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 KnownConfidence Interval for m: s KnownTI-83/84 CalculatorExample 71/84Example 71/84Example 71/84Example 71/84Example 71/84Example 71/84Example 71/84Example 71/84Example 71/84Example 71/84Example 71/84Example 71/84Example 71/84Example 71/84Example 71/84Slide 38Example 72/85Example 72/85Example 72/85Example 72/85Example 72/85Example 72/85Example 72/85Example 72/85Example 72/85Confidence and Margin of ErrorMargin of ErrorMargin of ErrorSample Size DeterminationSample Size DeterminationExample 73/86Example 73/86Example 73/86Slide 56Example 73/86STAT 210Lecture 32Confidence Intervals for m When s is KnownNovember 10, 2017Practice 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 – 245Top HatInferenceStatistical 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 statistical tests) for the population mean m.Top HatConfidence 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 m. Since m is unknown, we will never know for sure whether the interval contains it or not, but we typically choose a confidence level that is relatively high, such as 90%, 95%, 98% and 99%, so that our confidence of success is high.Note 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 theory of X derived 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 theory of X derived 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/ nThat has 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.Z*Confidence levelConfidence 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.The most common values are as follows:90% CI, Z* = 1.645 95% CI, Z* = 1.96098% CI, Z* = 2.326 99% CI, Z* = 2.576Confidence 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 100*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 the 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.”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.”In writing an interpretation, we do not use the term probability (confidence instead), we do not talk about the sample mean (population mean instead) and we do not talk about individual values.TI-83/84 CalculatorSee page 220 for instructors for using a calculatorto construct confidence intervals based
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