is not known and the sampling distribution not known and the sampling distribution of x bar is normal (parent is normal or the sample is sufficiently large) we must use a special probability distribution- The t-distribution or the Student t-distribution-family of probability distribution that are used to construct interval estimates of the population mean when the population standard deviation is unknown and the sampling distribution of x bar is normally or near normally distributed- The family of t-distributions is similar to the normal distribution except that it is shorter and wider (to allow for the fact that we are using an estimate of the population’s standard deviation)- The height and width of the t-distribution are determined by a parameter called degrees of freedomo A population mean degrees of freedom are n-1o As degrees of freedom increase the t-distribution becomes more normal (taller and narrower)o Values for the t-distribution for various combinations of a/2 and degrees of freedom increase are given- This is often referred to as an interval estimation or a confidence interval- (1-)100% is often referred to as confidence level or confidence coefficient- t /2, degrees of freedom is often referred to as margin of