Confidence Intervals for One Mean

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Confidence Intervals for One Mean"

Transcription

1 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 cofidece iterval about the mea whe the uderlyig data distributio is ormal. Cautio: This procedure assumes that the stadard deviatio of the future sample will be the same as the stadard deviatio that is specified. If the stadard deviatio to be used i the procedure is estimated from a previous sample or represets the populatio stadard deviatio, the Cofidece Itervals for Oe Mea with Tolerace Probability procedure should be cosidered. That procedure cotrols the probability that the distace from the mea to the cofidece limits will be less tha or equal to the value specified. Techical Details For a sigle mea from a ormal distributio with kow variace, a two-sided, 100(1 α)% cofidece iterval is calculated by z X ± 1 α / 2σ A oe-sided 100(1 α)% upper cofidece limit is calculated by z X + 1 α σ Similarly, the oe-sided 100(1 α)% lower cofidece limit is z X 1 α σ For a sigle mea from a ormal distributio with ukow variace, a two-sided, 100(1 α)% cofidece iterval is calculated by t X ± 1 α / 2, 1 A oe-sided 100(1 α)% upper cofidece limit is calculated by t X + ˆ σ, 1 ˆ σ 1 α 420-1

2 Cofidece Itervals for Oe Mea Similarly, the oe-sided 100(1 α)% lower cofidece limit is t X 1 α, 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 ˆ σ D = z1 α / σ 2 whe the stadard deviatio is kow, ad D t σ α = 1 / 2, 1 whe the stadard deviatio is ukow. These equatios 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. 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

3 Cofidece Itervals for Oe Mea 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. Cofidece 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. 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 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. Stadard Deviatio S (Stadard Deviatio) Eter a value (or rage of values) for the stadard deviatio. Roughly speakig, this value estimates the average absolute differece betwee each idividual ad every other idividual. You ca use the results of a pilot study, a previous study, or a ball park estimate based o the rage (e.g., Rage/4) to estimate this parameter. Kow Stadard Deviatio Check this box whe you wat to base your results o the ormal distributio. Whe the box is ot checked, calculatios are based o the t-distributio. The differece betwee the two distributios is egligible whe the sample sizes are large (>50)

4 Cofidece Itervals for Oe Mea Example 1 Calculatig Sample Size Suppose a study is plaed i which the researcher wishes to costruct a two-sided 95% cofidece iterval for the mea such that the width of the iterval is o wider tha 14 uits. The cofidece level is set at 0.95, but 0.99 is icluded for comparative purposes. The stadard deviatio estimate, based o the rage of data values, is 28. Istead of examiig oly the iterval half-width of 7, a series of half-widths from 5 to 9 will also be cosidered. The goal is to determie the ecessary sample size. Setup This sectio presets the values of each of the parameters eeded to ru this example. First, from the PASS Home widow, load the Cofidece Itervals for Oe Mea procedure widow by expadig Meas, the Oe Mea, the clickig o Cofidece Iterval, ad the clickig o Cofidece Itervals for Oe Mea. 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 Distace from Mea to Limit(s)... 5 to 9 by 1 S (Stadard Deviatio) Kow Stadard Deviatio... Not Checked Aotated Output Click the Calculate butto to perform the calculatios ad geerate the followig output. Numeric Results Numeric Results for Two-Sided Cofidece Itervals with Ukow Stadard Deviatio Target Actual Sample Distace Distace Stadard Cofidece Size from Mea from Mea Deviatio Level (N) to Limits to Limits (S) Refereces Hah, G. J. ad Meeker, W.Q Statistical Itervals. Joh Wiley & Sos. New York

5 Cofidece Itervals for Oe Mea 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. Summary Statemets A sample size of 123 produces a two-sided 95% cofidece iterval with a distace from the mea to the limits that is equal to whe the estimated stadard deviatio is This report shows the calculated sample size for each of the scearios. Chart Sectio These plots show the sample size versus the distace from the mea to the limits (precisio) for the two cofidece levels

6 Cofidece Itervals for Oe Mea Example 2 Validatio usig Moore ad McCabe Moore ad McCabe (1999) page 443 give a example of a sample size calculatio for a cofidece iterval o the mea whe the cofidece coefficiet is 95%, the stadard deviatio is kow to be 3, ad the margi of error is 2. The ecessary sample size is 9. Setup This sectio presets the values of each of the parameters eeded to ru this example. First, from the PASS Home widow, load the Cofidece Itervals for Oe Mea procedure widow by expadig Meas, the Oe Mea, the clickig o Cofidece Iterval, ad the clickig o Cofidece Itervals for Oe Mea. 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 Distace from Mea to Limit(s)... 2 S (Stadard Deviatio)... 3 Kow Stadard Deviatio... Checked Output Click the Calculate butto to perform the calculatios ad geerate the followig output. Numeric Results Target Actual Sample Distace Distace Stadard Cofidece Size from Mea from Mea Deviatio Level (N) to Limits to Limits (S) PASS also calculated the ecessary sample size to be

7 Cofidece Itervals for Oe Mea Example 3 Validatio usig Ostle ad Maloe Ostle ad Maloe (1988) page 536 give a example of a sample size calculatio for a cofidece iterval o the mea whe the cofidece coefficiet is 95%, the stadard deviatio is kow to be 7, ad the margi of error is 5. The ecessary sample size is 8. Setup This sectio presets the values of each of the parameters eeded to ru this example. First, from the PASS Home widow, load the Cofidece Itervals for Oe Mea procedure widow by expadig Meas, the Oe Mea, the clickig o Cofidece Iterval, ad the clickig o Cofidece Itervals for Oe Mea. 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 Distace from Mea to Limit(s)... 5 S (Stadard Deviatio)... 7 Kow Stadard Deviatio... Checked Output Click the Calculate butto to perform the calculatios ad geerate the followig output. Numeric Results Target Actual Sample Distace Distace Stadard Cofidece Size from Mea from Mea Deviatio Level (N) to Limits to Limits (S) PASS also calculated the ecessary sample size to be

Confidence Intervals for One Mean with Tolerance Probability

Confidence Intervals for One Mean with Tolerance Probability 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

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

Confidence Intervals for Paired Means

Confidence Intervals for Paired Means Chaper 496 Cofidece Iervals for Paired Meas Iroducio This rouie calculaes he sample size ecessary o achieve a specified disace from he paired sample mea erece o he cofidece limi(s) a a saed cofidece level

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

Definition. Definition. 7-2 Estimating a Population Proportion. Definition. Definition

Definition. Definition. 7-2 Estimating a Population Proportion. Definition. Definition 7- stimatig a Populatio Proportio I this sectio we preset methods for usig a sample proportio to estimate the value of a populatio proportio. The sample proportio is the best poit estimate of the populatio

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

Section 7-3 Estimating a Population. Requirements

Section 7-3 Estimating a Population. Requirements Sectio 7-3 Estimatig a Populatio Mea: σ Kow Key Cocept This sectio presets methods for usig sample data to fid a poit estimate ad cofidece iterval estimate of a populatio mea. A key requiremet i this sectio

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

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

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

CHAPTER 8: CONFIDENCE INTERVAL ESTIMATES for Means and Proportions

CHAPTER 8: CONFIDENCE INTERVAL ESTIMATES for Means and Proportions CHAPTER 8: CONFIDENCE INTERVAL ESTIMATES for Meas ad Proportios Itroductio: We wat to kow the value of a parameter for a populatio. We do t kow the value of this parameter for the etire populatio because

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

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

Hypothesis Tests Applied to Means

Hypothesis Tests Applied to Means The Samplig Distributio of the Mea Hypothesis Tests Applied to Meas Recall that the samplig distributio of the mea is the distributio of sample meas that would be obtaied from a particular populatio (with

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

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

Confidence Intervals for the Population Mean

Confidence Intervals for the Population Mean Cofidece Itervals Math 283 Cofidece Itervals for the Populatio Mea Recall that from the empirical rule that the iterval of the mea plus/mius 2 times the stadard deviatio will cotai about 95% of the observatios.

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

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

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

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

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

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

Confidence Intervals and Sample Size

Confidence Intervals and Sample Size 8/7/015 C H A P T E R S E V E N Cofidece Itervals ad Copyright 015 The McGraw-Hill Compaies, Ic. Permissio required for reproductio or display. 1 Cofidece Itervals ad Outlie 7-1 Cofidece Itervals for the

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

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

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

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

Section 7.2 Confidence Interval for a Proportion

Section 7.2 Confidence Interval for a Proportion Sectio 7.2 Cofidece Iterval for a Proportio Before ay ifereces ca be made about a proportio, certai coditios must be satisfied: 1. The sample must be a SRS from the populatio of iterest. 2. The populatio

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

7.1 Inference for a Population Proportion

7.1 Inference for a Population Proportion 7.1 Iferece for a Populatio Proportio Defiitio. The statistic that estimates the parameter p is the sample proportio cout of successes i the sample ˆp = cout of observatios i the sample. Assumptios for

More information

Confidence Intervals for the Mean of Non-normal Data Class 23, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom

Confidence Intervals for the Mean of Non-normal Data Class 23, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom Cofidece Itervals for the Mea of No-ormal Data Class 23, 8.05, Sprig 204 Jeremy Orloff ad Joatha Bloom Learig Goals. Be able to derive the formula for coservative ormal cofidece itervals for the proportio

More information

Statistics Lecture 14. Introduction to Inference. Administrative Notes. Hypothesis Tests. Last Class: Confidence Intervals

Statistics Lecture 14. Introduction to Inference. Administrative Notes. Hypothesis Tests. Last Class: Confidence Intervals Statistics 111 - Lecture 14 Itroductio to Iferece Hypothesis Tests Admiistrative Notes Sprig Break! No lectures o Tuesday, March 8 th ad Thursday March 10 th Exteded Sprig Break! There is o Stat 111 recitatio

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

Stat 104 Lecture 16. Statistics 104 Lecture 16 (IPS 6.1) Confidence intervals - the general concept

Stat 104 Lecture 16. Statistics 104 Lecture 16 (IPS 6.1) Confidence intervals - the general concept Statistics 104 Lecture 16 (IPS 6.1) Outlie for today Cofidece itervals Cofidece itervals for a mea, µ (kow σ) Cofidece itervals for a proportio, p Margi of error ad sample size Review of mai topics for

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

ME 101 Measurement Demonstration (MD 1) DEFINITIONS Precision - A measure of agreement between repeated measurements (repeatability).

ME 101 Measurement Demonstration (MD 1) DEFINITIONS Precision - A measure of agreement between repeated measurements (repeatability). INTRODUCTION This laboratory ivestigatio ivolves makig both legth ad mass measuremets of a populatio, ad the assessig statistical parameters to describe that populatio. For example, oe may wat to determie

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

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

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

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

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

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

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

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

Compare Multiple Response Variables

Compare Multiple Response Variables Compare Multiple Respose Variables STATGRAPHICS Mobile Rev. 4/7/006 This procedure compares the data cotaied i three or more Respose colums. It performs a oe-way aalysis of variace to determie whether

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

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

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

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

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

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

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

Laboratory: Case-Control Studies. Hypothesis Testing

Laboratory: Case-Control Studies. Hypothesis Testing Laboratory: Case-Cotrol Studies How may do I eed? is oe of the most commo questios addressed to a epidemiologist. The epidemiologist aswers with What questio are you attemptig to aswer? Sample size depeds

More information

Institute for the Advancement of University Learning & Department of Statistics

Institute for the Advancement of University Learning & Department of Statistics Istitute for the Advacemet of Uiversity Learig & Departmet of Statistics Descriptive Statistics for Research (Hilary Term, 00) Lecture 5: Cofidece Itervals (I.) Itroductio Cofidece itervals (or regios)

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

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

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

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

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

when n = 1, 2, 3, 4, 5, 6, This list represents the amount of dollars you have after n days. Note: The use of is read as and so on.

when n = 1, 2, 3, 4, 5, 6, This list represents the amount of dollars you have after n days. Note: The use of is read as and so on. Geometric eries Before we defie what is meat by a series, we eed to itroduce a related topic, that of sequeces. Formally, a sequece is a fuctio that computes a ordered list. uppose that o day 1, you have

More information

Hypothesis testing in a Nutshell

Hypothesis testing in a Nutshell Hypothesis testig i a Nutshell Summary by Pamela Peterso Drake Itroductio The purpose of this readig is to discuss aother aspect of statistical iferece, testig. A is a statemet about the value of a populatio

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

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

TIEE Teaching Issues and Experiments in Ecology - Volume 1, January 2004

TIEE Teaching Issues and Experiments in Ecology - Volume 1, January 2004 TIEE Teachig Issues ad Experimets i Ecology - Volume 1, Jauary 2004 EXPERIMENTS Evirometal Correlates of Leaf Stomata Desity Bruce W. Grat ad Itzick Vatick Biology, Wideer Uiversity, Chester PA, 19013

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

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

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

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

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

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

23.3 Sampling Distributions

23.3 Sampling Distributions COMMON CORE Locker LESSON Commo Core Math Stadards The studet is expected to: COMMON CORE S-IC.B.4 Use data from a sample survey to estimate a populatio mea or proportio; develop a margi of error through

More information

Hypothesis testing: one sample

Hypothesis testing: one sample Hypothesis testig: oe sample Describig iformatios Flow-chart for QMS 202 Drawig coclusios Forecastig Improve busiess processes Data Collectio Probability & Probability Distributio Regressio Aalysis Time-series

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

NPTEL STRUCTURAL RELIABILITY

NPTEL STRUCTURAL RELIABILITY NPTEL Course O STRUCTURAL RELIABILITY Module # 0 Lecture 1 Course Format: Web Istructor: Dr. Aruasis Chakraborty Departmet of Civil Egieerig Idia Istitute of Techology Guwahati 1. Lecture 01: Basic Statistics

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

ˆ p 2. ˆ p 1. ˆ p 3. p 4. ˆ p 8

ˆ p 2. ˆ p 1. ˆ p 3. p 4. ˆ p 8 Sectio 8 1C The Techiques of Hypothesis Testig A claim is made that 10% of the populatio is left haded. A alterate claim is made that less tha 10% of the populatio is left haded. We will use the techiques

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

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

THE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n

THE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n We will cosider the liear regressio model i matrix form. For simple liear regressio, meaig oe predictor, the model is i = + x i + ε i for i =,,,, This model icludes the assumptio that the ε i s are a sample

More information

Economics 140A Confidence Intervals and Hypothesis Testing

Economics 140A Confidence Intervals and Hypothesis Testing Ecoomics 140A Cofidece Itervals ad Hypothesis Testig Obtaiig a estimate of a parameter is ot the al purpose of statistical iferece because it is highly ulikely that the populatio value of a parameter is

More information

Case Study. Contingency Tables. Graphing Tabled Counts. Stacked Bar Graph

Case Study. Contingency Tables. Graphing Tabled Counts. Stacked Bar Graph Case Study Cotigecy Tables Bret Halo ad Bret Larget Departmet of Statistics Uiversity of Wiscosi Madiso October 4 6, 2011 Case Study Example 9.3 begiig o page 213 of the text describes a experimet i which

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

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

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

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

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

Table of standard normal deviates for Z ß...31

Table of standard normal deviates for Z ß...31 Statistical Power Ad Sample Size Calculatios... 1 Whe Do You Need Statistical Power Calculatios, Ad Why?... 1 Preparatio For The Questio What Is Statistical Power?... 1 Statistical Hypothesis Testig...

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 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

Section 9.2 Series and Convergence

Section 9.2 Series and Convergence Sectio 9. Series ad Covergece Goals of Chapter 9 Approximate Pi Prove ifiite series are aother importat applicatio of limits, derivatives, approximatio, slope, ad cocavity of fuctios. Fid challegig atiderivatives

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

hp calculators HP 12C Statistics - average and standard deviation Average and standard deviation concepts HP12C average and standard deviation

hp calculators HP 12C Statistics - average and standard deviation Average and standard deviation concepts HP12C average and standard deviation HP 1C Statistics - average ad stadard deviatio Average ad stadard deviatio cocepts HP1C average ad stadard deviatio Practice calculatig averages ad stadard deviatios with oe or two variables HP 1C Statistics

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

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

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

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

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

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

Chapter 5 Discrete Probability Distributions

Chapter 5 Discrete Probability Distributions Slides Prepared by JOHN S. LOUCKS St. Edward s Uiversity Slide Chapter 5 Discrete Probability Distributios Radom Variables Discrete Probability Distributios Epected Value ad Variace Poisso Distributio

More information