Confidence Intervals for One Mean with Tolerance Probability

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Confidence Intervals for One Mean with Tolerance Probability"

Transcription

1 Chapter 421 Cofidece Itervals for Oe Mea with Tolerace Probability Itroductio This procedure calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) with a give tolerace probability at a stated cofidece level for a cofidece iterval about a sigle mea whe the uderlyig data distributio is ormal. Techical Details For a sigle mea from a ormal distributio with ukow variace, a two-sided, 100(1 α)% cofidece iterval is calculated by t X ± / 2, 1 A oe-sided 100(1 α)% upper cofidece limit is calculated by t X + ˆ σ ˆ σ, 1 Similarly, the oe-sided 100(1 α)% lower cofidece limit is t X ˆ σ, 1 Each cofidece iterval is calculated usig a estimate of the mea plus ad/or mius a quatity that represets the distace from the mea to the edge of the iterval. For two-sided cofidece itervals, this distace is sometimes called the precisio, margi of error, or half-width. We will label this distace, D. The basic equatio for determiig sample size whe D has bee specified is D t σ α = 1 / 2,

2 Solvig for, we obtai = t / 2, 1 This equatio ca be solved for ay of the ukow quatities i terms of the others. The value α/2 is replaced by α whe a oe-sided iterval is used. There is a additioal subtlety that arises whe the stadard deviatio is to be chose for estimatig sample size. The sample sizes determied from the formula above produce cofidece itervals with the specified widths oly whe the future sample has a sample stadard deviatio that is o greater tha the value specified. As a example, suppose that 15 idividuals are sampled i a pilot study, ad a stadard deviatio estimate of 3.5 is obtaied from the sample. The purpose of a later study is to estimate the mea withi 10 uits. Suppose further that the sample size eeded is calculated to be 57 usig the formula above with 3.5 as the estimate for the stadard deviatio. The sample of size 57 is the obtaied from the populatio, but the stadard deviatio of the 57 idividuals turs out to be 3.9 rather tha 3.5. The cofidece iterval is computed ad the distace from the mea to the cofidece limits is greater tha 10 uits. This example illustrates the eed for a adjustmet to adjust the sample size such that the distace from the mea to the cofidece limits will be below the specified value with kow probability. Such a adjustmet for situatios where a previous sample is used to estimate the stadard deviatio is derived by Harris, Horvitz, ad Mood (1948) ad discussed i Zar (1984) ad Hah ad Meeker (1991). The adjustmet is D ˆ σ 2 2 t ˆ / 2, 1σ = F1 γ ; 1, m 1 D where 1 γ is the probability that the distace from the mea to the cofidece limit(s) will be below the specified value, ad m is the sample size i the previous sample that was used to estimate the stadard deviatio. The correspodig adjustmet whe o previous sample is available is discussed i Kupper ad Hafer (1989) ad Hah ad Meeker (1991). The adjustmet i this case is t = 2 ˆ / 2, 1σ γ, 1 D 2 χ1 1 where, agai, 1 γ is the probability that the distace from the mea to the cofidece limit(s) will be below the specified value. Each of these adjustmets accouts for the variability i a future estimate of the stadard deviatio. I the first adjustmet formula (Harris, Horvitz, ad Mood, 1948), the distributio of the stadard deviatio is based o the estimate from a previous sample. I the secod adjustmet formula, the distributio of the stadard deviatio is based o a specified value that is assumed to be the populatio stadard deviatio

3 Fiite Populatio Size The above calculatios assume that samples are beig draw from a large (ifiite) populatio. Whe the populatio is of fiite size (N), a adjustmet must be made. The adjustmet reduces the stadard deviatio as follows: σ fiite = σ 1 N This ew stadard deviatio replaces the regular stadard deviatio i the above formulas. Cofidece Level The cofidece level, 1 α, has the followig iterpretatio. If thousads of samples of items are draw from a populatio usig simple radom samplig ad a cofidece iterval is calculated for each sample, the proportio of those itervals that will iclude the true populatio mea is 1 α. Procedure Optios This sectio describes the optios that are specific to this procedure. These are located o the Desig tab. For more iformatio about the optios of other tabs, go to the Procedure Widow chapter. Desig Tab The Desig tab cotais most of the parameters ad optios that you will be cocered with. Solve For Solve For This optio specifies the parameter to be solved for from the other parameters. Oe-Sided or Two-Sided Iterval Iterval Type Specify whether the iterval to be used will be a oe-sided or a two-sided cofidece iterval. Populatio Populatio Size This is the umber of idividuals i the populatio. Usually, you assume that samples are draw from a very large (ifiite) populatio. Occasioally, however, situatios arise i which the populatio of iterest is of limited size. I these cases, appropriate adjustmets must be made. This optio sets the populatio size

4 Cofidece ad Tolerace Cofidece Level (1 Alpha) The cofidece level, 1 α, has the followig iterpretatio. If thousads of samples of items are draw from a populatio usig simple radom samplig ad a cofidece iterval is calculated for each sample, the proportio of those itervals that will iclude the true populatio mea is 1 α. Ofte, the values 0.95 or 0.99 are used. You ca eter sigle values or a rage of values such as 0.90, 0.95 or 0.90 to 0.99 by Tolerace Probability This is the probability that a future iterval with sample size N ad the specified cofidece level will have a distace from the mea to the limit(s) that is less tha or equal to the distace specified. If a tolerace probability is ot used, as i the 'Cofidece Itervals for Oe Mea' procedure, the sample size is calculated for the expected distace from the mea to the limit(s), which assumes that the future stadard deviatio will also be the oe specified. Usig a tolerace probability implies that the stadard deviatio of the future sample will ot be kow i advace, ad therefore, a adjustmet is made to the sample size formula to accout for the variability i the stadard deviatio. Use of a tolerace probability is similar to usig a upper boud for the stadard deviatio i the 'Cofidece Itervals for Oe Mea' procedure. Values betwee 0 ad 1 ca be etered. The choice of the tolerace probability depeds upo how importat it is that the distace from the iterval limit(s) to the mea is at most the value specified. You ca eter a rage of values such as or 0.70 to 0.95 by Sample Size N (Sample Size) Eter oe or more values for the sample size. This is the umber of idividuals selected at radom from the populatio to be i the study. You ca eter a sigle value or a rage of values. Precisio Distace from Mea to Limit(s) This is the distace from the cofidece limit(s) to the mea. For two-sided itervals, it is also kow as the precisio, half-width, or margi of error. You ca eter a sigle value or a list of values. The value(s) must be greater tha zero

5 Stadard Deviatio Stadard Deviatio Source This procedure permits two sources for estimates of the stadard deviatio: S is a Populatio Stadard Deviatio This optio should be selected if there is o previous sample that ca be used to obtai a estimate of the stadard deviatio. I this case, the algorithm assumes that future sample obtaied will be from a populatio with stadard deviatio S. S from a Previous Sample This optio should be selected if the estimate of the stadard deviatio is obtaied from a previous radom sample from the same distributio as the oe to be sampled. The sample size of the previous sample must also be etered uder 'Sample Size of Previous Sample'. Stadard Deviatio Populatio Stadard Deviatio S (Stadard Deviatio) Eter a estimate of the stadard deviatio (must be positive). I this case, the algorithm assumes that future samples obtaied will be from a populatio with stadard deviatio S. Oe commo method for estimatig the stadard deviatio is the rage divided by 4, 5, or 6. You ca eter a rage of values such as or 1 to 10 by 1. Press the Stadard Deviatio Estimator butto to load the Stadard Deviatio Estimator widow. Stadard Deviatio Stadard Deviatio from Previous Sample S (SD Estimated from a Previous Sample) Eter a estimate of the stadard deviatio from a previous (or pilot) study. This value must be positive. A rage of values may be etered. Press the Stadard Deviatio Estimator butto to load the Stadard Deviatio Estimator widow. Sample Size of Previous Sample Eter the sample size that was used to estimate the stadard deviatio etered i S (SD Estimated from a Previous Sample). This value is etered oly whe 'Stadard Deviatio Source:' is set to 'S from a Previous Sample'

6 Example 1 Calculatig Sample Size A researcher would like to estimate the mea weight of a populatio with 95% cofidece. It is very importat that the mea weight is estimated withi 15 grams. Data available from a previous study are used to provide a estimate of the stadard deviatio. The estimate of the stadard deviatio is 45.1 grams, from a sample of size 14. The goal is to determie the sample size ecessary to obtai a two-sided cofidece iterval such that the mea weight is estimated withi 15 grams. Tolerace probabilities of 0.70 to 0.95 will be examied. Setup This sectio presets the values of each of the parameters eeded to ru this example. First, from the PASS Home widow, load the procedure widow by expadig Meas, the Oe Mea, the clickig o Cofidece Iterval, ad the clickig o Cofidece Itervals for Oe Mea with Tolerace Probability. You may the make the appropriate etries as listed below, or ope Example 1 by goig to the File meu ad choosig Ope Example Template. Optio Value Desig Tab Solve For... Sample Size Iterval Type... Two-Sided Populatio Size... Ifiite Cofidece Level Tolerace Probability to 0.95 by 0.05 Distace from Mea to Limit(s) Stadard Deviatio Source... S from a Previous Sample S Sample Size of Previous Sample Aotated Output Click the Calculate butto to perform the calculatios ad geerate the followig output. Numeric Results Numeric Results for Two-Sided Cofidece Itervals Target Actual Sample Distace Distace Stadard Cofidece Size from Mea from Mea Deviatio Tolerace Level (N) to Limits to Limits (S) Probability Sample size for estimate of S from previous sample = 14. Refereces Hah, G. J. ad Meeker, W.Q Statistical Itervals. Joh Wiley & Sos. New York. Zar, J. H Biostatistical Aalysis. Secod Editio. Pretice-Hall. Eglewood Cliffs, New Jersey. Harris, M., Horvitz, D. J., ad Mood, A. M 'O the Determiatio of Sample Sizes i Desigig Experimets', Joural of the America Statistical Associatio, Volume 43, No. 243, pp

7 Report Defiitios Cofidece level is the proportio of cofidece itervals (costructed with this same cofidece level, sample size, etc.) that would cotai the populatio mea. N is the size of the sample draw from the populatio. Distace from Mea to Limit is the distace from the cofidece limit(s) to the mea. For two-sided itervals, it is also kow as the precisio, half-width, or margi of error. Target Distace from Mea to Limit is the value of the distace that is etered ito the procedure. Actual Distace from Mea to Limit is the value of the distace that is obtaied from the procedure. The stadard deviatio of the populatio measures the variability i the populatio. Tolerace Probability is the probability that a future iterval with sample size N ad correspodig cofidece level will have a distace from the mea to the limit(s) that is less tha or equal to the specified distace. Summary Statemets The probability is that a sample size of 49 will produce a two-sided 95% cofidece iterval with a distace from the mea to the limits that is less tha or equal to if the populatio stadard deviatio is estimated to be by a previous sample of size 14. This report shows the calculated sample size for each of the scearios. Plots Sectio This plot shows the sample size versus the tolerace probability

8 Example 2 Validatio usig Hah ad Meeker Hah ad Meeker (1991) page 139 give a example of a sample size calculatio for a two-sided cofidece iterval o the mea whe the cofidece level is 95%, the populatio stadard deviatio is assumed to be 2500, the distace from the mea to the limit is 1500, ad the tolerace probability is The ecessary sample size is 19. Setup This sectio presets the values of each of the parameters eeded to ru this example. First, from the PASS Home widow, load the procedure widow by expadig Meas, the Oe Mea, the clickig o Cofidece Iterval, ad the clickig o Cofidece Itervals for Oe Mea with Tolerace Probability. You may the make the appropriate etries as listed below, or ope Example 2 by goig to the File meu ad choosig Ope Example Template. Optio Value Desig Tab Solve For... Sample Size Iterval Type... Two-Sided Populatio Size... Ifiite Cofidece Level Tolerace Probability Distace from Mea to Limit(s) Stadard Deviatio Source... S is a Populatio Stadard Deviatio S Output Click the Calculate butto to perform the calculatios ad geerate the followig output. Numeric Results Numeric Results for Two-Sided Cofidece Itervals Target Actual Sample Distace Distace Stadard Cofidece Size from Mea from Mea Deviatio Tolerace Level (N) to Limits to Limits (S) Probability PASS also calculated the ecessary sample size to be

9 Example 3 Validatio usig Zar Zar (1984) pages give a example of a sample size calculatio for a two-sided cofidece iterval o the mea whe the cofidece level is 95%, the stadard deviatio is estimated to be by a previous sample of size 25, the distace from the mea to the limit is 1.5, ad the tolerace probability is The ecessary sample size is 53. Setup This sectio presets the values of each of the parameters eeded to ru this example. First, from the PASS Home widow, load the procedure widow by expadig Meas, the Oe Mea, the clickig o Cofidece Iterval, ad the clickig o Cofidece Itervals for Oe Mea with Tolerace Probability. You may the make the appropriate etries as listed below, or ope Example 3 by goig to the File meu ad choosig Ope Example Template. Optio Value Desig Tab Solve For... Sample Size Iterval Type... Two-Sided Populatio Size... Ifiite Cofidece Level Tolerace Probability Distace from Mea to Limit(s) Stadard Deviatio Source... S from a Previous Sample S Sample Size of Previous Sample Output Click the Calculate butto to perform the calculatios ad geerate the followig output. Numeric Results Numeric Results for Two-Sided Cofidece Itervals Target Actual Sample Distace Distace Stadard Cofidece Size from Mea from Mea Deviatio Tolerace Level (N) to Limits to Limits (S) Probability PASS also calculated the ecessary sample size to be

10 Example 4 Validatio usig Harris, Horvitz, ad Mood Harris, Horvitz, ad Mood (1948) pages give a example of a sample size calculatio for a two-sided cofidece iterval o the mea whe the cofidece level is 99%, the stadard deviatio is estimated to be 3 by a previous sample of size 9, the distace from the mea to the limit is 2, ad the tolerace probability is The ecessary sample size is 49. Setup This sectio presets the values of each of the parameters eeded to ru this example. First, from the PASS Home widow, load the procedure widow by expadig Meas, the Oe Mea, the clickig o Cofidece Iterval, ad the clickig o Cofidece Itervals for Oe Mea with Tolerace Probability. You may the make the appropriate etries as listed below, or ope Example 4 by goig to the File meu ad choosig Ope Example Template. Optio Value Desig Tab Solve For... Sample Size Iterval Type... Two-Sided Populatio Size... Ifiite Cofidece Level Tolerace Probability Distace from Mea to Limit(s)... 2 Stadard Deviatio Source... S from a Previous Sample S... 3 Sample Size of Previous Sample... 9 Output Click the Calculate butto to perform the calculatios ad geerate the followig output. Numeric Results Numeric Results for Two-Sided Cofidece Itervals Target Actual Sample Distace Distace Stadard Cofidece Size from Mea from Mea Deviatio Tolerace Level (N) to Limits to Limits (S) Probability PASS also calculated the ecessary sample size to be

Confidence Intervals for One Mean

Confidence Intervals for One Mean Chapter 420 Cofidece Itervals for Oe Mea Itroductio This routie calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) at a stated cofidece level for a

More information

Confidence Intervals for Linear Regression Slope

Confidence Intervals for Linear Regression Slope Chapter 856 Cofidece Iterval for Liear Regreio Slope Itroductio Thi routie calculate the ample ize eceary to achieve a pecified ditace from the lope to the cofidece limit at a tated cofidece level for

More information

Overview. Learning Objectives. Point Estimate. Estimation. Estimating the Value of a Parameter Using Confidence Intervals

Overview. Learning Objectives. Point Estimate. Estimation. Estimating the Value of a Parameter Using Confidence Intervals Overview Estimatig the Value of a Parameter Usig Cofidece Itervals We apply the results about the sample mea the problem of estimatio Estimatio is the process of usig sample data estimate the value of

More information

5: Introduction to Estimation

5: Introduction to Estimation 5: Itroductio to Estimatio Cotets Acroyms ad symbols... 1 Statistical iferece... Estimatig µ with cofidece... 3 Samplig distributio of the mea... 3 Cofidece Iterval for μ whe σ is kow before had... 4 Sample

More information

Z-TEST / Z-STATISTIC: used to test hypotheses about. µ when the population standard deviation is unknown

Z-TEST / Z-STATISTIC: used to test hypotheses about. µ when the population standard deviation is unknown Z-TEST / Z-STATISTIC: used to test hypotheses about µ whe the populatio stadard deviatio is kow ad populatio distributio is ormal or sample size is large T-TEST / T-STATISTIC: used to test hypotheses about

More information

Determining the sample size

Determining the sample size Determiig the sample size Oe of the most commo questios ay statisticia gets asked is How large a sample size do I eed? Researchers are ofte surprised to fid out that the aswer depeds o a umber of factors

More information

Chapter 7: Confidence Interval and Sample Size

Chapter 7: Confidence Interval and Sample Size Chapter 7: Cofidece Iterval ad Sample Size Learig Objectives Upo successful completio of Chapter 7, you will be able to: Fid the cofidece iterval for the mea, proportio, ad variace. Determie the miimum

More information

Chapter 10. Hypothesis Tests Regarding a Parameter. 10.1 The Language of Hypothesis Testing

Chapter 10. Hypothesis Tests Regarding a Parameter. 10.1 The Language of Hypothesis Testing Chapter 10 Hypothesis Tests Regardig a Parameter A secod type of statistical iferece is hypothesis testig. Here, rather tha use either a poit (or iterval) estimate from a simple radom sample to approximate

More information

Center, Spread, and Shape in Inference: Claims, Caveats, and Insights

Center, Spread, and Shape in Inference: Claims, Caveats, and Insights Ceter, Spread, ad Shape i Iferece: Claims, Caveats, ad Isights Dr. Nacy Pfeig (Uiversity of Pittsburgh) AMATYC November 2008 Prelimiary Activities 1. I would like to produce a iterval estimate for the

More information

Definition. A variable X that takes on values X 1, X 2, X 3,...X k with respective frequencies f 1, f 2, f 3,...f k has mean

Definition. A variable X that takes on values X 1, X 2, X 3,...X k with respective frequencies f 1, f 2, f 3,...f k has mean 1 Social Studies 201 October 13, 2004 Note: The examples i these otes may be differet tha used i class. However, the examples are similar ad the methods used are idetical to what was preseted i class.

More information

Confidence Intervals. CI for a population mean (σ is known and n > 30 or the variable is normally distributed in the.

Confidence Intervals. CI for a population mean (σ is known and n > 30 or the variable is normally distributed in the. Cofidece Itervals A cofidece iterval is a iterval whose purpose is to estimate a parameter (a umber that could, i theory, be calculated from the populatio, if measuremets were available for the whole populatio).

More information

Quadrat Sampling in Population Ecology

Quadrat Sampling in Population Ecology Quadrat Samplig i Populatio Ecology Backgroud Estimatig the abudace of orgaisms. Ecology is ofte referred to as the "study of distributio ad abudace". This beig true, we would ofte like to kow how may

More information

Research Method (I) --Knowledge on Sampling (Simple Random Sampling)

Research Method (I) --Knowledge on Sampling (Simple Random Sampling) Research Method (I) --Kowledge o Samplig (Simple Radom Samplig) 1. Itroductio to samplig 1.1 Defiitio of samplig Samplig ca be defied as selectig part of the elemets i a populatio. It results i the fact

More information

Descriptive Statistics Summary Tables

Descriptive Statistics Summary Tables Chapter 201 Descriptive Statistics Summary Tables Itroductio This procedure is used to summarize cotiuous data. Large volumes of such data may be easily summarized i statistical tables of meas, couts,

More information

1. C. The formula for the confidence interval for a population mean is: x t, which was

1. C. The formula for the confidence interval for a population mean is: x t, which was s 1. C. The formula for the cofidece iterval for a populatio mea is: x t, which was based o the sample Mea. So, x is guarateed to be i the iterval you form.. D. Use the rule : p-value

More information

Measures of Spread and Boxplots Discrete Math, Section 9.4

Measures of Spread and Boxplots Discrete Math, Section 9.4 Measures of Spread ad Boxplots Discrete Math, Sectio 9.4 We start with a example: Example 1: Comparig Mea ad Media Compute the mea ad media of each data set: S 1 = {4, 6, 8, 10, 1, 14, 16} S = {4, 7, 9,

More information

Using Excel to Construct Confidence Intervals

Using Excel to Construct Confidence Intervals OPIM 303 Statistics Ja Stallaert Usig Excel to Costruct Cofidece Itervals This hadout explais how to costruct cofidece itervals i Excel for the followig cases: 1. Cofidece Itervals for the mea of a populatio

More information

Standard Errors and Confidence Intervals

Standard Errors and Confidence Intervals Stadard Errors ad Cofidece Itervals Itroductio I the documet Data Descriptio, Populatios ad the Normal Distributio a sample had bee obtaied from the populatio of heights of 5-year-old boys. If we assume

More information

Key Ideas Section 8-1: Overview hypothesis testing Hypothesis Hypothesis Test Section 8-2: Basics of Hypothesis Testing Null Hypothesis

Key Ideas Section 8-1: Overview hypothesis testing Hypothesis Hypothesis Test Section 8-2: Basics of Hypothesis Testing Null Hypothesis Chapter 8 Key Ideas Hypothesis (Null ad Alterative), Hypothesis Test, Test Statistic, P-value Type I Error, Type II Error, Sigificace Level, Power Sectio 8-1: Overview Cofidece Itervals (Chapter 7) are

More information

Analyzing Longitudinal Data from Complex Surveys Using SUDAAN

Analyzing Longitudinal Data from Complex Surveys Using SUDAAN Aalyzig Logitudial Data from Complex Surveys Usig SUDAAN Darryl Creel Statistics ad Epidemiology, RTI Iteratioal, 312 Trotter Farm Drive, Rockville, MD, 20850 Abstract SUDAAN: Software for the Statistical

More information

Review for Test 3. b. Construct the 90% and 95% confidence intervals for the population mean. Interpret the CIs.

Review for Test 3. b. Construct the 90% and 95% confidence intervals for the population mean. Interpret the CIs. Review for Test 3 1 From a radom sample of 36 days i a recet year, the closig stock prices of Hasbro had a mea of $1931 From past studies we kow that the populatio stadard deviatio is $237 a Should you

More information

1 Computing the Standard Deviation of Sample Means

1 Computing the Standard Deviation of Sample Means Computig the Stadard Deviatio of Sample Meas Quality cotrol charts are based o sample meas ot o idividual values withi a sample. A sample is a group of items, which are cosidered all together for our aalysis.

More information

I. Chi-squared Distributions

I. Chi-squared Distributions 1 M 358K Supplemet to Chapter 23: CHI-SQUARED DISTRIBUTIONS, T-DISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad t-distributios, we first eed to look at aother family of distributios, the chi-squared distributios.

More information

Normal Distribution.

Normal Distribution. Normal Distributio www.icrf.l Normal distributio I probability theory, the ormal or Gaussia distributio, is a cotiuous probability distributio that is ofte used as a first approimatio to describe realvalued

More information

Case Study. Normal and t Distributions. Density Plot. Normal Distributions

Case Study. Normal and t Distributions. Density Plot. Normal Distributions Case Study Normal ad t Distributios Bret Halo ad Bret Larget Departmet of Statistics Uiversity of Wiscosi Madiso October 11 13, 2011 Case Study Body temperature varies withi idividuals over time (it ca

More information

Estimating the Mean and Variance of a Normal Distribution

Estimating the Mean and Variance of a Normal Distribution Estimatig the Mea ad Variace of a Normal Distributio Learig Objectives After completig this module, the studet will be able to eplai the value of repeatig eperimets eplai the role of the law of large umbers

More information

Statistical inference: example 1. Inferential Statistics

Statistical inference: example 1. Inferential Statistics Statistical iferece: example 1 Iferetial Statistics POPULATION SAMPLE A clothig store chai regularly buys from a supplier large quatities of a certai piece of clothig. Each item ca be classified either

More information

Confidence Intervals

Confidence Intervals Cofidece Itervals Cofidece Itervals are a extesio of the cocept of Margi of Error which we met earlier i this course. Remember we saw: The sample proportio will differ from the populatio proportio by more

More information

Chapter 7 Methods of Finding Estimators

Chapter 7 Methods of Finding Estimators Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of

More information

9.8: THE POWER OF A TEST

9.8: THE POWER OF A TEST 9.8: The Power of a Test CD9-1 9.8: THE POWER OF A TEST I the iitial discussio of statistical hypothesis testig, the two types of risks that are take whe decisios are made about populatio parameters based

More information

AQA STATISTICS 1 REVISION NOTES

AQA STATISTICS 1 REVISION NOTES AQA STATISTICS 1 REVISION NOTES AVERAGES AND MEASURES OF SPREAD www.mathsbox.org.uk Mode : the most commo or most popular data value the oly average that ca be used for qualitative data ot suitable if

More information

CHAPTER 7: Central Limit Theorem: CLT for Averages (Means)

CHAPTER 7: Central Limit Theorem: CLT for Averages (Means) CHAPTER 7: Cetral Limit Theorem: CLT for Averages (Meas) X = the umber obtaied whe rollig oe six sided die oce. If we roll a six sided die oce, the mea of the probability distributio is X P(X = x) Simulatio:

More information

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,

More information

Confidence Intervals for Two Proportions

Confidence Intervals for Two Proportions PASS Samle Size Software Chater 6 Cofidece Itervals for Two Proortios Itroductio This routie calculates the grou samle sizes ecessary to achieve a secified iterval width of the differece, ratio, or odds

More information

Section 11.3: The Integral Test

Section 11.3: The Integral Test Sectio.3: The Itegral Test Most of the series we have looked at have either diverged or have coverged ad we have bee able to fid what they coverge to. I geeral however, the problem is much more difficult

More information

Inference on Proportion. Chapter 8 Tests of Statistical Hypotheses. Sampling Distribution of Sample Proportion. Confidence Interval

Inference on Proportion. Chapter 8 Tests of Statistical Hypotheses. Sampling Distribution of Sample Proportion. Confidence Interval Chapter 8 Tests of Statistical Hypotheses 8. Tests about Proportios HT - Iferece o Proportio Parameter: Populatio Proportio p (or π) (Percetage of people has o health isurace) x Statistic: Sample Proportio

More information

Hypothesis testing. Null and alternative hypotheses

Hypothesis testing. Null and alternative hypotheses Hypothesis testig Aother importat use of samplig distributios is to test hypotheses about populatio parameters, e.g. mea, proportio, regressio coefficiets, etc. For example, it is possible to stipulate

More information

Statistical Methods. Chapter 1: Overview and Descriptive Statistics

Statistical Methods. Chapter 1: Overview and Descriptive Statistics Geeral Itroductio Statistical Methods Chapter 1: Overview ad Descriptive Statistics Statistics studies data, populatio, ad samples. Descriptive Statistics vs Iferetial Statistics. Descriptive Statistics

More information

Now here is the important step

Now here is the important step LINEST i Excel The Excel spreadsheet fuctio "liest" is a complete liear least squares curve fittig routie that produces ucertaity estimates for the fit values. There are two ways to access the "liest"

More information

Review for 1 sample CI Name. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Review for 1 sample CI Name. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Review for 1 sample CI Name MULTIPLE CHOICE. Choose the oe alterative that best completes the statemet or aswers the questio. Fid the margi of error for the give cofidece iterval. 1) A survey foud that

More information

One-sample test of proportions

One-sample test of proportions Oe-sample test of proportios The Settig: Idividuals i some populatio ca be classified ito oe of two categories. You wat to make iferece about the proportio i each category, so you draw a sample. Examples:

More information

Hypergeometric Distributions

Hypergeometric Distributions 7.4 Hypergeometric Distributios Whe choosig the startig lie-up for a game, a coach obviously has to choose a differet player for each positio. Similarly, whe a uio elects delegates for a covetio or you

More information

Practice Problems for Test 3

Practice Problems for Test 3 Practice Problems for Test 3 Note: these problems oly cover CIs ad hypothesis testig You are also resposible for kowig the samplig distributio of the sample meas, ad the Cetral Limit Theorem Review all

More information

The following example will help us understand The Sampling Distribution of the Mean. C1 C2 C3 C4 C5 50 miles 84 miles 38 miles 120 miles 48 miles

The following example will help us understand The Sampling Distribution of the Mean. C1 C2 C3 C4 C5 50 miles 84 miles 38 miles 120 miles 48 miles The followig eample will help us uderstad The Samplig Distributio of the Mea Review: The populatio is the etire collectio of all idividuals or objects of iterest The sample is the portio of the populatio

More information

This document contains a collection of formulas and constants useful for SPC chart construction. It assumes you are already familiar with SPC.

This document contains a collection of formulas and constants useful for SPC chart construction. It assumes you are already familiar with SPC. SPC Formulas ad Tables 1 This documet cotais a collectio of formulas ad costats useful for SPC chart costructio. It assumes you are already familiar with SPC. Termiology Geerally, a bar draw over a symbol

More information

4.1 Sigma Notation and Riemann Sums

4.1 Sigma Notation and Riemann Sums 0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas

More information

x : X bar Mean (i.e. Average) of a sample

x : X bar Mean (i.e. Average) of a sample A quick referece for symbols ad formulas covered i COGS14: MEAN OF SAMPLE: x = x i x : X bar Mea (i.e. Average) of a sample x i : X sub i This stads for each idividual value you have i your sample. For

More information

Topic 5: Confidence Intervals (Chapter 9)

Topic 5: Confidence Intervals (Chapter 9) Topic 5: Cofidece Iterval (Chapter 9) 1. Itroductio The two geeral area of tatitical iferece are: 1) etimatio of parameter(), ch. 9 ) hypothei tetig of parameter(), ch. 10 Let X be ome radom variable with

More information

Incremental calculation of weighted mean and variance

Incremental calculation of weighted mean and variance Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically

More information

Chapter 7 - Sampling Distributions. 1 Introduction. What is statistics? It consist of three major areas:

Chapter 7 - Sampling Distributions. 1 Introduction. What is statistics? It consist of three major areas: Chapter 7 - Samplig Distributios 1 Itroductio What is statistics? It cosist of three major areas: Data Collectio: samplig plas ad experimetal desigs Descriptive Statistics: umerical ad graphical summaries

More information

Multi-server Optimal Bandwidth Monitoring for QoS based Multimedia Delivery Anup Basu, Irene Cheng and Yinzhe Yu

Multi-server Optimal Bandwidth Monitoring for QoS based Multimedia Delivery Anup Basu, Irene Cheng and Yinzhe Yu Multi-server Optimal Badwidth Moitorig for QoS based Multimedia Delivery Aup Basu, Iree Cheg ad Yizhe Yu Departmet of Computig Sciece U. of Alberta Architecture Applicatio Layer Request receptio -coectio

More information

1 Hypothesis testing for a single mean

1 Hypothesis testing for a single mean BST 140.65 Hypothesis Testig Review otes 1 Hypothesis testig for a sigle mea 1. The ull, or status quo, hypothesis is labeled H 0, the alterative H a or H 1 or H.... A type I error occurs whe we falsely

More information

7. Sample Covariance and Correlation

7. Sample Covariance and Correlation 1 of 8 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 7. Sample Covariace ad Correlatio The Bivariate Model Suppose agai that we have a basic radom experimet, ad that X ad Y

More information

GCSE STATISTICS. 4) How to calculate the range: The difference between the biggest number and the smallest number.

GCSE STATISTICS. 4) How to calculate the range: The difference between the biggest number and the smallest number. GCSE STATISTICS You should kow: 1) How to draw a frequecy diagram: e.g. NUMBER TALLY FREQUENCY 1 3 5 ) How to draw a bar chart, a pictogram, ad a pie chart. 3) How to use averages: a) Mea - add up all

More information

Chapter 5 Unit 1. IET 350 Engineering Economics. Learning Objectives Chapter 5. Learning Objectives Unit 1. Annual Amount and Gradient Functions

Chapter 5 Unit 1. IET 350 Engineering Economics. Learning Objectives Chapter 5. Learning Objectives Unit 1. Annual Amount and Gradient Functions Chapter 5 Uit Aual Amout ad Gradiet Fuctios IET 350 Egieerig Ecoomics Learig Objectives Chapter 5 Upo completio of this chapter you should uderstad: Calculatig future values from aual amouts. Calculatig

More information

Properties of MLE: consistency, asymptotic normality. Fisher information.

Properties of MLE: consistency, asymptotic normality. Fisher information. Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout

More information

3.1 Measures of Central Tendency. Introduction 5/28/2013. Data Description. Outline. Objectives. Objectives. Traditional Statistics Average

3.1 Measures of Central Tendency. Introduction 5/28/2013. Data Description. Outline. Objectives. Objectives. Traditional Statistics Average 5/8/013 C H 3A P T E R Outlie 3 1 Measures of Cetral Tedecy 3 Measures of Variatio 3 3 3 Measuresof Positio 3 4 Exploratory Data Aalysis Copyright 013 The McGraw Hill Compaies, Ic. C H 3A P T E R Objectives

More information

Output Analysis (2, Chapters 10 &11 Law)

Output Analysis (2, Chapters 10 &11 Law) B. Maddah ENMG 6 Simulatio 05/0/07 Output Aalysis (, Chapters 10 &11 Law) Comparig alterative system cofiguratio Sice the output of a simulatio is radom, the comparig differet systems via simulatio should

More information

Math C067 Sampling Distributions

Math C067 Sampling Distributions Math C067 Samplig Distributios Sample Mea ad Sample Proportio Richard Beigel Some time betwee April 16, 2007 ad April 16, 2007 Examples of Samplig A pollster may try to estimate the proportio of voters

More information

Unit 20 Hypotheses Testing

Unit 20 Hypotheses Testing Uit 2 Hypotheses Testig Objectives: To uderstad how to formulate a ull hypothesis ad a alterative hypothesis about a populatio proportio, ad how to choose a sigificace level To uderstad how to collect

More information

1 Correlation and Regression Analysis

1 Correlation and Regression Analysis 1 Correlatio ad Regressio Aalysis I this sectio we will be ivestigatig the relatioship betwee two cotiuous variable, such as height ad weight, the cocetratio of a ijected drug ad heart rate, or the cosumptio

More information

Spss Lab 7: T-tests Section 1

Spss Lab 7: T-tests Section 1 Spss Lab 7: T-tests Sectio I this lab, we will be usig everythig we have leared i our text ad applyig that iformatio to uderstad t-tests for parametric ad oparametric data. THERE WILL BE TWO SECTIONS FOR

More information

where: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return

where: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return EVALUATING ALTERNATIVE CAPITAL INVESTMENT PROGRAMS By Ke D. Duft, Extesio Ecoomist I the March 98 issue of this publicatio we reviewed the procedure by which a capital ivestmet project was assessed. The

More information

LECTURE 13: Cross-validation

LECTURE 13: Cross-validation LECTURE 3: Cross-validatio Resampli methods Cross Validatio Bootstrap Bias ad variace estimatio with the Bootstrap Three-way data partitioi Itroductio to Patter Aalysis Ricardo Gutierrez-Osua Texas A&M

More information

15.075 Exam 3. Instructor: Cynthia Rudin TA: Dimitrios Bisias. November 22, 2011

15.075 Exam 3. Instructor: Cynthia Rudin TA: Dimitrios Bisias. November 22, 2011 15.075 Exam 3 Istructor: Cythia Rudi TA: Dimitrios Bisias November 22, 2011 Gradig is based o demostratio of coceptual uderstadig, so you eed to show all of your work. Problem 1 A compay makes high-defiitio

More information

Sampling Distribution And Central Limit Theorem

Sampling Distribution And Central Limit Theorem () Samplig Distributio & Cetral Limit Samplig Distributio Ad Cetral Limit Samplig distributio of the sample mea If we sample a umber of samples (say k samples where k is very large umber) each of size,

More information

STATISTICAL METHODS FOR BUSINESS

STATISTICAL METHODS FOR BUSINESS STATISTICAL METHODS FOR BUSINESS UNIT 7: INFERENTIAL TOOLS. DISTRIBUTIONS ASSOCIATED WITH SAMPLING 7.1.- Distributios associated with the samplig process. 7.2.- Iferetial processes ad relevat distributios.

More information

A Mathematical Perspective on Gambling

A Mathematical Perspective on Gambling A Mathematical Perspective o Gamblig Molly Maxwell Abstract. This paper presets some basic topics i probability ad statistics, icludig sample spaces, probabilistic evets, expectatios, the biomial ad ormal

More information

Stat 104 Lecture 2. Variables and their distributions. DJIA: monthly % change, 2000 to Finding the center of a distribution. Median.

Stat 104 Lecture 2. Variables and their distributions. DJIA: monthly % change, 2000 to Finding the center of a distribution. Median. Stat 04 Lecture Statistics 04 Lecture (IPS. &.) Outlie for today Variables ad their distributios Fidig the ceter Measurig the spread Effects of a liear trasformatio Variables ad their distributios Variable:

More information

USING STATISTICAL FUNCTIONS ON A SCIENTIFIC CALCULATOR

USING STATISTICAL FUNCTIONS ON A SCIENTIFIC CALCULATOR USING STATISTICAL FUNCTIONS ON A SCIENTIFIC CALCULATOR Objective:. Improve calculator skills eeded i a multiple choice statistical eamiatio where the eam allows the studet to use a scietific calculator..

More information

Repeated sampling in Successive Survey

Repeated sampling in Successive Survey Statistics istitutio Repeated samplig i Successive Survey (RSSS) Xiaolu Cao 15-högskolepoägsuppsats iom Statistik III, HT011 Supervisor: Mikael Möller Abstract: I this thesis, the idea of samplig desig

More information

Ch 7.1 pg. 364 #11, 13, 15, 17, 19, 21, 23, 25

Ch 7.1 pg. 364 #11, 13, 15, 17, 19, 21, 23, 25 Math 7 Elemetary Statistics: A Brief Versio, 5/e Bluma Ch 7.1 pg. 364 #11, 13, 15, 17, 19, 1, 3, 5 11. Readig Scores: A sample of the readig scores of 35 fifth-graders has a mea of 8. The stadard deviatio

More information

CHAPTER 3 THE TIME VALUE OF MONEY

CHAPTER 3 THE TIME VALUE OF MONEY CHAPTER 3 THE TIME VALUE OF MONEY OVERVIEW A dollar i the had today is worth more tha a dollar to be received i the future because, if you had it ow, you could ivest that dollar ad ear iterest. Of all

More information

PSYCHOLOGICAL STATISTICS

PSYCHOLOGICAL STATISTICS UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION B Sc. Cousellig Psychology (0 Adm.) IV SEMESTER COMPLEMENTARY COURSE PSYCHOLOGICAL STATISTICS QUESTION BANK. Iferetial statistics is the brach of statistics

More information

MEI Structured Mathematics. Module Summary Sheets. Statistics 2 (Version B: reference to new book)

MEI Structured Mathematics. Module Summary Sheets. Statistics 2 (Version B: reference to new book) MEI Mathematics i Educatio ad Idustry MEI Structured Mathematics Module Summary Sheets Statistics (Versio B: referece to ew book) Topic : The Poisso Distributio Topic : The Normal Distributio Topic 3:

More information

Statistical Inference: Hypothesis Testing for Single Populations

Statistical Inference: Hypothesis Testing for Single Populations Chapter 9 Statistical Iferece: Hypothesis Testig for Sigle Populatios A foremost statistical mechaism for decisio makig is the hypothesis test. The cocept of hypothesis testig lies at the heart of iferetial

More information

1 Introduction to reducing variance in Monte Carlo simulations

1 Introduction to reducing variance in Monte Carlo simulations Copyright c 007 by Karl Sigma 1 Itroductio to reducig variace i Mote Carlo simulatios 11 Review of cofidece itervals for estimatig a mea I statistics, we estimate a uow mea µ = E(X) of a distributio by

More information

INVESTMENT PERFORMANCE COUNCIL (IPC)

INVESTMENT PERFORMANCE COUNCIL (IPC) INVESTMENT PEFOMANCE COUNCIL (IPC) INVITATION TO COMMENT: Global Ivestmet Performace Stadards (GIPS ) Guidace Statemet o Calculatio Methodology The Associatio for Ivestmet Maagemet ad esearch (AIM) seeks

More information

3. Covariance and Correlation

3. Covariance and Correlation Virtual Laboratories > 3. Expected Value > 1 2 3 4 5 6 3. Covariace ad Correlatio Recall that by takig the expected value of various trasformatios of a radom variable, we ca measure may iterestig characteristics

More information

Estimating Probability Distributions by Observing Betting Practices

Estimating Probability Distributions by Observing Betting Practices 5th Iteratioal Symposium o Imprecise Probability: Theories ad Applicatios, Prague, Czech Republic, 007 Estimatig Probability Distributios by Observig Bettig Practices Dr C Lych Natioal Uiversity of Irelad,

More information

STA 2023 Practice Questions Exam 2 Chapter 7- sec 9.2. Case parameter estimator standard error Estimate of standard error

STA 2023 Practice Questions Exam 2 Chapter 7- sec 9.2. Case parameter estimator standard error Estimate of standard error STA 2023 Practice Questios Exam 2 Chapter 7- sec 9.2 Formulas Give o the test: Case parameter estimator stadard error Estimate of stadard error Samplig Distributio oe mea x s t (-1) oe p ( 1 p) CI: prop.

More information

Mann-Whitney U 2 Sample Test (a.k.a. Wilcoxon Rank Sum Test)

Mann-Whitney U 2 Sample Test (a.k.a. Wilcoxon Rank Sum Test) No-Parametric ivariate Statistics: Wilcoxo-Ma-Whitey 2 Sample Test 1 Ma-Whitey 2 Sample Test (a.k.a. Wilcoxo Rak Sum Test) The (Wilcoxo-) Ma-Whitey (WMW) test is the o-parametric equivalet of a pooled

More information

Descriptive statistics deals with the description or simple analysis of population or sample data.

Descriptive statistics deals with the description or simple analysis of population or sample data. Descriptive statistics Some basic cocepts A populatio is a fiite or ifiite collectio of idividuals or objects. Ofte it is impossible or impractical to get data o all the members of the populatio ad a small

More information

Confidence intervals and hypothesis tests

Confidence intervals and hypothesis tests Chapter 2 Cofidece itervals ad hypothesis tests This chapter focuses o how to draw coclusios about populatios from sample data. We ll start by lookig at biary data (e.g., pollig), ad lear how to estimate

More information

Notes on Hypothesis Testing

Notes on Hypothesis Testing Probability & Statistics Grishpa Notes o Hypothesis Testig A radom sample X = X 1,..., X is observed, with joit pmf/pdf f θ x 1,..., x. The values x = x 1,..., x of X lie i some sample space X. The parameter

More information

Biology 171L Environment and Ecology Lab Lab 2: Descriptive Statistics, Presenting Data and Graphing Relationships

Biology 171L Environment and Ecology Lab Lab 2: Descriptive Statistics, Presenting Data and Graphing Relationships Biology 171L Eviromet ad Ecology Lab Lab : Descriptive Statistics, Presetig Data ad Graphig Relatioships Itroductio Log lists of data are ofte ot very useful for idetifyig geeral treds i the data or the

More information

This is arithmetic average of the x values and is usually referred to simply as the mean.

This is arithmetic average of the x values and is usually referred to simply as the mean. prepared by Dr. Adre Lehre, Dept. of Geology, Humboldt State Uiversity http://www.humboldt.edu/~geodept/geology51/51_hadouts/statistical_aalysis.pdf STATISTICAL ANALYSIS OF HYDROLOGIC DATA This hadout

More information

Modified Line Search Method for Global Optimization

Modified Line Search Method for Global Optimization Modified Lie Search Method for Global Optimizatio Cria Grosa ad Ajith Abraham Ceter of Excellece for Quatifiable Quality of Service Norwegia Uiversity of Sciece ad Techology Trodheim, Norway {cria, ajith}@q2s.tu.o

More information

Hypothesis testing using complex survey data

Hypothesis testing using complex survey data Hypotesis testig usig complex survey data A Sort Course preseted by Peter Ly, Uiversity of Essex i associatio wit te coferece of te Europea Survey Researc Associatio Prague, 5 Jue 007 1 1. Objective: Simple

More information

Simple Linear Regression

Simple Linear Regression Simple Liear Regressio We have bee itroduced to the otio that a categorical variable could deped o differet levels of aother variable whe we discussed cotigecy tables. We ll exted this idea to the case

More information

Solving Logarithms and Exponential Equations

Solving Logarithms and Exponential Equations Solvig Logarithms ad Epoetial Equatios Logarithmic Equatios There are two major ideas required whe solvig Logarithmic Equatios. The first is the Defiitio of a Logarithm. You may recall from a earlier topic:

More information

Measures of Central Tendency

Measures of Central Tendency Measures of Cetral Tedecy A studet s grade will be determied by exam grades ( each exam couts twice ad there are three exams, HW average (couts oce, fial exam ( couts three times. Fid the average if the

More information

Gregory Carey, 1998 Linear Transformations & Composites - 1. Linear Transformations and Linear Composites

Gregory Carey, 1998 Linear Transformations & Composites - 1. Linear Transformations and Linear Composites Gregory Carey, 1998 Liear Trasformatios & Composites - 1 Liear Trasformatios ad Liear Composites I Liear Trasformatios of Variables Meas ad Stadard Deviatios of Liear Trasformatios A liear trasformatio

More information

Systems Design Project: Indoor Location of Wireless Devices

Systems Design Project: Indoor Location of Wireless Devices Systems Desig Project: Idoor Locatio of Wireless Devices Prepared By: Bria Murphy Seior Systems Sciece ad Egieerig Washigto Uiversity i St. Louis Phoe: (805) 698-5295 Email: bcm1@cec.wustl.edu Supervised

More information

Sample size for clinical trials

Sample size for clinical trials Outcome variables for trials British Stadards Istitutio Study Day Sample size for cliical trials Marti Blad Prof. of Health Statistics Uiversity of York http://martiblad.co.uk A outcome variable is oe

More information

Page 1. Real Options for Engineering Systems. What are we up to? Today s agenda. J1: Real Options for Engineering Systems. Richard de Neufville

Page 1. Real Options for Engineering Systems. What are we up to? Today s agenda. J1: Real Options for Engineering Systems. Richard de Neufville Real Optios for Egieerig Systems J: Real Optios for Egieerig Systems By (MIT) Stefa Scholtes (CU) Course website: http://msl.mit.edu/cmi/ardet_2002 Stefa Scholtes Judge Istitute of Maagemet, CU Slide What

More information

Chapter 10 Student Lecture Notes 10-1

Chapter 10 Student Lecture Notes 10-1 Chapter 0 tudet Lecture Notes 0- Basic Busiess tatistics (9 th Editio) Chapter 0 Two-ample Tests with Numerical Data 004 Pretice-Hall, Ic. Chap 0- Chapter Topics Comparig Two Idepedet amples Z test for

More information

CHAPTER 3 DIGITAL CODING OF SIGNALS

CHAPTER 3 DIGITAL CODING OF SIGNALS CHAPTER 3 DIGITAL CODING OF SIGNALS Computers are ofte used to automate the recordig of measuremets. The trasducers ad sigal coditioig circuits produce a voltage sigal that is proportioal to a quatity

More information

A Guide to the Pricing Conventions of SFE Interest Rate Products

A Guide to the Pricing Conventions of SFE Interest Rate Products A Guide to the Pricig Covetios of SFE Iterest Rate Products SFE 30 Day Iterbak Cash Rate Futures Physical 90 Day Bak Bills SFE 90 Day Bak Bill Futures SFE 90 Day Bak Bill Futures Tick Value Calculatios

More information

Non-life insurance mathematics. Nils F. Haavardsson, University of Oslo and DNB Skadeforsikring

Non-life insurance mathematics. Nils F. Haavardsson, University of Oslo and DNB Skadeforsikring No-life isurace mathematics Nils F. Haavardsso, Uiversity of Oslo ad DNB Skadeforsikrig Mai issues so far Why does isurace work? How is risk premium defied ad why is it importat? How ca claim frequecy

More information