Slide 1Practice ProblemsAdditional Reading and ExamplesSlide 4InferenceInferenceMotivating ExampleMotivating ExampleMotivating ExampleConfidence IntervalsConfidence IntervalsMotivating ExampleConfidence IntervalsMotivating ExampleConfidence IntervalsMotivating ExampleConfidence IntervalsMotivating ExampleConfidence IntervalsConfidence IntervalsMotivating ExampleSlide 22Confidence IntervalConfidence IntervalConfidence IntervalConfidence IntervalConfidence IntervalConfidence LevelConfidence IntervalConfidence Interval - InterpretationMotivating ExampleSlide 32Confidence IntervalConfidence IntervalExample 56Example 56Example 56Example 56Example 56Example 56InferenceTests of SignificanceMotivating ExampleTests of SignificanceTests of SignificanceTests of SignificanceMotivating ExampleTests of SignificanceMotivating ExampleTests of SignificanceTests of SignificanceTests of SignificanceTests of SignificanceSlide 54Slide 55STAT 210Lecture 24Confidence Intervals and HypothesesOctober 24, 2016Practice ProblemsPages 187 and 188Relevant problems: VII.1 through VII.4Recommended problems: VII.1 through VII.4Additional Reading and ExamplesRead pages 185 and 186Clicker2InferenceStatistical inference involves using statistics computed from data collected in a sample to make statements (inferences) about unknown population parameters.Two types of statistical inference are estimation of parameters using confidence intervals and statistical tests about parameters.In this chapter we learn the basic concepts associated with confidence intervals and with statistical tests.InferenceThe first step in any inference procedure is to state the practical question that needs to be answered.This involves specifying the population of interest, and then the specific parameter that inferences need to be made about.Motivating ExampleSuppose one is interested in determining the mean IQ of all students at this university.What is the population of interest?What is the parameter of interest?Motivating ExampleSuppose one is interested in determining the mean IQ of all students at this university.What is the population of interest?The population of interest is all VCU students.What is the parameter of interest?The parameter of interest is the mean IQ of all VCU students.Motivating ExampleSuppose one is interested in determining the mean IQ of all students at this university.To determine the exact value of the mean IQ of all VCU students we would need to determine the IQ of EVERY VCU student.This is probably not possible. Hence there is a need to make inferences about this unknown mean IQ of all VCU students.Confidence IntervalsConfidence intervals are statistical procedures that allow for the estimation of unknown population parameters.The procedure involves both the calculation of the interval, and then the interpretation of the interval. The interpretation is the statistical inference.Confidence IntervalsTo estimate an unknown population parameter we begin by selecting a sample from the population.We would prefer for the sample to be selected randomly as opposed to haphazardly or voluntarily – more on this in later chapters.Once the sample is selected, we then collect the necessary information from those in the sample.Motivating ExampleSuppose one is interested in determining the mean IQ of all students at this university.The population of interest is all VCU students. The parameter of interest is the mean IQ of all VCU students.We could randomly select a sample of 100 VCU students and then determine the IQ of each of these 100 VCU students.Confidence IntervalsThe data collected from the sample is used to compute a statistic, and this statistic becomes the starting point for the confidence interval and hence the statistical inference.Motivating ExampleWe could randomly select a sample of 100 VCU students and then determine the IQ of each of these 100 VCU students.The mean IQ of the 100 VCU students in the sample could be computed, and this sample mean X becomes the starting point for the confidence interval.Confidence IntervalsThe data collected from the sample is used to compute a statistic, and this statistic becomes the starting point for the confidence interval and hence the statistical inference.The value computed from the sample data collected is referred to as the point estimate of the unknown population parameter.Motivating ExampleWith the parameter of interest being m = the mean IQ of all VCU students.Then X = the mean IQ of the 100 VCU students in the sample is the point estimate of the unknown mean IQ of all VCU students.Confidence IntervalsIf the sample is representative of the population, then one would expect that the point estimate will be a good estimate of the population parameter and hence would be very close to the actual (but unknown) value of the parameter. However, it is very unlikely that the actual value of the population parameter will be exactly equal to the point estimate value.Motivating ExampleWith the parameter of interest being m = the mean IQ of all VCU students.Then X = the mean IQ of the 100 VCU students in the sample is the point estimate of the unknown mean IQ of all VCU students. Suppose X = 106.If the random sample of 100 VCU students is representative of the population of all VCU students, then one would expect that the unknown mean IQ of all VCU students (m) will be close to 106, but it probably will not equal 106 exactly.Confidence IntervalsTherefore, since the point estimate should be close to the population parameter but will likely not equal it exactly, to the point estimate we subtract and add a quantity called a margin of error to create an interval of values in which we hope the true value of the parameter will be contained.This interval of values generated by subtracting and adding the margin of error is called a confidence interval.Confidence IntervalsTo the point estimate we subtract and add a quantity called a margin of error to create a confidence interval, in which we hope the true value of the parameter will be contained.The margin of error is considered a practical upper bound for the distance between the point estimate and the unknown parameter being estimated.Motivating ExampleWith the parameter of interest being m = the mean IQ of all VCU students.Then X = the mean IQ of the 100 VCU students in the sample is the point estimate of the unknown mean IQ of all VCU students. Suppose X = 106.Suppose (based on calculations to be discussed later) the margin of
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