3301 Topic 10 Confidence Intervals

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1 CONFIDENCE INTERVALS (C.I.)

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UTA CSE IE 3301

Transcript of 3301 Topic 10 Confidence Intervals

  • *CONFIDENCE INTERVALS (C.I.)

  • *Statistical Inference STATISTICAL INFERENCE

    PARAMETER ESTIMATION HYPOTHESIS TESTING

    POINT INTERVALESTIMATION ESTIMATION

  • *Point EstimationPOINT ESTIMATION General symbol parameter, statisticStatistic, or estimator, estimates a parameter estimates eg. estimates

  • *Point EstimationProperties of estimators

    Unbiasedness

    Mathematical theory required to see proof is unbiased if Eg. is UE of because Note: s is not UE of because is UE of

  • *Point Estimation2. Relative Efficiency (R.E.)

    The more efficient estimator among two or more for a parameter has smaller variance. 3. An optimal estimator of is that i.e., MVUE.

    with the smallest MSE

  • *Point EstimationMethods of Point Estimation

    Maximum LikelihoodLeast SquaresMoments

  • *Confidence Intervals (C.I.)A C.I. is an estimate of a population parameter expressed not as a single value (point estimate) but as a range of values (interval) within which we would expect to find the parameterWe attach a level of confidence (1) to the C.I. Typical values for (1) % are 90, 95 or 99 %

  • *C.I. for : Known - 1Assume sample is selected from normal population or sample size n is large

    has a ND

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    C.I. for : Known - 2

  • *C.I. for : Known - 3C.I. For is

    This can also be written as

    where is the standard error (s.e.) or standard deviation of the estimate

  • *C.I. for : Known - 4 has the formstatistic table value s.e. of estimate

    This formula is valid only ifx ~ ND regardless of sample size, n, orn 30 when x is not normally distributed

  • *Interpretation of C.I.(1)% C.I. for a population parameter (e.g. ) means that if we were to compute (1)% C.I. for many samples of the same size (n), then (1)% of them would contain the true parameter ()

  • *C.I. for : Known - 6What factors influence the width of the C.I.Confidence coefficient (1)Sample size nPopulation varianceC.I. is narrower when(1) is smallern is larger2 is smaller

  • *C.I. For : Known - 7When is used as estimate of , the error of estimation isWe can be (1)% confident that the error will not exceed

  • *Sample Size (n) Needed To Estimate : Known - 1We may want to estimate with error no greater than some value, e, with (1)% confidence e may be written as

  • *Sample Size (n) Needed To Estimate : Known- 2The sample size, n, is found by rearranging the previous formula for e , giving

    Note that the value of e is specified by the experimenter

  • *C.I. For : Unknown - 1When is unknown, then

    has a Students t

    distribution with (n-1) degrees of freedoms = sample standard deviation

  • *C.I. For : Unknown and(1)% C.I. for is: or

    where is the t-value, with n-1 d.f. having an area to its rightUse t-table to find

  • *Page 286; 9.6: The heights of a random sample of 50 students has a mean of 174.5 cm and a s.d. of 6.9 cm. Find a 98% C.I. for the true mean height .Here: n=50; =174.5; s = 6.9; ;

    Solution: 98% C.I. is

    Conclusion: We are 98% confident that the true mean population height lies in the interval 172.15-176.85.

  • *C.I. For the Difference Between Means of 2 Independent Populations (12) - 1There are 3 different formulas, depending on the condition of the population variancesThe conditions are and are known and are unknown but and are unknown but

  • *C.I. For (12) - 2: Data Set UpWe would like to estimate the difference in the average running times (min) of films from two companies (x1 and x2)

    Here x1 and x2 are independent- they are from different companies

    x1

    103

    94

    110

    87

    98

    x2

    97

    82

    123

    92

    175

    88

    118

  • *C.I. For (12): Variances Known - 1If x1 and x2 are ND, then

  • *C.I. For (12): Variance Known - 2As for single C.I. = statistics table value s.e. statistic

    or

  • *C.I. For (12): Variance Known- ExampleWork example 9.6

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  • *C.I. For (12): Variance Unknown But Equal - 1If and are unknown then use their estimates andIf , then pool the separate variances to get a single estimate of the common variance follows at t-distribution with (n1-1)+(n2-1) degrees of freedom

  • *C.I. For (12): Variance Unknown But Equal - 2C.I. is

    or

  • *C.I. For (12): Variance Unknown But Equal - ExampleWork example 9.7, page 256

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  • *C.I. For (12): Variance Unknown And Unequal - 1 now follows a t-distribution with degrees of freedom, where

    is rounded to the nearest whole number

  • *C.I. For (12): Variance Unknown And Unequal - 2C.I. is

  • *Work example 9.8, page 258C.I. For (12): Variance Unknown And Unequal- Example

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  • *C.I. For Mean of Paired Difference, d - 1The objective is to estimate the difference in two treatments or conditions that occur in pairsExamplesthe strengths of the right hand (x1) and left hand (x2)body weight before (x1) and after (x2) a diet program designed to help people lose weight

  • *C.I. For Mean of Paired Difference, d - 2The data therefore have the following propertiesThey are paired; e.g. left and right arm strengthsThey are not independentThe variances of the populations are unknown; so the mean difference has a t-distribution

  • *C.I. For Mean of Paired Difference, d - 4Data Set UpA sample of strengths (kg) from 6 persons may be written as follows:

    x1

    40

    36

    34

    40

    36

    30

    x2

    42

    35

    32

    38

    30

    27

  • *C.I. For Mean of Paired Difference, d- 5The differences are as follows: di: -2, 1, 2, 2, 6, 3Note that because x1 and x2 are naturally paired, we do not use but instead

  • *C.I. For Mean of Paired Difference, d- 6C.I. is

    where is the estimate of dsd is the standard deviation of the paired differences, din is the number of pairs or the sample size from each population

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  • *C.I. For a Proportion (p) - 1Suppose we wished to estimate the proportion of students who own a VCR set:From a sample of 50 students we found 34 who owned VCR setsThe sample estimate of the proportion, is where x = 1 for each of the 34 students; x is a Binomial variable

  • *C.I. For a Proportion (p) - 2By the central limit theorem, for sufficiently large n, is normally distributed So

    We cannot use this statistic since we dont know the value of p. However, for large n, we can substitute for p and use the statistic

  • *C.I. For a Proportion, (p) - 3The C.I. is

  • *C.I. For a Proportion, (p) - 4For small n and p close to 0 or 1 this C.I. formula is not validThe formula is valid for and > 5

  • *C.I. For a Proportion, (p) - 5Work example 9.10, page 265

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  • *C.I. For Difference in Two Proportions, (p1p2) - 1For sufficiently large sample sizes from two Binomial populations and as before Thereforewith mean, andvariance,

  • *C.I. For Difference in Two Proportions, (p1p2) - 2Therefore

    For large sample sizes p1, q1, p2 and q2 may be replaced by their sample estimates

  • *C.I. For Difference in Two Proportions, (p1p2) - 3The C.I. is

  • *C.I. For Difference in Two Proportions, (p1p2)- 4Work example 9.13, page 269

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  • *C.I. For Variance of a Population ( 2)- 1The point estimate of 2 is s2 with n-1 degrees of freedomSince we dont know the distribution of s2 we convert s2 to 2 and use distribution of 2

  • *C.I. For Variance of a Population ( 2)- 2

    The above expression yields the (1)% C.I. for 2

  • *C.I. For Variance of a Population ( 2)- 3C.I. for 2 is

  • *C.I. For 2: ExampleWork example 9.14, page 272

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  • *C.I. For Ratio of Two Population Variances ( ) - 1Two variances are compared via their ratios not their differences

    Note that with (n1-1) and (n2-1) d.f.The d.f.s (n1-1) and (n2-1) will be represented by 1 and 2

  • *C.I. For Ratio of Two Population Variances ( ) - 2

    Replacing F by and rearranging gives

  • *C.I. For Ratio of Two Population Variances ( ) - 3The C.I. for is therefore:

  • *C.I. For Ratio of Two Population Variances ( ) - 4Note in the C.I. formula:The order of the degrees of freedom: if we are estimating the order is 2, 1 If you have no table for thenuse

    For example

  • *C.I. For : ExampleWork example 9.15, page 274

  • Using C.I. to Test HypothesesCalculate a 95% C.I. and use it to test the hypothesis that is significant.Solution: The C.I was calculated (not shown here) to be 0.8-1.0H0: = 0.0Ha: 0.0T.S.: C.I. is .80<
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  • *Tolerance Limits - 1See page 249, W&M, 6th edition (your textbook)Tolerance limits refer to the range of values that contain a certain amount of measurements from a population(1)% tolerance limits are the limiting values that we are (1)% certain contain a specified proportion (1) of measurements

  • *Tolerance Limits - 2(1)% T.L. for (1)% of measurements is:where and s are computed from a sample, and k is a value depending on , (1) and n k is found in table A.7, page 692, W&M, 6th edition.

  • *Tolerance Limits: Example Work example 9.5, page 250

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  • *Sample Size (n) for Estimating p- 1We may want to estimate a population proportion, p, with error not greater than a specified value, n, with (1)% confidence e may be written as

  • *Sample Size (n) for Estimating p- 2Sample size, n, required is found by rearranging the previous formula for e. The result is

    Note that the value of e is specified by the experimenter

  • *Sample Size (n) for Estimating p - 2In practice we must use an estimate for since we will not know until the sample is taken. The estimate is 1/41/4 represents the greatest value for ; that is, when andAny other value will give an n value that is larger than necessary The formula for n, above, will then reduce to

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  • *Slide 50 - Reverse v1 and v2 at leftf not FReference to textbook on slide 54\correct sl 26put hat in 40

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