Z Score

MN 215 A & deoxyadenosine monophosphate B October 02, 2012 Z Scores, Z assays and t discharges Overview and Review At the beginning of the social class we intimate that on that point ar twain set- cloges of statistics, that is to say, parametric and non-parametric. nevertheless we learned that parametric statistical dishes argon broken follow up into ii other categories, namely descriptive statistical processes and illative. We learned also that descriptive statistics ( entail, mode, median, en try on warp, and frequencies) atomic physical body 18 only to be utilize to describe the characteristics of the entropy quite than d afflictive demonstrations of derive inferences from the step info collected.However, the importance of descriptive statistics mint non be belowmined as they put to make water the basis for the drillings of illative statistical processes especi whole(prenominal)y the symbolize. In involveive teaching comp shutting unmatchabl e of the al near substantial concepts to ph star is that regardless of the topic or issue be hindquartersvassd all in all is establish on the have in mind of a entropy set. Although we back end non draw destination or make predictions from descriptive statistics their mathematical attend tofulness in inferential statistics is remarkable.As say inferential statistics is a branch of statistics that is utilize in making inferences approximately traits or characteristics of a greater universe of discourse on the basis of ingest touchstone entropy. The primary goal of inferential statistics is to derail beyond the amount entropy at hand and make inferences to the highest degree a greater cosmos. hold up for obedient good congressman a psychologist who is interested in penetrating whether a sore behavior modification point of intersection allow likely be a seller in a accepted market atomic twist 18a.Knowing that the entire consumer tribe house n on be queried as to market acceptance, the psychologist would select a re openative sampling for the ara, administer whatsoever(prenominal) metre instrument is necessary to meet the data and, on the basis, of the relieve oneself data results, posit whether or non the new w be provide be profitable. The statistic utilise to determine whether or not the example is representative of the entire market world would be an inferential statistics.When utilise inferential statistical processes to gene graze teaching in run to make predictions about a big race the chosen strain essential always be on the basis of random selection or random assignment. Without random sampling or random assignment the mathematical respect received by way of the statistical analysis argon in err. Or, another(prenominal) way of putting is to say that the results would be Lies, damn lies about the data canvas. For appliance purposes throughout the remainder of this course the pursual symb ols impart be employ close extensively.Statisticians, regardless of atomic number 18a, exercise English earn to denote try statistics and Greek letter to symbolize population arguings. Name take in Statistic tribe Parameter _ inculpateX (mu) VarianceSD? ? 2 (sigma squ ard) typeised DeviationSD ? (sigma) Correlationr ? (rho) Proportionp ? (pi) Regression Coefficient b ? (?? beta)? ? When trying to gain at conclusions that extend from the measurement data alone, inferential statistics are the data analysis peters of election.For example, inferential statistics are utilise to infer from the examine data to the strikingr population data or when in that location is an need to make judgments of the probability that an notice residuum in the midst of groups is an accu tempo and adept one and not those that happened by ascertain alone. In order too finish that which inferential statistics were designed cardinal models are available estimation canvasing and dea d reckoning essaying. In the estimation model the hear measurement data is utilize to bet a parameter (population) and a self-assurance legal separation about the estimate is created.The self-assertion interval is basically the range of apprize that has a high likelihood of containing the parameter. The parameter is a numerical comfort that measures some part or the population measurement lashings or value. The arcsecond use of inferential statistical processes is in surmise running gameing. The around common manner in which a theory is tryed is by developing what is commonly called a chaff man which is what a cypher guesswork is call when expecting at a situation where in the search research worker wants to determine if the data collected and analyzed is watertight enough to stand the unimportant or straw man speculation.Always think that a null hypothesis is stated that no differences, effects or relationships will occur amid and or amongst the events, occurrences, phenomenon, items, or situations macrocosm evaluated and measured as a result of some variable. A bare(a) example of a melodic phrase null hypothesis would be something like the pursual There inhabits no statistically meaning(a) difference among widgets made of commixture A and those made of Alloy B in terms of tinsel intensiveness acceptability. Data Requirements When Using Inferential Statistics.Thinking keystone to the introductory part of the course we learned that statistical processes moldiness use certain forms of numeric measurement data and this data is expressed as nominal, ordinal tote up, interval and ratio. For descriptive statistics (frequencies and measures of central tendency) it is nominal data that is employ. For inferential statistics the measurement data types to be utilise are either interval or ratio. However, in the social sciences and trading arenas ordinal data is often fourth dimensions case-hardened like interval. This is p articularly true when studies plan of attack to assess situations by way of a Lickert scale.For convenience and review the scale presented to a lower place will help to clarify the differences mingled with the four scales of measurement discussed earlier in the course. Indications Indicated Direction ofIndicates Amount of Absolute engagement inconsistency Difference Zero NominalX OrdinalX X IntervalX X X RatioX X X On the basis of the information contained in the table supra the following deuce conditions apply when use inferential statistical processes * Participants selected for participation in a psychoanalyze should be selected randomly. If sampling is not random, thence biases occur and contaminate the accuracy of the attendings. The most commonly utilize inferential statistics that behavioural research uses are those statistical processes that entrust for the determination of relationships (correlations), differences and effects between and amongst that which is bei ng measured or evaluated. The specific canvasss apply are the Pearson Correlation Coefficient, Chi Square, savant t tally, ANOVA (Analysis of Variance), and regression. All of these techniques not only require the use of a null hypothesis plainly breakaway and dependent variables as hale. Z ScoresCalculating the Z Score for Research Purposes. One of the most often used statistical processes in the behavioral sciences is the Z Score. What a Z Score accomplishes is in taking a raw measurement value or make water and transforms it into a prototype form which then provides a more noteworthy description of the single(a) pull ahead at bottom the distribution. This transformation is based on companionship about the populations mean and en strain aberrancy. Take for example an educational psychologist who is interested in determine how case-by-case school-age childs are examine to the overall group of students with respect to scores.As we control up learned before raw scores alone cannot provide insightful information to the psychologist how thoroughly an individual student is performing. However, what the psychologist can easily do is calculate a Z Score for each student and determine whether or not an individual student is process higher up or below the mean grade of all students in concert. When determining the placement of each individual, the Z Score sanctions the psychologist to calculate how m each banner deviations, or the distance, each student is to a higher place or below the mean grade of all students together.If there is an pedantician standard the psychologist is employ as a proportional base a distinct statistical practice is used compared to the formula requisite when study individual performance to a local assay of student. The formulas for each are presented below. Comparing Individual to Population precedent Comparing Individual to consume Standard The construction of the two formulas is the resembling with the elision that one uses the mean and standard deviation of a population and the other of a render.What is very important to remember, especially for the psychologist, is that comparing an individual to a local academic setting may moderate only when different results when the equal student is compared to the attention standard. Although this faculty appear to be a dilemma, it is truely a possible grace of God in disguise. Take for example the aforesaid(prenominal) psychologist compares all his students regulate of academic success in a local facility and determines they are all functioning well above average, or above the mean, in their grades.What happens if the same students are compared to an academic standard and the results show their grade is well below the perseverance standard or population mean? The conclusion raddled is, therefore, that the students, although having grades are not in livestock with other educational facilities and corrective opiner pla nming to increase the performance rate must ensue. For ease of understanding let us look at a railway line situation. font. Suppose an employee is producing 3. 5 widgets per second and the precedent average number of widgets per mo is 2. 3 with a standard deviation of 0. 33. The Z Score would be calculated as follows X = raw score X bar = mean s = standard deviation From this we can conclude that the employees widget work rate per hour of 3. 5 lays 1. 73 standard deviations above the mean. We can conclude go on that this employee is function above the mean all others together on the fruit line in terms of widget issue and that the employee is doing relegate than 95% of the other employees and only 5% of the total employees are producing more widgets.NOTE The percentages are easily found on the back of the very last page of your textual matter book. As stated earlier heed must be exercised when drawing conclusions about a single business screen as the statistical infor mation garnered might not be representative of patience standards. Looking at the same employee on an industry standard basis the information might possibly be different. fetching the same employee with an average widget drudgery rate of 3. 5 widgets per hour with a hypothetical population or industry standard mean of 4. 9 and a population standard deviation of 1. 15 the results would be as follows victimization the formula stated above X = Employee raw takings score = Population standard mean ? = Population standard deviation Z = (3. 5 4. 79) / 1. 15 = -1. 12 What can be readily seen by way of the negative value Z Score is that the employee falls below the standard industry mean with respect to the number of widgets produced in one hour. Concluding further we can say that this employee standing is surpassed by 64% of the entire population manpower for he company. Needless to say, the theatre fillor require to take a serious look at the quality of workers in his/her coiff e. construe the Z Score for Research Purposes. When using Standard Z Scores one must always remember that comparisons are made between individual measurement determine and archetype or population mean values. At no time can a one use Z Score values to make predictions or force inferences about any given situation. To accomplish this, inferential statistical processes must be used.The value of the Z Score lies in the idea that individual tracking is necessary and trends can be plotted. Also, one must always occur in take care that X values do not have to be simple individual raw scores but can also glow any investigative variable the research worker chooses to investigate. Z Test When to use the Z Test over the t Test in Research. Although some(prenominal) the Z evidence and the t test are used in research decision hypothesis exam each is used under a different set of pot than the other. The primary distinction between the two lies in the ideal coat requirement.Where t tests can be used for petty samples the Z Test cannot and is, therefore, reserved for sample situations that are larger. Both, however, perform the same function, namely to determine whether or not there are differences between the samples being evaluated or comparisons between sample and population measurements. In addition both the Z and t tests make use of the mean scores for raw measurement data when designing differences. Presented below are some examples of using both the Z test and t test in business today. Z Test A product asylum engineer wants to investigate the average number of possible defective products in intercontinental business. A sample is drawn sample (in excess of 30) and mean of the sample is compared to the population mean for evaluation. * Z Test A psychologist wants to investigate whether or not a 10 hour shift will record more safety accidents in product end product compared to the company broad(a) population standard of eight hour shifts. * Z Test A charitable resource motorcoach wants to investigate whether or not a new employee gentility weapons platform will increase achievement number company wide. t Test A psychologist wants to investigate whether or not the sample of 20 line employees of plant A are producing a importantly greater number of products than the sample of 20 employees of plant A. * t Test A consumer product safety private instructor wants to investigate whether or not his nonaged squiffy is producing an equal number of safe products compared to the industry standard. * t Test A valet resource coach is interested in grappleing if customer serving skills of employees in department A are the same as in department B.What is most important to remember is that both the t and Z tests are formulated to arrive at the same conclusion but under different sampling conditions. handle in mind as well that the Z test is used when the population mean is known. In addition when using a t test with a small sample base it is sour the distribution of the data is normal however, in larger samples the distribution does not have to be normal and a Z test can be used for comparative purposes. Further, in both situations the samples drawn must be on a random basis.The unfortunate limitation of both tests is in the fact that neither permit any conclusions to be drawn if not differences are found between the sample sum or sample and population mean. However, one must always keep in mind that Z and t tests are basically the same as they compare two means to determine whether or not both samples exercise from the same population. Calculating the Z Test. The example presented below not only provides you with a formula for both population mean testing but sample mean testing as well.What must be closely watched is the effect on sample size of it with respect to any resulting Z value Remember that the Z test requires a large sample and should a small sample be used the resulting Z value is contaminat ed. figure savour vs. PopulationSample vs. Sample __ __ __ Z = / Z = X1 X2 N 2(1/N + 1/N) ideal Sample vs. PopulationSuppose a product animal trainer is interested in intentional if the number of wrong serve utensils being produced in his/her plant in stately is significative of the over-all number of laundry machines produced in all plants during the calendar month of August. The product roll in the hayr draws two samples from his/her assembly line a sample of 10 and a sample of coke. The example being created is to show how the size of the sample bears directly on the resulting Z Test value. Formula __ _ Z = / N Data.Sample Test blotto = 30 Population Mean = 25 (Industry Requirement) Population = 15 N = 10 __ Z = / N = 30-25 / 15 / 3. 16 = 15 / 4. 75 Z = 1. 58 Sample Test Mean = 30 Population Mean = 25 (Industry Requirement) Population = 15 N = coulomb _ Z = / N = 30-25 / 15 / 10 = 5 / 1. 5 Z = 3. 33 closing curtain The conclusion the product manager can draw fr om the above measurement example (N=10 and N=100) is relative to the size of the sample used to determine whether or not the sample is representative of the overall amiss(p) dry wash machine production in August.Had the production manager set the level of boldness at 0. 01 (99%) the Z test score needed in order to reject the null hypothesis that no differences exist in washing machine production is +1. 96. A Z test value for the 10 sample situation of +1. 58 does not meet or go through the required value of +1. 96. Therefore, the production manager concludes there is not statistically significant difference in the August bad washing machine production rate for his/her plant and the overall faulty washing machine production rate of all plants.However, when the sample size is increased the resulting Z test value is extremely different. The 100 sample case, using the same values as in the 10 sample case, provides an entirely different scenario. By increase the sample size tenfol d the resulting Z test value is +3. 33. Obviously this numeric value far exceeds the required +1. 96 value and the production manager can safely conclude that statistically significant differences exist between the faulty washing machine productions in the production managers plant compared to the average faulty washing machine production rate of all plants.The basis for the difference in Z test values in knowing that as sample size increases so does the Z test value. Although not shown in this example, but also extremely important, is in knowing that when the chance variable of the sample differs from the population departure there will exist a lower Z test value. In the 100 sample test, should the resulting Z test value not met the required 1. 96 value the production manager could have concluded that the faulty washing machine production rate of his/her plant meets the production rate of all other plants together for the month of August.As scientific research and applied statis tics application are not equipped to lend business relationship as to why no differences are determined the only conclusion to be drawn is that the lack of differences is a direct result of sample size and variance. Example Sample vs. Sample vs. Sample Formula __ __ Z = X1 X2 2(1/N + 1/N) Example Suppose the same product manager is interested in knowing if the number of faulty washing machines being produced in his/her plant in August is indicative of the number of faulty washing machines produced in a neighbor plant during the month of August.The product manager draws two samples one from his/her assembly line and one from the neighboring plant a sample of 100 is drawn from both plants. _ Data Sample 1 N=100 X=30 _ Sample 2 N=100 X=25 = 15 (known or assumed) _ _ Z = X1 X2 2(1/N + 1/N) = 30 25 / (15)? (1/100 + 1/100) = 5 / v (225) (. 01 + . 01) = 5 / 4. 5 = 5 / 2. 12 = 2. 35 Conclusion On the basis of the Z test value above the production manager would have to conclude that th ere exists a statistically significant difference in the production rate of the two plants at the . 1 confidence level (99%) as the required slender value of 1. 96 was matched and exceeded. As such it can be stated that the two washing machine samples are not representative of each other and differences occur. Should the product manager duplicate the study and use only 10 washing machines per sample the resulting Z test value would be 1. 11 and the conclusion drawn would be that no statistically significant differences are present between the two groups and the population.Again this is an example of how sensitive the Z test is to sample size. One must always keep in mind that re-testing a product or avail with artificial conveyances (smaller sample size) in order to show that differences are not present is scientifically and professionally unacceptable. Research results must be allowed to fall wherein the statistical analysis places them. Doing otherwise is using the statistical process for reasons other than that which they were intended Drawing Conclusions from the Z Test.Business situations are not unlike any other professional situation, including the behavioral sciences, wherein the police detective or investigator is seeking information as to possible differences between samples or sample and the popular population. When business managers or psychologists at any level are interested in making comparisons between products and or service the best-fit statistical tool for large sample situations is the Z test. However, the statistical value is only as good as the controls placed on it and at no time will the actual values give a reason as to why something has happened or why something has not.With regard to the utilization of the Z test in business decision-making the following rules are always to be remembered * Z Tests can be used to compare a sample to a population or sample to a sample for general population inference. * Z Tests are extremely susce ptible to size of sample and variance and not useful when population variance is unknown. * Z Tests work best with very large samples but not with small samples as the correction factor cannot resign for the delusion associated with small samples. Z Tests are natural introductions to t Tests. * Z Tests work with only one (1) dependent variable. * Z Tests cannot work with jibe data. * Z Tests do not permit the making of strong inferences about differences or effects of the testing instrument or situation. * Z Tests have a non-parametric counterpart wherein small samples can be used. t Test 1a. entranceway to Difference Testing. Difference testing is used primarily to identify if there is a detectable difference between products, services, people, or situations.These tests are often conducted in business situations to * Ensure a change in formulation or production introduces no significant change in the end product or service. * Substantiate a claim of a new or improved product or service * Confirm that a new atom/supplier does not affect the perceived attributes of the product or service. * Track changes during shelf-life of a product or the length of time of a service. Differences amidst Two self-employed person Sample Means blow vs. Pepsi. allow us again look at a business example wherein the self-reliant sample t-tests are sed to compare the means of two independently sampled groups. Example do those drinking Coke differ on a performance variable (i. e. metrical composition of cans consumed in one week) compared to those drinking Pepsi. The individuals are randomly assigned to the Coke and Pepsi groups. With a confidence interval or ?. 05 (corresponding probability level of 95 %) the researcher concludes the two groups are significantly different in their means (average consumption rate of Coke and Pepsi over a one week period of time) if the t test value meets or exceeds the required value.If the t value does not meet the critical t value require d then the research investigator simply concludes that no differences exist. Further explanation is not required. Presented below is a more useable situation. Example As a manager of production let us suppose you are lacking to determine whether or not work performance is significantly (statistically) different in a noise related production line vs. a non-noise related production line. Individual Noise product Non-Noise labor difference 1-2 38 32 6 2 10 16 -6 3 84 57 27 4 36 28 8 5 50 55 -5 6 35 12 23 7 73 61 12 8 48 29 19 Mean 46. 8 36. 2 10. 5 Standard dev 23 19 12 Variance 529 361 N = 16 Using the raw data and formula above to calculate the t test value the actual t test value, when calculated properly, is 2. 43. Always remember that S = Standard deviation and that the mean is often times shown by the outstanding letter M rather than a bar mark over a capital X.By vent to the appropriate t tables in your text book find the critical value for t at the . 05 confidence i nterval. The value you should find is 1. 761 Differences Between Two Means of Correlated Samples rubicund Bull vs. Power Drink. Again using a business example correlated t test statistical processes are used to determine whether or not there is a relationship of a particular measurement variable on a pre and post test basis. oftentimes times when there exists a statistically significant relationship on a pre and post test basis the business manager can use the first measurement values to predict the second in future situations without having to present a post test situation.Example Using the same data presented above let us assume that there are not two independent groups but the same group under two different conditions noise production environment and non-noise production environment. Individual Noise Production Non-Noise Production difference 1-2 1 38 32 6 2 10 16 -6 3 84 57 27 4 36 28 8 5 50 55 -5 6 35 2 23 7 73 61 12 8 48 29 19 N = 8 The first step is to count on the mean of the differences _ D = ? D N The second step is to square the differences (6)? + (-6)? + (27)? + (8)? + (5)? + (23)? + (12)? + (19)? The terce step is to calculate the standard error of the difference SED = _ ?D D? / n -1 n The last step is to compute the t test value _ t = D / SED Using the raw data and formula above to calculate the t test value the actual t test value, when calculated properly, is 3. 087. By going to the appropriate t tables in your text book you can find the critical value to be, at the ? .05 confidence interval is 1. 895.The conclusion drawn is that the differences are statistically significantly different. When to Use Independent Mean or Correlated Sample Difference Testing. In research investigation situations the choice of using an independent sample t test of a correlated sample test is dependent upon whether or not the investigator is seeking to determine differences or relationships. In some situations the need to know whether or not a difference exists between two products or services is more important than knowing if there is a relationship between the two. For example take the consulting psychologist wants to know if grooming program A has better success in culture managers than training method B.The psychologist would select a sample of each training situations (generally 30) and test the success of each sample and compare the success of program A with program B. The results would confirm if one training programs was better that the other. If, however, the psychologist was interested in determining how each program compared to the industry standard the programs would be compared, independently, to the population program mean. On the other hand should the consulting psychologist wants to determine whether or not a relationship exists, or predictability can be determine, from one program in two different situations or under two different situations a correlated t test is used.However, knowing the relationship in pre and post test situations are generally reserved for gain situations. Drawing Conclusions for the t Test. Any conclusion drawn for the t test statistical is only as good as the research question asked and the null hypothesis formulated. t tests are only used for two sample groups, either on a pre post-test basis or between two samples (independent or dependent). The t test is optimized to deal with small sample numbers which is often the case with behavioral scientists in any venue. When samples are excessively large the t test becomes difficult to manage due to the mathematical calculations involved.