Slide 1Test 5 ResultsPractice ProblemsAdditional Reading and ExamplesTest 6Slide 6InferenceConfidence IntervalsConfidence IntervalsConfidence IntervalsConfidence Interval for mConfidence Intervals: s UnknownConfidence Interval for mSlide 14Confidence Interval for m: s UnknownConfidence Interval for m: s UnknownSlide 17Confidence Interval for mTI-83/84 CalculatorExample 87/74Example 87/74Example 87/74Example 87/74Example 87/74Example 87/74Slide 26Example 87/74Example 87/74Example 87/74Example 87/74Example 88/75Example 88/75Example 88/75Example 88/75Example 88/75Example 88/75Example 88/75Example 89/76Example 89/76Example 89/76Example 89/76Example 89/76Example 89/76Example 89/76Slide 45Example 89/76Example 89/76Example 89/76Example 89/76Motivating ExampleMotivating ExampleMotivating ExampleMotivating Example SolutionMotivating Example SolutionSlide 55STAT 210Lecture 33Confidence Intervals for m When s is UnknownNovember 14, 2016Test 5 ResultsMean: 73.0 Median: 82.5Max: 100 Min: 12 n: 120Score Frequency Relative Frequency90’s 42 35.00%80’s 24 20.00%70’s 11 9.17%60’s 12 10.00% <60 31 25.83%Practice ProblemsSailboat: Pages 220 through 230Relevant problems: VIII.19 (a) and (b), VIII.21 (a), VIII.22 (a) and (b), VIII.23 (a), VIII.24 (a), (b), and (c), and VIII.25Recommended problems: VIII.21 (a) and VIII.23 (a)Hummingbird: Pages 250 through 260Relevant problems: IX.19 (a) and (b), IX.21 (a), IX.22 (a) and (b), IX.23 (a), IX.24 (a), (b), and (c), and IX.25Recommended problems: IX.21 (a) and IX.23 (a)Additional Reading and ExamplesSailboat: Read pages 213 through 218Pay particular attention to page 216Hummingbird: Read pages 243 through 248Pay particular attention to page 246Test 6Monday, November 21Questions for the first 10 minutes, then test – papers due promptly at the end of class!Covers Chapter 9 in Hummingbird (pages 219 – 260) or Chapter 8 in Sailboat (pages 189 – 230)Combination of multiple choice questions and written/short answer questions and problems.Formulas provided; Bring a calculator!Practice Tests and Formula Sheet on Blackboard.ClickerInferenceStatistical 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.Confidence 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 in the last lecture and the confidence interval application discussed today will be based on the sampling distribution theory of X derived earlier in the chapter, and hence both will require the following two assumptions.Confidence Interval for mAssumptions: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 applyConfidence Intervals: s Unknown The difference between last lecture’s problems and the following problems is previously we assumed that the population standard deviation s was known, but now we will assume the more realistic situation that the population standard deviation s is unknown (meaning we only know the sample standard deviation s).Confidence Interval for mX + Z* (s/ n )“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 ).”However, when s is unknown the formula above based on Z is not appropriate, and instead we must use a formula based on a t-distribution.ClickerConfidence Interval for m: s UnknownGOAL: Estimate the population mean mThe point estimate of m remains the sample mean X, however the margin of error changes to use the t-distribution instead of the Z-distribution. When the population standard deviation s is unknown, a C% confidence interval estimate for m isX + t*df (s/ n )Confidence Interval for m: s UnknownWhen the population standard deviation s is unknown, a C% confidence interval estimate for m isX + t*df (s / n )The t* value follows a t-distribution with degrees of freedom one less than the sample size, so df = n – 1. These values are looked up in the t-table as for Z*, except instead of using the bottom row labeled Z* the row of the table appropriate for the degrees of freedom is used.Suppose we want a95% CI and the samplesize is 24. Then df =24 – 1 = 23, and thet* value is 2.069.Confidence Interval for mThe definition of confidence level, the interpretation of the interval, and the properties are the same as they were when s was known and we used the Z* critical value.TI-83/84 CalculatorSee the end of the chapter for instructions on using a calculator to compute confidence intervals based on t-distributions.1. Hit STAT2. Scroll over to TESTS3. Choose 8:TInterval4. Next to Inpt choose Stats5. Enter required components and hit CalculateExample 87/74Population of interest: ???Parameter of interest: ???Example 87/74Population of interest: all members of the health clubParameter of interest: m = mean amount of time that members spend per week at the clubExample 87/74Parameter of interest: m = mean amount of time that members spend per week at the clubGoal: estimate m using a 95% confidence intervalAssumptions: ???Example 87/74Parameter of interest: m = mean amount of time that members spend per week at the clubGoal: estimate m using a 95% confidence intervalAssumptions: 1. We have a simple random sample 2. The population is known to be approximately
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