Math 143 Sprig 2012 Test 2 Iformatio 1 Test 2 will be give i class o Thursday April 5. Material Covered The test is cummulative, but will emphasize the recet material (Chapters 6 8, 10 11, ad Sectios 12.1 3). Note that we omitted a couple sectios (8.6 ad 11.5) ad that there was sme additioal iformatio about computig meas ad variaces for liear trasformatios ad combiatios of radom variables. You are resposible for material covered i the text, i the problem sets, ad i class. Format Test questios will be desiged to try to see how well you uderstad the material, ot how well you ca perform various procedures midlessly. A variety of questio formats may be used. Some items may be be tested usig short aswers (a couple seteces to a paragraph), multiple choice, or true/false. Istructios Read through these prior to comig to the test ad follow them whe you take your test. 1. Always show your work ad explai your reasoig. Aswers without work or reasoig will ot receive full credit. Use mathematical otatio (especially the equals sig) correctly. Do t be afraid to use words i your explaatios. If you get a ureasoable aswer, be sure to say so. Give a brief explaatio about how you kow your aswer is wrog (for example, the mea I calculated is less tha 10, but I ca see from the data that it sould be at least 20). The go o to other problems ad come back ad try to fix the error if you have time at the ed of the test period. Eve if you caot do a problem completely, show me what you do kow. 2. Test restrictios. The test is closed book No otes are allowed. You may use RStudio (brig your ow laptop) or your calculator. There may be portios of the test where you are ot allowed to use techology. Do ot write i purple o the exam. (The exam will be graded i purple.)
Math 143 Sprig 2012 Test 2 Iformatio 2 Cotet Here is a list of thigs you should be sure you kow how to do. It is ot iteded to be a exhaustive list, but it is a importat list. You should be able to: Uderstad, use ad explai the statistical vocabulary/termiology. Here are some examples: populatio, sample, parameter, statistic, samplig distributio, stadard error, p-value, cofidece iterval, cofidece level, critical value, sigificace level, z-score, observatioal study, experimet, paired desig (paried t, for example), idepedet samples desig (2-sample t, for example), resistat, robust,... Work with radom variables. This icludes: Usig the basic rules of probability to determie probabilities of evets. Computig the mea ad stadard deviatio of a radom variable ad kowig what they tell you. Iterpretig area as probability i a graph of a distributio. Recogizig situatios that are described by biomial, ormal, ad t distributios. Beig able to use rules for meas ad variaces to determie the mea ad variace of a more complicated radom variable from meas ad variaces of simpler radom variables. Uderstad the issues ivolved i collectig good data ad the desig of studies, icludig the distictios betwee observatioal studies ad radomized experimets. matchig study desigs with appropriate aalysis methods. Work with ormal, t, ad biomial distributios. This icludes beig able to use the 68-95-99.7 Rule ad/or techology to fid percetages, z-scores, critical values, etc. Uderstad the basic framework for hypothesis testig ad how to iterpret p-values. Uderstad the basic framework for cofidece itervals ad how to iterpret the cofidece level. Perform ad iterpret all of the cofidece itervals ad hypothesis tests covered so far. (You should be able to do these usig RStudio ad by had.) Be aware of the assumptios that must be true to make use of various statistical procedures ad the degree to which the procedures are robust. Uderstad how to make ad iterpret graphical represetatios of data (histogram, desity plot, boxplot, bar graph, scatter plot, ormal-quatile plot) ad whe each might be appropriate or iappropriate to use. Note that the test will be a sample from the possible topics, it will ot be exhaustive.
Math 143 Sprig 2012 Test 2 Iformatio 3 Example Problems A umber of extra problems have bee assiged with each problem set. The followig problem is oe that I have used very frequetly o tests. 1. What do I do? I each of the followig situatios, preted you wat to kow some iformatio ad you are desigig a statistical study to fid out about it. Give the followig THREE pieces of iformatio for each: (i), what variables you would eed to have i your data set (ii) whether they are categorical or quatitative, ad (iii) what statistical procedure you would use to aalyze the results. Select your procedures from the followig list: 1-proportio (a.k.a. biomial test), Chi-squared goodess of fit, 1-sample t, Paired t, 2-sample t, oe of these. Record your aswers i the table. Part a) has bee doe as a example. (a) You wat to kow if boys or girls score better o readig tests i Ket Couty grade schools. Aswer Table: Variable(s) a) readig score o stadardized test [quat] geder (male or female) [cat] Procedure 2-sample t Notes: Ofte more tha oe desig is possible, so there may be multiple correct aswers. But you should ot choose a desig that is clearly iferior to aother desig we have already studied. You should ot choose oe of these if there is a reasoable desig that ca be aalyzed by a method we already kow about. Noe of these should mea that oe of the listed procedures will suffice.
Math 143 Sprig 2012 Test 2 Iformatio 4 Iferetial Statistics Summary Sheet (z ad t) Usually we will use the followig otatio (subscripts idicate multiple populatios/samples): parameters (populatio) p, proportio (of a categorical variable) µ, mea (of quatitative variable) σ, stadard deviatio statistics (sample), sample size x, cout (of a categorical variable) ˆp = x, proportio (of a categorical variable) p = x+2 +4, Plus-four proportio (of a categorical variable) x, mea (of quatitative variable) s, stadard deviatio samplig distributio SE, stadard error (stadard deviatio of the samplig distributio) µˆp, µ x, mea of samplig distributio (for ˆp ad x, respectively) The procedures ivolvig the z (ormal) ad t distributios are all very similar. To do a hypothesis test, compute t or z = data value hypothesis value SE ad compare with the appropriate distributio (usig tables or a computer)., To compute a cofidece iterval, determie the critical value for the desired level of cofidece (z or t ), the the cofidece iterval is data value ± (critical value)(se). Note: Each of these procedures is oly valid whe certai assumptios are met. I particular, remember The sample must be a simple radom sample (or somethig very close to it). The sample sizes must be large eough. Procedures ivolvig meas are geerally sesitive to outliers, because outliers ca have a large effect o the mea (ad stadard deviatio). Procedures ivolvig the t statistic geerally assume a populatio that is ormal or early ormal. These procedures are sesitive to to skewess (for small sample sizes) ad to outliers (for ay size). You ca t do good statistics from bad data. The margi of error i a cofidece iterval, for example, oly accouts for radom samplig variability, ot for errors i experimetal desig.
Summary of Formulas for Sigificace Tests ad Cofidece Itervals [z ad t distributios] Situatio parameters statistics (computed from data) distributios (usually approximate) SE 1-Proportio proportio p p σ = p(1 p) x = cout ˆp = x p = x+2 +4 = + 4 x ˆp z Biom(, p) Norm(p, σ ) Norm(p, SE) Norm(0, 1) p0(1 p0) [for hyp. test] ˆp(1 ˆp) p (1 p ) [Wald CI] [Plus-4 CI] 2-Proportio 2 Categorical Variables diff. of two proportios p1 p2 [If H0 : p1 = p2 the use pooled est. for p (= p1 = p2) ] p1, p2 (σ1, σ2) x1, x2 (success couts) ˆp1 = x1, ˆp 2 = x2, 1 2 s1 = ˆp1(1 ˆp1), s2 = ˆp2(1 ˆp2) [ˆp = x 1 + x2 1 + 2 = pooled perc.] ˆp1 ˆp2 z Norm(p1 p2, SE) Norm(0, 1) s 2 1 + s2 2 1 2 = [ ˆp1(1 ˆp1) 1 + ˆp2(1 ˆp2) 2 ˆp(1 ˆp)( 1 1 + 1 2 )] 1-Sample t 1 Quatitative Variable mea value: µ Paired t σ, µ x, s, df = 1 t T (df) s 2 Quatitative Variables mea value of differeces: µdiff σ diff, µ diff x diff, s diff, df = 1 t T (df) s diff Welch 2-Sample t 1 Quatitative Variable differece of two meas: µ1 µ2 σ1, σ2 x1, s1, x2, s2, df mi(1 1, 2 1) df 1 + 2 2 s 2 t T (df) 1 + s2 2 1 2 2-Sample t 1 Quatitative Variable differece of two meas: µ1 µ2 pooled est. of st dev σ = σ1 = σ2 x1, s1, x2, s2 (df1)s 2 1 + (df 2)s 2 2 sp = df1 + df2 df1 = 1 1 df2 = 2 1 df = df1 + df2 t T (df) sp 1 + 1 1 2