One-sample Inference for students(1)

One-sample Inference for students(1) - One-sample Inference...

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© 2010 Radha Bose FSU Department of Statistics One-sample Inference — 1 In this section we will be introduced to HYPOTHESIS TESTING and ESTIMATION. We will look specifically at procedures for a population mean μ , but the general concepts covered here will carry over to other types of procedures . Under the broad heading of Hypothesis Testing, we will look at p- value approach tests, interpretation of the p-value, confidence interval approach tests, and the relationship between the two approaches. Under the broad heading of Estimation, we will look at point estimate, confidence interval estimate, interpretations of margin of error and confidence level, effect of changing margin of error or confidence level, and required sample size. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ESTIMATION Do this before coming to class. (1) Go to http://www.random.org/integers/ . (2) Generate 10 random integers between 1 and 100 and write them below. 85 , 86 , 96 , 8 , 74 , 21 , 49 , 16 , 91 , 5 (3) Calculate the mean of your sample: = x 53.1 (4) Perform the following calculations ( ) 89 . 17 = ME : x ME - = 35.2 x ME + = 71.0 this is a set of interval endpoints We will now study the use and interpretation of such intervals. Note that for the exercise above, the population mean is 1 2 3 98 99 100 50.5 100 X μ + + + + + + = = and the population standard deviation is ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 1 50.5 2 50.5 3 50.5 98 50.5 99 50.5 100 50.5 28.866. 100 X σ - + - + - + + - + - + - = =
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© 2010 Radha Bose FSU Department of Statistics One-sample Inference — 2 Information about the graph below Population: whole numbers 1—100 Popn Mean and Std Devn: 866 . 28 , 5 . 50 = = X X σ μ Sample size: 10 = n Mean and Std Devn of X : 10 866 . 28 , 5 . 50 = = = = n X X X X σ σ μ μ 95% Margin of Error: 89 . 17 10 866 . 28 96 . 1 96 . 1 = × = = X ME σ No. of samples (= no. of intervals): 49 No. and % of intervals that contain μ : 46 , ≈ 94% (which is close to 95%)
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© 2010 Radha Bose FSU Department of Statistics One-sample Inference — 3 TYPES OF ESTIMATE Point estimate — single number used as estimate of popn parameter, x is the best point estimate of μ . Confidence interval estimate — interval of numbers used as estimate of popn parameter, quoted as interval of the form ) , ( bound upper bound lower with associated confidence level % C . C is a measure of how certain we are that our method has produced an interval that actually contains the value of the parameter being estimated. lower bound upper bound __________(_________________________|_________________________)__________ x ME - x x ME + this distance is the margin of error, this is the point estimate, it is half the length of the interval it is the midpoint of the interval (upper bound—lower bound)/2 (lower bound + upper bound)/2
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© 2010 Radha Bose FSU Department of Statistics One-sample Inference — 4 CONFIDENCE INTERVAL ESTIMATES Since the procedures here depend on a Normal sampling distribution of X , we have to assume that we have a SRS of subjects and that the population distribution of observations has a certain shape (depending on the sample size): sample size in question shape of parent population 15 < n is roughly bell-shaped with no outliers 40 15 < n is roughly symmetric with no outliers 40 n has no outliers When σ is unknown, we use s as an estimate.
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