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VCU STAT 210 - Lecture32(2) (1)

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Slide 1Practice ProblemsAdditional Reading and ExamplesSlide 4InferenceSlide 6Confidence IntervalsConfidence IntervalsConfidence IntervalsConfidence IntervalsConfidence Interval for m: s KnownConfidence Interval for m: s KnownConfidence Interval for m: s KnownConfidence Interval for m: s KnownConfidence Interval for m: s KnownConfidence Interval for m: s KnownSlide 17Confidence Interval for m: s KnownConfidence Interval for m: s KnownConfidence Interval for m: s KnownTI-83/84 CalculatorExample 84/71Example 84/71Example 84/71Example 84/71Example 84/71Example 84/71Example 84/71Example 84/71Example 84/71Example 84/71Example 84/71Example 84/71Example 84/71Slide 35Example 85/72Example 85/72Example 85/72Example 85/72Example 85/72Example 85/72Example 85/72Example 85/72Example 85/72Confidence and Margin of ErrorMargin of ErrorMargin of ErrorSample Size DeterminationSample Size DeterminationExample 86/73Example 86/73Example 86/73Slide 53Example 86/73Motivating ExampleMotivating ExampleMotivating Example SolutionSlide 58STAT 210Lecture 32Confidence Intervals for m When s is KnownNovember 10, 2016Practice ProblemsSailboat: Pages 220 through 230Relevant problems: VIII.7 – VIII.10, and VIII.15 (a) and (b)Recommended problems: VIII.8 and VIII.10Hummingbird: Pages 250 through 260Relevant problems: IX.7 – IX.10, and IX.15 (a) and (b)Recommended problems: IX.8 and IX.10Additional Reading and ExamplesSailboat: Read pages 213 – 218Pay special attention to pages 214 – 215Hummingbird: Read page 243 – 248Pay special attention to pages 244 – 245ClickerInferenceStatistical inference involves using statistics computed from data collected in a sample to make statements (inferences) about unknown population parameters.In this chapter we will discuss statistical inferences (confidence intervals and tests of significance) for the population mean m.ClickerConfidence IntervalsTo estimate m, we will select a simple random sample from the population and compute the sample mean X for the data in the sample. This sample mean X will be the point estimate of m. To this point estimate we will subtract and add a margin of error, creating an interval of values that we hope the unknown population mean m is between.This interval is referred to as a confidence interval.Confidence IntervalsThe term confidence refers to the amount of confidence that we have that our interval will contain the true value of m. Since m is unknown, we will never know for sure whether our interval contains m or not, but we typically choose a confidence level that is relatively high, such as 90%, 95%, 98%, or 99%, so that our confidence of success is high.Note that the only way to have 100% confidence is to actually know the value of m.Confidence IntervalsBoth the confidence interval application discussed today and the confidence interval application discussed in the next lecture will be based on the sampling distribution of X discussed in the last lecture, and hence both will require the following two assumptions.Confidence IntervalsBoth the confidence interval application discussed today and the confidence interval application discussed in the next lecture will be based on the sampling distribution of X discussed in the last lecture, and hence both will require the following two assumptions.1. We must have a simple random sample from the population.2. The population must be normal, or the sample size must be large enough for the Central Limit Theorem to apply.Confidence Interval for m: s KnownAssumptions:1. We have a simple random sample from the population.2. Either (i) the population is normal or (ii) the sample size is large enough for the Central Limit Theorem to apply 3. The population standard deviation s is known.Confidence Interval for m: s Known1. We have a simple random sample from the population.2. Either (i) the population is normal or (ii) the sample size is large enough for the Central Limit Theorem to apply 3. The population standard deviation s is known.With these assumptions, the sampling distribution of X is:X ~ N(m, s/ n )Confidence Interval for m: s KnownSo X ~ N(m, s/ n )We can then apply the Z-score transformation to create the statistic X - mZ = s/ nwhich follows a standard normal distribution: Z ~ N(0, 1).Confidence Interval for m: s KnownSo X ~ N(m, s/ n )We can then apply the Z-score transformation X - mZ = s/ nSolving this for the unknown population mean m yields the following confidence interval formula:X + Z* (s/ n )Confidence Interval for m: s KnownSolving this for the unknown population mean m yields the following confidence interval formula:X + Z* (s/ n )In this expression the symbol + means subtract and add. We can get a lower limit L = X - Z* (s/ n ) and an upper limit U = X + Z* (s/ n ).The quantity Z* (s/ n ) is the margin of error.Confidence Interval for m: s KnownX + Z* (s/ n )The value of Z* depends on the amount of confidence stated and is determined from the t-table on page 340.One looks up the confidence level across the bottom row, and then reads the Z* value from the row directly above it.Z*Confidence levelConfidence Interval for m: s KnownX + Z* (s/ n )The interpretation of the confidence interval is the statistical inference and should be stated as follows.“We have C% confidence that the population mean m falls between the lower limit X – Z* (s/ n ) and the upper limit X + Z* (s/ n ).”Confidence Interval for m: s KnownX + Z* (s/ n )For example, if estimating m = the mean age of all students at this university with a 95% confidence interval, and if the lower limit is 20.5 and the upper limit is 28.5, then we write: “We have 95% confidence that the mean age of all students at this university falls between 20.5 and 28.5 years.”Confidence Interval for m: s KnownX + Z* (s/ n )“We have 95% confidence that the mean age of all students at this university falls between 20.5 and 28.5 years.”In writing an interpretation, we do not use the term “probability” (use “confidence” instead), we do not talk about the sample mean (rather the population mean) and we do not talk about individual values.TI-83/84 CalculatorSee the end of the chapter for instructions for using a calculator to construct confidence intervals based on a Z-distribution.1. Select STAT2. Choose TESTS3. Choose option 7:ZInterval4. Next to Inpt: choose Stats5. Enter values and hit CalculateExample 84/71Population of interest:


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VCU STAT 210 - Lecture32(2) (1)

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