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Statistics 11 Hypothesis Testing Discover the relationships that exist between events/things Accomplished by: Asking questions Getting answers In accord with certain rules... the scientific method. Question is a hypothesis Answer is obtained by testing the hypothesis Which gives the general model………

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Statistics 12.

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3 With some IMPORTANT restrictions about –How the hypothesis is formed. How the hypothesis is tested…….. Forming hypotheses is an "everyday-everybody" activity I will do better on examinations if relax the night before Is a "hypothesis"... a statement of a relationship OK, BUT NOT a scientific hypothesis

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Statistics 14 A scientific hypothesis must meet certain criterion A scientific hypothesis must be: Specific Empirically testable Strictly related to some experimental procedure

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Statistics 15 Moreover, a scientific hypothesis actually consists of two separate mutually exclusive hypotheses A null hypothesis An alternative hypothesis

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Statistics 16 Null Hypothesis A statement reflecting the possibility that there are no differences between the objects and/or events that are being observed In formal terms: Ho: µ 1 = µ 2 Where: µ 1 and µ 2 are the mean or average of several observations

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Statistics 17 Alternative Hypothesis A statement reflecting the possibility that there are differences between the objects and/or events that are being observed In formal terms: H 1 : µ 1 <> µ 2 or H 1 : µ 1 µ 2

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Statistics 18 Testing between the null and alternative hypothesis Accomplished through collection of data Data must be scientifically acceptable, i.e. –Observable –Public –Replicable The test concentrates on the null hypothesis which you either –Reject –Fail to reject

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Statistics 19 If there are no differences between your observations you –Fail to reject the null hypothesis and –Disregard the alternative hypothesis If there are differences between your observations you –Reject the null hypothesis and Accept the alternative hypothesis

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Statistics 110 Some things to note about hypothesis testing Failing to reject the null hypothesis –Does not mean that the null hypothesis is TRUE –The null hypothesis can never be proven –You can only fail to reject it Rejecting the null hypothesis –Means you accept the alternative hypothesis –It does not establish the validity of a relationship Validity is a function of experimental design

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Statistics 111 When testing a hypothesis Two possible outcomes re: null hypothesis Two possible states of real world Thus four possible decisions –Two are incorrect... i.e. errors

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Statistics 112 The Real World Ho: trueHo: false Your Decision Reject Ho Type I error alpha (p level) Correct Power 1 - beta Do not reject Ho Correct 1 - alpha Type II error beta

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Statistics 113 Some things to file for future reference Type I error –You can directly "set" this –It is the chance (probability of the making the error) you are willing to accept when you test your hypothesis. Type II error –You cannot directly "set" this You can attempt to control it through good experimental design.

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Statistics 114 Alpha Level or the level of significance is a probability value that is used to define the very unlikely sample outcomes if the null hypothesis is true. Critical region is composed of extreme sample values that are very unlikely to be obtained if the null hypothesis is true. The boundaries for the critical region are determined by the alpha level. If sample data fall in the critical region, the null hypothesis is rejected.

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Statistics 115 Estimating Population Parameters from Samples Sample mean Unlikely to be exactly equal to population mean BUT Not more likely to be greater Not more likely to be less So sample mean is an unbiased estimate of population mean

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Statistics 116 Sample standard deviation Unlikely to be exactly equal to population standard deviation BUT More likely to be less Is usually an under estimate of population parameter So sample standard deviation is a biased estimate of the population standard deviation

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Statistics 117 To understand why this is so you must understand the nature and concept of a sampling distribution What it all means When you take a sample

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Statistics 118 Sample mean is unbiased estimate of population mean –But no reason to suspect it is higher or lower than population mean Sample standard deviation is a biased estimate of population standard deviation –But it is more likely to be smaller than population variance and standard deviation

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Statistics 119 And so must correct any estimate of the population variance increase it (i.e. use "n-1" when calculating the estimate)

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Statistics 120

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Statistics 121 Parametric Tests- Tests that do make assumptions and test hypotheses about population parameters. z & t ANOVA F test Involves an assessment of whether your observed data is related to your independent variable

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Statistics 122 Or is simply what might be expected by chance random sampling –i.e. no relation between Independent variable Dependent measure

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Statistics 123 Requires knowing or estimating population parameters Mean: (μ) Standard deviation: (σ) Assumption of normality For example: consider Pat (individual score) = 64 Population mean (μ) = 50 Population standard deviation (σ) = 8

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Statistics 124 And if population is normal, then you know ~68 (68.26) % data points between + 1 ~95 (95.44) % data points between + 2 ~99 (99.74) % data points between + 3 And remember: these are percentages, not absolute values Areas under the normal curve

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Statistics 125 Remember the z-distribution ? Provided areas (proportion of scores) under a normal distribution according to And if convert Pat's raw score to a z-score

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Statistics 126 And then look up Pat's z-score in the Z-table Meaning that ~96% (1.00 -.0401 =.9599) of scores in distribution are below Pat. (page 699, G&W). OR PUT OTHERWISE –If we were to randomly select a score from Pat's distribution The probability that the score would be greater than Pat's would only be 4 in 100

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Statistics 127 And we have done what we set out to do Accomplished a statistical test … i.e. comparing …. –Observed data and What would be expected by chance. And thus, Pat's score is significant at p <.05

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Statistics 128 P-value The P-value is the probability, when Ho is true, of a test statistic value at least as contradictory to Ho as the value actually observed. The smaller the P-value, the more strongly the data contradict Ho. The P-value is denoted by P. The P-value summarizes the evidence in the data about the null hypothesis. A moderate to large P-value means that the data are consistent with Ho.

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Statistics 129 Eg. P-value.26 or.83 indicates that the observed data would not be unusual if Ho were true. However, a P-value such as.001 means that such data would be very unlikely, if Ho were true. The P-value is the primary reported result of a significance test. If the P-value is sufficiently small, one rejects Ho and accepts H 1.

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Statistics 130 Standard Error of the Mean When comparing an individual to a population needed to know two things about the population Mean: Standard deviation: Only slightly different when comparing a sample to a population

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Statistics 131 Since you are not concerned with a single individual but with a sample of individuals The "population" of interest is not –A population of individuals but rather –A population of samples, i.e. a SAMPLING DISTRIBUTION

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Statistics 132 And the measure of "variability" is not –The standard deviation of a population but rather –The standard deviation of a sampling distribution, i.e. the STANDARD ERROR OF THE MEAN

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Statistics 133 The calculation details The standard error of the mean σ m = σ ∕ √n where: n = sample size The z comparison Z = (M – μ) ∕ σ m Which is not really different than what we did when comparing an individual to a population

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Statistics 134 Suppose:Herd of 10 cows (n=10) Mean milk production is 1.8 gal/cow Question:How unique is this herd? μ = 1.5,σ =.55, n=10, M=1.8

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Statistics 135 Thus herd is pretty unique since the likelihood that any random sample of 10 cows from the population would produce more milk is less than 5 times in 100 Or in "statistics" –The probability of selecting a herd of greater milk producers is p < 0.05

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Statistics 136

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Statistics 137 Problem: compare a sample to a population Method: 1. Use population parameters to calculate the standard error of the mean of a sampling distribution. σ m = σ ∕ √n

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Statistics 138 2. Use the standard error of the mean to compare sample mean with population mean by calculating a z-score Z = (M – μ) ∕ σ m

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Statistics 139 3. Use z-table to determine the probability that a random sample would yield a mean greater than the mean of the sample

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Statistics 140 A word on the logic and requirements of the statistic The "uniqueness" of your sample is the probability that another random sample of the same size would have the same mean as your sample. Or put otherwise, is your sample mean, is what would be expected by chance, a random selection? The more unique your sample, the more likely it reflects a relationship between:

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Statistics 141 Your independent variable Your dependent measure

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Statistics 142 Two requirements The population is normally distributed You know the population Mean Standard deviation

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Statistics 143 The t statistic -An alternative to z MUST know the population mean But can estimate population standard deviation from sample data A sample standard deviation is given by (as you know)

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Statistics 144 And so an estimated standard error of the mean is: s m = s ∕ √n And to use the estimated standard error of the mean to compare your sample to the population must make one adjustment –Adjustment is necessary to account for the fact that you are estimating

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Statistics 145 Comparisons When Estimating Population Parameters The adjustment part Estimating the standard deviation requires a different sampling distribution Sampling distribution is the t-distribution

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Statistics 146 The t-distribution Normal distribution More platokurtic than z-distribution Tails more elevated

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Statistics 147 And comparison becomes t = (M – μ) ∕ s m Thus: Because use estimate of population standard deviation to estimate standard error of mean Must use t-distribution to get probability of randomly selecting a sample with a mean similar to the mean of your sample But not conceptually different -- just an adjustment

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Statistics 148 For Example… Suppose:Mean milk production of your herd of 10 cows is: 1.8, 1.7, 2.4, 2.3, 1.1, 1.7, 1.5, 2.4,1.9, 1.2 gals (Mean = 1.8 gal/cow) Question:How unique is your herd? μ = 1.5 gal/day *given σ = unknown σ m =unknown because do not know population standard deviation

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Statistics 149 QUESTION now is what does that "t-value" mean i.e. What is the probability of a random sample of 10 cows being like your cows To find out, consult a t-table

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Statistics 150

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Statistics 151 The t-Distribution Z-table gives exact probabilities t-table gives ranges of probabilities

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Statistics 152 Enter table with degrees of freedom of your sample –Degrees of freedom Number of values in a calculation that are free to vary –That is: –The degrees of freedom for a mean of 10 values is 9... because –If the mean of 10 numbers is, for example, 5 –Nine of the numbers "free" to be any value but when these are established, the 10th number is determined if the mean is to be 5

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Statistics 153 Determine the probability of your t-observed by the tabled t-values that it falls between, For example: With 10 data points there are 9 degrees of freedom (df=9) If the t-observed statistic is 2.04

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Statistics 154 The t-table gives a probability of that occurring by chance between 0.05 and 0.02 two-tailed (between 5 and 2 times in 100) And for your cows The observed t-value of 2.04, df=9 gives a tabled probability of p > 0.05

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Statistics 155 Which is traditionally not sufficient to reject the Null hypothesis –Any event that has probability of occurring 5 times or more in 100 is considered by most psychologists an indication of a chance event And thus your cows are "just average old cows" Well maybe NOT……….

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Statistics 156 Directionality of Statistical Tests Statistical tests have a property called "directionality" Nondirectional, called "two-tailed" tests Directional, called "one-tailed" tests

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Statistics 157 Directionality is determined BEFORE you run your experiment Based upon prior knowledge or data You predict of the outcome of your observations, the affect of your independent variable –Your independent variable "will improve performance" –Your independent variable "will interfere with learning" –etc

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Statistics 158 Your ability to predict an outcome means that you are better able to determine whether an event is a chance occurrence More likely to reject Null hypothesis In statistical terms the region of the sampling distribution indicating that an event is something different than what would be expected by chance is larger

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Statistics 159 And cows again If you had valid reasons to predict that your cows produced more milk You would use a directional test, i.e. one-tailed test And you would reject the Null hypothesis at p <.05 that your herd was not different than what you would expect from another random sample of cows

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Statistics 160

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Statistics 161 Z-test a statistical test used to decide if a sample mean does or does not come from a specified population, when the standard deviation of the population is known. When the standard deviation of the population is unknown then a t-test is performed. Hypothesis testing, the goal is to decide whether to reject the null hypothesis.

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Statistics 162 Alpha level, traditionally set at.05 where, also, the acceptance and rejection regions are determined. Critical value, the absolute value of that defines the rejection region(s). Non-directional (two-tailed), where rejection of the sample mean is either above or below hypothesized population mean. Directional (one-tailed), where rejection of the sample mean is determined prior to experimentation.

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Statistics 163 Compared a Sample to a population: When population parameters are known….. Sample to population: assume a normal population and known standard deviation Z = (M – μ) ∕ σ m

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Statistics 164 When population parameters are unknown…. Sample to population: assume a normal population and unknown standard deviation t = (M – μ) ∕ s m

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