Hypothesis Tests Applied to Means

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Hypothesis Tests Applied to Means"

Transcription

1 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 mea µ ad stadard deviatio ), if a ifiite umber of radom samples were draw from the populatio ad the mea from each of these samples was calculated. By the Cetral Limit Theorem (CLT), for ay give populatio, (with mea µ ad stadard deviatio ) the distributio of sample meas for a particular sample size,, approaches a ormal distributio, with a mea idetical to that of the populatio (i.e. µ ) ad a stadard deviatio of, as the sample size gets larger ad approaches ifiity. The CLT tells us that what the mea ad stadard deviatio will be for ay give sample size ad also tells us that the shape of the samplig distributio of the mea approaches ormality as the sample size icrease, regardless of the shape of the populatio that the sample mea is assumed to come from. Hypothesis Testig whe the Stadard Deviatio of the Populatio () is Kow I the example provided earlier (i.e. the high school couselor who wated to determie if the course he developed actually icreased SAT scores) the stadard deviatio of the populatio was kow. This is ofte the case whe usig stadardized tests as a depedet variable i a research study because such tests are give to a represetative radom sample of the populatio to determie orms. Coceptually, testig whether the mea of a sample (that was likely provided some sort of treatmet) differs from the mea of the populatio is o differet tha testig how likely it is to have obtaied a particular observatio from a populatio. However, we eed to use the stadard deviatio of the samplig distributio of the mea, kows as the stadard error of the mea or simply the stadard error. Example Dr. Frost is a researcher i a large school district that aually evaluates their studets academic progress i mathematics usig the Iowa Test of Basic Skills (ITBS). I the testig maual for the ITBS she discovers that the average performace of sixth grade studets i the atio o this test (i.e. the orm) is 85 ad that the stadard deviatio of scores i the atio is 0. She wats to kow if studets i her school district are performig at the same level. How could she go about testig this? What if the stadard deviatio of scores i the atio was oly 30? What if it was oly 0? Techically, or mathematically, oe could always use the stadard error of the mea because whe, it simply represets the stadard deviatio of the populatio.

2 Cofidece itervals are aother way to covey the results of a research study. The mea obtaied from a sample is a ubiased estimate of the mea i the populatio ad ca be cosidered a poit estimate of the mea of the populatio from which the sample was draw. Poit estimates use a sigle umber to estimate a ukow quatity. Cofidece itervals use a rage of values to estimate a ukow quatity. I the case of estimatig the mea of the populatio from which a particular sample was draw, a cofidece iterval represets a rage of possible populatio meas that may correspod to the give sample. I geeral, a cofidece iterval is obtaied by usig the sample statistic, its associated stadard error, ad the critical values associated with Type I error rate you are willig to risk. The critical values are the values of the test statistic that must be exceeded to reject the ull hypothesis. For example, if we are willig to risk makig a Type I error 5% of the time (i.e. α 0.05) the the critical value for coductig a z-test (used whe is kow) is ±.96. Note that cofidece itervals always use the critical value associated with a two-tailed test. We use these values to solve for the populatio parameter we are tryig to estimate. I the case of a z-test: z. 05 µ ±. 96 µ So a 95% cofidece iterval would be expressed by ±.96( ). What would a 95% cofidece iterval be for the true mathematics achievemet test scores for the sixth grade studets i Dr. Frost s school district? I geeral, a (-α) cofidece iterval for a z-test is obtaied by: ± z (α/) ( ) sample mea ± z (α/) (stadard error of the mea) What would a 90% cofidece iterval be for the true mathematics achievemet test scores for the sixth grade studets i Dr. Frost s school district? What would a 99% cofidece iterval be? Eve more geerally, a (-α) cofidece iterval for ay statistical test is obtaied by: sample estimate ± critical value (α/) (stadard error of the estimate) Hypothesis Testig whe the Stadard Deviatio of the Populatio () is NOT Kow Hypothesis Testig with Oe Sample: The Oe-Sample t Test I most practical research situatios that do ot utilize a stadardized test as a depedet variable, the stadard deviatio of the depedet variable i the populatio is ukow. Therefore, it must be estimated from the sample. Obviously, similar to the fact that the mea obtaied from ay oe sample is oe of a ifiite umber of sample meas that could have bee obtaied had aother sample bee

3 3 draw, the stadard deviatio obtaied from ay oe sample is oe of a ifiite umber of possible stadard deviatios that could have bee obtaied had aother sample bee draw. The samplig distributio of the stadard deviatio is coceptually the same as the samplig distributio of the mea. However, this distributio is ot ormal, like the samplig distributio of the mea is, rather it is positively skewed, ad the smaller the sample size the more positively skewed it is. Therefore, the stadard deviatio obtaied from ay oe sample is more likely to uderestimate the true stadard deviatio i the populatio, rather tha overestimate it. This is especially true for smaller observatios. As we saw earlier, the magitude of the stadard deviatio of the populatio has a direct effect o the magitude of the test statistic that will be obtaied. The larger the stadard deviatio is i the populatio the more difficult it is to detect differeces betwee a sample mea ad the populatio mea. This makes sese coceptually, because if a distributio is more variable tha there is a larger rage of likely values that ca be obtaied from ay oe sample. O the other had, the smaller the stadard deviatio i the sample is the easier it is to detect differeces betwee the sample mea ad the populatio mea. This also makes sese coceptually, because if a distributio is less variable tha there is a smaller rage of likely values that ca be obtaied from ay oe sample. Therefore, whe usig a estimate of the stadard deviatio i the populatio from a sample we must accout for the fact that we are more likely to reject a ull hypothesis that is true (i.e. commit a Type I error). This is true because we kow that smaller stadard deviatios lead to larger test statistics which are less likely to occur uder the ull hypothesis AND we kow that the stadard deviatio we obtai from ay oe sample is likely to uderestimate the true stadard deviatio i the populatio. The t-distributio ca be used istead of the stadard ormal distributio (i.e. z- distributio) whe we do ot kow the value of the populatio stadard deviatio ad we wat to test if a sample mea differs from some hypothesized populatio if the sample was draw from a ormal populatio OR the sample size is large eough to assume that the samplig distributio of the mea is ormal. I geeral, the t-distributio has fatter tails ad fewer values ear the mea. As sample size icreases, the t-distributio more closely resembles the stadard ormal distributio. Similar to the ormal distributio, this distributio represets a family of distributios. However, whereas the ormal distributio differs depedig o the mea ad stadard deviatio i the populatio the all t-distributios are all assumed to be stadardized, with a mea of zero ad a stadard deviatio of oe, ad differ depedig o sample size. The t-distributio that should be used depeds o the degrees of freedom. For the t- distributio the degrees of freedom are the umber of observatios whose values could be chaged if the mea must remai costat which is because the mea is used to estimate the stadard deviatio i the populatio. Techically, the t-statistic is almost idetical to the z-statistic, except the stadard deviatio estimated from the sample replaces the populatio stadard deviatio that was previously assumed to be kow.

4 4 t µ s µ s A 95% cofidece iterval for the true populatio mea associated with the sample draw is also calculated similarly ad ca be expressed by: ± t s ) Example (. 05 Professor Dyett, the Health Director at Marquette Uiversity, believes that studets at Marquette are very health coscious ad therefore cosume less sugar tha most people i the U.S. She kows that the average perso i the U.S. cosumes about 00 lbs. of raw sugar per year, mostly i the form of soft driks, cadies, ad pastries. Although she believes studets at Marquette cosume less sugar tha ormal she also wats to kow if they are cosumig more sugar tha ormal. To test her belief she radomly samples 5 studets ad asks them to log their sugar cosumptio for the ext year. She fids that the average sugar cosumptio i her sample is 80. lbs with a stadard deviatio of 35.5 lbs. What ca she coclude? Use both a poit estimate to test her hypothesis ad a iterval estimate. Hypothesis Testig with Two Idepedet Samples: The Idepedet Samples t-test Up util this poit we have oly cosidered sigle-sample techiques. However the more iterestig research questios typically ivolve two or more samples. For example, oe might wat to compare whether studets taught usig oe method score higher o a test tha those taught usig aother method. As aother example oe might wat to compare lefthaded people ad right-haded people to see if oe group is more creative tha the other. Wheever two idepedet samples are compared it is likely that meas obtaied from the two groups will differ by some amout due to chace aloe. Therefore, we eed to determie if the differece is large eough to coclude that the two samples are from two differet populatios. I geeral, the commo form of the test statistics that deal with hypotheses about meas ca be thought of as a ratio of the () differece betwee iformatio obtaied from the sample ad that assumed to be true i the populatio uder the ull hypothesis ad () the differece that would be expected to occur by chace aloe. Because we have two differet groups we eed symbolism to differetiate betwee the populatio parameters ad sample statistics obtaied from each of the groups so we use µ to represet the populatio mea from oe of the groups ad µ to represet the populatio

5 5 mea from the other group. Similarly, we use to represet the sample mea from oe of the groups ad to represet the sample mea from the other group. I additio, it is possible that the two samples are comprised of a differet umber of observatios so we use to represet the sample size of oe sample ad to represet the sample size of the other sample It is also possible that the two samples come from populatios that differ i their variability so we use to represet the variace for the populatio that oe of the groups is from ad to represet the stadard deviatio for the populatio that oe of the groups is from. We use to represet the variace for the populatio that the other group is from ad to represet the stadard deviatio for the populatio that oe of the groups is from. Similarly we use to represet the estimate of variace from oe of the samples ad s to represet s the estimate of the stadard deviatio from oe of the samples. We use s to represet the estimate of variace from the other sample ad s to represet the estimate of the stadard deviatio from the other sample. The ull hypothesis that we are testig i this case is: H 0 : µ µ or, equivaletly, H 0 : µ µ 0. To test this hypothesis we have to cosider the samplig distributio of the differece betwee meas. This distributio is approximately ormally distributed with a mea of µ µ. The stadard deviatio of this distributio is the stadard error of the differece betwee meas ad ca be determied by utilizig the law of variaces that states that the variace of a sum or differece of two idepedet variables is simply the sum of the two variaces. It should be oted that this is oly true if the two variables are idepedet. Therefore, sice the variace of the first samplig distributio of meas is ad the variace of the secod samplig distributio of meas is samplig distributio of the differece betwee the two meas is simply stadard deviatio of this distributio is equivalet to. the the variace of the ad the. Note that this is ot mathematically If we kew the two populatio variaces (recall the variace is simply the square of the stadard deviatio, which is typically reported rather tha the variace) we could simply calculate a z-score usig the observed differece betwee the meas i our sample, the hypothesized differece betwee the meas i the populatio (typically assumed to be zero) ad the stadard error of the differece betwee meas. Specifically,

6 6 ( z ) ( µ µ ) ( ) 0 However, we typically do ot kow the two populatio variaces so we have to estimate them from our samples. The problem with simple replacig with s ad with s is that we do t kow the exact samplig distributio of a t-statistic that is calculated i this way. I other words, we do t kow the degrees of freedom to use if we simply replace with s ad with s to calculate a test statistic. Oe way to get aroud this problem is to assume that the two populatios that the two samples are draw from have equal variaces, (i.e. ). This assumptio is kow as the homogeeity of variace assumptio ad if it is met our problem is solved because we do kow the samplig distributio of a test statistic that is calculated uder this assumptio. Specifically, we kow that it follows a t-distributio with degrees of freedom. If this assumptio is met the the sample variaces should be similar. Typically, for small samples ( < 0), if oe of the sample variaces is more tha 4 times larger tha the other tha the assumptio has bee violated. For larger samples, oe should be cocered if oe variace is more tha twice as large as the other. Oe ca also coduct a statistical test, kow as Levee s test, to see if this assumptio has bee met. The test computes the differece betwee the observed score i each group with its absolute deviatio from the mea (i.e., where i deotes the particular ij j observatio ad j deotes the group) ad coducts a t-test for idepedet samples o the differece scores for the two groups. However, if make this assumptio the we have a slightly differet problem to solve i that we have two estimates of the populatio variace, s ad s which oe should we use? Ituitively, you might thik that we should simply average the two estimates but this ca oly be doe if both groups have the same umber of observatios (i.e. equal sample sizes, ) because if the sample sizes are ot equal the the two estimates will ot be equally represetative of the variability i the populatio. Specifically, we kow that a larger sample should produce a more stable estimate of the populatio variace tha a smaller sample so we use weighted average of the two estimates to obtai a pooled variace estimate. This estimate gives more weight to the estimate of the variace that is obtaied form the larger sample ad is expressed by: s P ( ) s ( ) s It is this pooled estimate that should be used whe calculatig a test statistic to determie if the meas obtaied from two idepedet samples are similar eough to coclude that they

7 7 may have bee draw from the same populatio. This test statistic follows a t-distributio with degrees of freedom. Two degrees of freedom are lost because two sample meas are utilized whe estimatig the two sample stadard deviatio. Specifically: t s P s P s sp A 95% cofidece iterval for the differece betwee two mea obtaied from idepedet samples is expressed by: ( ) ± t ( ) Example.05 s A researcher was iterested i determiig if readig comprehesio of dyslexic childre was the same uder ormal ad reduced visual cotrast. So she radomly assiged 4 childre to oe of twp groups. Oe group was give a readig comprehesio test usig ormal text ad aother group was give a readig comprehesio test usig text that was covered by a plastic sheet desiged to reduce the visual cotrast of the text. However five childre i the secod group were uable to participate. The average readig comprehesio test scores for the first group was 75.0 ( 00) while the average readig comprehesio test scores for the secod group was 56.7 ( s s 33.3). What should she coclude? Use both poit ad iterval estimates. The fact that meas obtaied from two groups are foud to statistically differ from each other does ot mea that they differ from each other i a meaigful way, kow as the effect size. Cohes d is a effect size measure that expresses the differece betwee two meas i terms of stadard deviatio uits. Specifically, it ca be estimated by: d s P What is the effect size measure from the study described above? If the homogeeity of variace assumptio has bee violated The Welsh-Satterthwaite solutio ca be used which tries to estimate the appropriate degrees of freedom (df) for the test statistic. It is based o the idea that the true df for the t-statistic must fall somewhere betwee the smaller of either ad ad. Sice the magitude of the critical t-value is larger with a smaller umber of df (makig it more difficult to reject the

8 8 ull) we ca use the smallest umber of degrees of freedom to coduct the statistical test. If this test is sigificat the it would surely be sigificat if a larger umber of df were used. If it is ot the the correct umber of df ca be calculated usig the formula i your book. The stadard t-test for idepedet samples (i.e. assumig homogeeity of variace ad that the sample is draw from a populatio that is ormally distributed, which implies the samplig distributio of the differece betwee the meas will be ormally distributed) is said to be robust to violatios of the assumptios, especially for equal sample sizes. I other words, moderate violatios of the assumptios will ot drastically affect the magitude of the test statistic obtaied. However, if oe believes that the assumptios have bee drastically violated they should cosider alterate statistical tests. Depedet or Paired Sample t test Sometimes we may wat to determie whether or ot there has bee some chage i our sample over time, i which case we would collect data from respodets o more tha oe occasio, kow as repeated measures desig. Whe this desig is used it is importat to esure that a subject s secod respose is ot affected by their first respose or that time is ifluecig performace i subjects. Other times we wat to determie if two related samples that ca be matched up i some way, such as data obtaied from spouses or twis, differ, kow as a matched sample desig. This type of desig ca also be used to match subjects o some extraeous variable a researcher wishes to cotrol for, such as IQ. This helps to esure that differeces foud i the depedet variable for the two groups are ot caused by differeces i the extraeous variable. I either of these desigs a paired sample t-test is the appropriate statistical test to coduct The advatage of usig either of these research desigs is to miimize the possibility that the subjects i oe group are substatially differet from the other to begi with (before treatmet) which would bias the results. Similar to the t-test for idepedet samples, the ull hypothesis that we are testig i this case is: H 0 : µ µ or, equivaletly, H 0 : µ µ 0. However the t-test for depedet samples is coducted o differece scores which simply represet the differece betwee scores obtaied at the two differet poits i time or betwee the two matched respodets. If there is truly o differece betwee the two depedet samples tha the average of differece scores would be expected to be zero. This reduces the complexity of our test because our data is reduced to oe observatio per perso/matched pair so we are really testig a hypothesis usig oe sample of data. The ull hypothesis that we are testig ca be reduced to: H 0 : µ D µ µ 0. The test statistic i this case ca be expressed by: D µ D D 0 t sd sd ad a cofidece iterval is expressed by: D ± t s ). 05( D

9 9 The degrees of freedom for our test statistic are the same as they were for the oe-sample t- test ad represet the umber of pairs of observatios, or. Example Dr. Fredrick believes that the eviromet is more importat tha geetics i ifluecig itelligece. He locates pairs of idetical twis that have bee reared apart, where oe twi has bee raised i a eriched eviromet ad the other twi has bee raised i a impoverished eviromet, ad admiisters a stadardized IQ test to each twi. He obtais the followig data: Pair Eriched Impoverished Eviromet Eviromet Differece The mea of the observed differece scores is 3.75 ad the stadard deviatio of these scores is What is the ull hypothesis ad what does this mea from a substative perspective? How does this relate to his research hypothesis? Coduct a depedet sample t-test. Ca Dr. Frederick coclude that the eviromet is more importat tha geetics from the results of his study? Why or why ot?

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

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

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

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

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

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

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

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

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

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

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

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

Chapter 14 Nonparametric Statistics

Chapter 14 Nonparametric Statistics Chapter 14 Noparametric Statistics A.K.A. distributio-free statistics! Does ot deped o the populatio fittig ay particular type of distributio (e.g, ormal). Sice these methods make fewer assumptios, they

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

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

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

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

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

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

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

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

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

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

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

Lesson 17 Pearson s Correlation Coefficient

Lesson 17 Pearson s Correlation Coefficient Outlie Measures of Relatioships Pearso s Correlatio Coefficiet (r) -types of data -scatter plots -measure of directio -measure of stregth Computatio -covariatio of X ad Y -uique variatio i X ad Y -measurig

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

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

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

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

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

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

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

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

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

Lesson 15 ANOVA (analysis of variance)

Lesson 15 ANOVA (analysis of variance) Outlie Variability -betwee group variability -withi group variability -total variability -F-ratio Computatio -sums of squares (betwee/withi/total -degrees of freedom (betwee/withi/total -mea square (betwee/withi

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

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

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

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

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

University of California, Los Angeles Department of Statistics. Distributions related to the normal distribution

University of California, Los Angeles Department of Statistics. Distributions related to the normal distribution Uiversity of Califoria, Los Ageles Departmet of Statistics Statistics 100B Istructor: Nicolas Christou Three importat distributios: Distributios related to the ormal distributio Chi-square (χ ) distributio.

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

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

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

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

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

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

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

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

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

Correlation. example 2

Correlation. example 2 Correlatio Iitially developed by Sir Fracis Galto (888) ad Karl Pearso (8) Sir Fracis Galto 8- correlatio is a much abused word/term correlatio is a term which implies that there is a associatio betwee

More information

0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5

0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5 Sectio 13 Kolmogorov-Smirov test. Suppose that we have a i.i.d. sample X 1,..., X with some ukow distributio P ad we would like to test the hypothesis that P is equal to a particular distributio P 0, i.e.

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

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

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

Probability & Statistics Chapter 9 Hypothesis Testing

Probability & Statistics Chapter 9 Hypothesis Testing I Itroductio to Probability & Statistics A statisticia s most importat job is to draw ifereces about populatios based o samples take from the populatio Methods for drawig ifereces about parameters: ) Make

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

Robust and Resistant Regression

Robust and Resistant Regression Chapter 13 Robust ad Resistat Regressio Whe the errors are ormal, least squares regressio is clearly best but whe the errors are oormal, other methods may be cosidered. A particular cocer is log-tailed

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

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

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

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

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

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

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

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

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

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

In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008

In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008 I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces

More information

Example Consider the following set of data, showing the number of times a sample of 5 students check their per day:

Example Consider the following set of data, showing the number of times a sample of 5 students check their  per day: Sectio 82: Measures of cetral tedecy Whe thikig about questios such as: how may calories do I eat per day? or how much time do I sped talkig per day?, we quickly realize that the aswer will vary from day

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

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

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

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

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

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

Discrete Random Variables and Probability Distributions. Random Variables. Chapter 3 3.1

Discrete Random Variables and Probability Distributions. Random Variables. Chapter 3 3.1 UCLA STAT A Applied Probability & Statistics for Egieers Istructor: Ivo Diov, Asst. Prof. I Statistics ad Neurology Teachig Assistat: Neda Farziia, UCLA Statistics Uiversity of Califoria, Los Ageles, Sprig

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

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

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

Recursion and Recurrences

Recursion and Recurrences Chapter 5 Recursio ad Recurreces 5.1 Growth Rates of Solutios to Recurreces Divide ad Coquer Algorithms Oe of the most basic ad powerful algorithmic techiques is divide ad coquer. Cosider, for example,

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

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

Covariance and correlation

Covariance and correlation Covariace ad correlatio The mea ad sd help us summarize a buch of umbers which are measuremets of just oe thig. A fudametal ad totally differet questio is how oe thig relates to aother. Stat 0: Quatitative

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

OMG! Excessive Texting Tied to Risky Teen Behaviors

OMG! Excessive Texting Tied to Risky Teen Behaviors BUSIESS WEEK: EXECUTIVE EALT ovember 09, 2010 OMG! Excessive Textig Tied to Risky Tee Behaviors Kids who sed more tha 120 a day more likely to try drugs, alcohol ad sex, researchers fid TUESDAY, ov. 9

More information

Sequences II. Chapter 3. 3.1 Convergent Sequences

Sequences II. Chapter 3. 3.1 Convergent Sequences Chapter 3 Sequeces II 3. Coverget Sequeces Plot a graph of the sequece a ) = 2, 3 2, 4 3, 5 + 4,...,,... To what limit do you thik this sequece teds? What ca you say about the sequece a )? For ǫ = 0.,

More information

Module 4: Mathematical Induction

Module 4: Mathematical Induction Module 4: Mathematical Iductio Theme 1: Priciple of Mathematical Iductio Mathematical iductio is used to prove statemets about atural umbers. As studets may remember, we ca write such a statemet as a predicate

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

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

8.1 Arithmetic Sequences

8.1 Arithmetic Sequences MCR3U Uit 8: Sequeces & Series Page 1 of 1 8.1 Arithmetic Sequeces Defiitio: A sequece is a comma separated list of ordered terms that follow a patter. Examples: 1, 2, 3, 4, 5 : a sequece of the first

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

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

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

Lecture 10: Hypothesis testing and confidence intervals

Lecture 10: Hypothesis testing and confidence intervals Eco 514: Probability ad Statistics Lecture 10: Hypothesis testig ad cofidece itervals Types of reasoig Deductive reasoig: Start with statemets that are assumed to be true ad use rules of logic to esure

More information

Chapter 9: Correlation and Regression: Solutions

Chapter 9: Correlation and Regression: Solutions Chapter 9: Correlatio ad Regressio: Solutios 9.1 Correlatio I this sectio, we aim to aswer the questio: Is there a relatioship betwee A ad B? Is there a relatioship betwee the umber of emploee traiig hours

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

Unit 8: Inference for Proportions. Chapters 8 & 9 in IPS

Unit 8: Inference for Proportions. Chapters 8 & 9 in IPS Uit 8: Iferece for Proortios Chaters 8 & 9 i IPS Lecture Outlie Iferece for a Proortio (oe samle) Iferece for Two Proortios (two samles) Cotigecy Tables ad the χ test Iferece for Proortios IPS, Chater

More information

Problem Set 1 Oligopoly, market shares and concentration indexes

Problem Set 1 Oligopoly, market shares and concentration indexes Advaced Idustrial Ecoomics Sprig 2016 Joha Steek 29 April 2016 Problem Set 1 Oligopoly, market shares ad cocetratio idexes 1 1 Price Competitio... 3 1.1 Courot Oligopoly with Homogeous Goods ad Differet

More information