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Introduction to Probability and Statistics 11th Week (5/24) Hypothesis Testing
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Hypothesis in statistics, is a claim or statement about a property of a populationHypothesis Testing is to test the claim or statementExample: A conjecture is made that “the average starting salary for computer science gradate is $30,000 per year”.
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Question:How can we justify/test this conjecture? A. What do we need to know to justifythis conjecture? B. Based on what we know, how shouldwe justify this conjecture?
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Answer to A:Randomly select, say 100, computerscience graduates and find out theirannual salaries---- We need to have some sampleobservations, i.e., a sample set!
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Answer to B:That is what we will learn in thischapter---- Make conclusions based on thesample observations
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Statistical Reasoning Analyze the sample set in an attempt to distinguish between results that can easily occur and results that are highly unlikely.
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Statistical DecisionsDecisions about populations on the basis of sample information.Ex) We may wish to decide on the basis of sample data whether a newserum is really effective in curing a disease, or whether one educationalprocedure is better than another
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Definitions Null Hypothesis (denoted H 0): is the statement being tested in a test of hypothesis. Alternative Hypothesis (H 1): is what is believe to be true if the null hypothesis is false.
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Null Hypothesis: H0 Must contain condition of equality =, ≥, or ≤ Test the Null Hypothesis directly Reject H 0 or fail to reject H 0
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Alternative Hypothesis: H1 Must be true if H0 is false ≠, <, > ‘opposite’ of Null Example: H0 : µ = 30 versus H1 : µ > 30
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Statistical Hypotheses and Null Hypotheses•Statistical hypotheses: Assumptions or guesses about the populations involved. (Such assumptions, which may or may not be true)•Null hypotheses (H0): Hypothesis that there is no difference between the procedures. We formulate it if we want to decide whether one procedure is better than another.•Alternative hypotheses (H1): Any hypothesis that differs from a given null hypothesisExample 1. For example, if the null hypothesis is p = 0.5, possible alternative hypotheses are p =0.7, or p ≠ 0.5.
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Concepts of Hypothesis Testing (1)…• The two hypotheses are called the null hypothesis and the other the alternative or research hypothesis. The usual notation is: pronounced H “nought”• H0: — the ‘null’ hypothesis• H1: — the ‘alternative’ or ‘research’ hypothesis• The null hypothesis (H0) will always state that the parameter equals the value specified in the alternative 11.14 hypothesis (H1)
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Stating Your Own HypothesisIf you wish to support your claim, the claim must be stated so that it becomes the alternative hypothesis.
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Important Notes:H0 must always contain equality; however some claims are not stated using equality. Therefore sometimes the claim and H0 will not be the same.Ideally all claims should be stated that they are Null Hypothesis so that the most serious error would be a Type I error.
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Tests of Hypotheses and Significance“Significant”: If on the supposition that a particular hypothesis is true we find that results observed in a random sample differ markedly from those expected under the hypothesis on the basis of pure chance using sampling theory, we would say that the observed differences are significant•We would be inclined to reject the hypothesis if the observed differences are significant.• Tests of hypotheses, tests of significance, or decision rules: Procedures that enable us to decide whether to accept or reject hypotheses or to determine whether observed samples differ significantly from expected results
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Type I ErrorThe mistake of rejecting the null hypothesis when it is true.The probability of doing this is called the significance level, denoted by α (alpha).Common choices for α: 0.05 and 0.01Example: rejecting a perfectly good parachute and refusing to jump
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Type II Errorthe mistake of failing to reject the null hypothesis when it is false.denoted by ß (beta)Example: failing to reject a defective parachute and jumping out of a plane with it.
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Table 7-2 Type I and Type II Errors True State of Nature The null The null hypothesis is hypothesis is true false We decide to Type I error Correct reject the (rejecting a true decision null hypothesis null hypothesis) Decision We fail to Type II error Correct reject the (failing to reject decision null hypothesis a false null hypothesis)
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Type I and Type II Errors•Type I error: If we reject a hypothesis when it happens to be true.•Type II error: If we accept a hypothesis when it should be rejected.•In order for any tests of hypotheses or decision rules to be good, they must be designed so as to minimize errors of decision.• An attempt to decrease one type of error is accompanied in general by an increase in the other type of error. The only way to reduce both types of error is to increase the sample size, which may or may not be possible.
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Significant Differences Hypothesis testing is designed to detect significant differences: differences that did not occur by random chance. In the “one sample” case: we compare a random sample (from a large group) to a population. We compare a sample statistic to a population parameter to see if there is a significant difference.
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Level of Significance ( 유의수준 )• Level of significance: In testing a given hypothesis, the maximum probability with which we would be willing to risk a Type I error is called the level of significance
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Level of Significance•In practice a level of significance of 0.05 or 0.01 is customary, although other values are used.• If for example a 0.05 or 5% level of significance is chosen in designing a test of a hypothesis, then there are about 5 chances in 100 that we would reject the hypothesis when it should be accepted, i.e., whenever the null hypotheses is true, we are about 95% confident that we would make the right decision. In such cases we say that the hypothesis has been rejected at a 0.05 level of significance, which means that we could be wrong with probability 0.05.
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DefinitionTest Statistic:is a sample statistic or value based on sample dataExample: x – µx z = σ/ n
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DefinitionCritical Region : is the set of all values of the test statistic that would cause a rejection of the null hypothesis
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Critical Region• Set of all values of the test statistic that would cause a rejection of the null hypothesis Critical Region
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Critical Region• Set of all values of the test statistic that would cause a rejection of the • null hypothesis Critical Region
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Critical Region• Set of all values of the test statistic that would cause a rejection of the null hypothesis Critical Regions
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DefinitionCritical Value: is the value (s) that separates the critical region from the values that would not lead to a rejection of H 0
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Critical ValueValue (s) that separates the critical region from the values that would not lead to a rejection of H 0 Critical Value ( z score )
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Critical ValueValue (s) that separates the critical region from the values that would not lead to a rejection of H 0 Reject H0 Fail to reject H0 Critical Value ( z score )
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Tests Involving the Normal Distribution- Level of confidence : 0.05•The critical region (or region of rejection of the hypothesis or the region of significance): The set of z scores outside the range -1.96 to 1.96 constitutes• The region of acceptance of the hypothesis (or the region of nonsignificance) : The set of z scores inside the range -1.96 to 1.96 could
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Tests Involving the Normal Distribution• Decision Rule • When the level of confidence is 0.01, a value 2.58 should be instead of 1.96.
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Two-tailed TestH0: µ = 200 α is divided equally between the two tails of the criticalH1: µ ≠ 200 region
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Two-tailed Test H0: µ = 200 α is divided equally between the two tails of the critical H1: µ ≠ 200 regionMeans less than or greater than
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Two-tailed Test H0: µ = 200 α is divided equally between the two tails of the critical H1: µ ≠ 200 regionMeans less than or greater than Reject H0 Fail to reject H0 Reject H0 200 Values that differ significantly from 200
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One-Tailed and Two-Tailed Tests•Two-tailed tests or two-sided tests: When we display interest in extreme values of the statistic S or its corresponding z score on both sides of the mean, i.e., in both tails of the distribution.• One-tailed tests or one-sided tests: When we are interested only in extreme values to one side of the mean, i.e., in one tail of the distribution, as, for example, when we are testing the hypothesis that one process is better than another (which is different from testing whether one process is better or worse than the other).
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P Value• The null hypothesis H0 will be an assertion that a populationparameter has a specific value, and the alternative hypothesis H1 will be one of the following assertions:(i) The parameter is greater than the stated value (right-tailed test).(ii) The parameter is less than the stated value (left-tailed test).(iii) The parameter is either greater than or less than the stated value (two- tailed test).• P value of the test: The probability that a value of S in the direction(s) of H1 and as extreme as the one that actually did occur would occur if H0 were true.
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P Value•Small P values provide evidence for rejecting the null hypothesis in favor of the alternative hypothesis, and large P values provide evidence for not rejecting the null hypothesis in favor of the alternative hypothesis.•The P value and the level of significance do not provide criteria for rejecting or not rejecting the null hypothesis by itself, but for rejecting or not rejecting the null hypothesis in favor of the alternative hypothesis.• When the test statistic S is the standard normal random variable, the table in Appendix C is sufficient to compute the P value, but when S is one of the t, F, or chi-square random variables, all of which have different distributions depending on their degrees of freedom, either computer software or more extensive tables will be needed to compute the P value.
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Special Tests of Significance for Large Samples: Means
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Special Tests of Significance for Large Samples: Means
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Special Tests of Significance for Large Samples: Means
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Special Tests of Significance for Large Samples: Means
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Our Problem: The education department at a university has been accused of “grade inflation” so education majors have much higher GPAs than students in general. GPAs of all education majors should be compared with the GPAs of all students. There are 1000s of education majors, far too many to interview. How can this be investigated without interviewing all education majors?
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What we know: The average GPA for all students is 2.70. µ = 2.70 This value is a parameter. To the right is the X = 3.00 statistical information for a random sample s= 0.70 of education majors: N= 117
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Questions to ask: Is there a difference between the parameter (2.70) and the statistic (3.00)? Could the observed difference have been caused by random chance? Is the difference real (significant)?
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Two Possibilities:1. The sample mean (3.00) is the same as the pop. mean (2.70). The difference is trivial and caused by random chance.1. The difference is real (significant). Education majors are different from all students.
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The Null and Alternative Hypotheses:1. Null Hypothesis (H0) The difference is caused by random chance. The H0 always states there is “no significant difference.” In this case, we mean that there is no significant difference between the population mean and the sample mean.1. Alternative hypothesis (H1) “The difference is real”. (H1) always contradicts the H0. One (and only one) of these explanations must be true. Which one?
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Test the Explanations We always test the Null Hypothesis. Assuming that the H0 is true: What is the probability of getting the sample mean (3.00) if the H0 is true and all education majors really have a mean of 2.70? In other words, the difference between the means is due to random chance. If the probability associated with this difference is less than 0.05, reject the null hypothesis.
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Test the Hypotheses Use the .05 value as a guideline to identify differences that would be rare or extremely unlikely if H0 is true. This “alpha” value delineates the “region of rejection.” Use the Z score formula for single samples and Appendix A to determine the probability of getting the observed difference. If the probability is less than .05, the calculated or “observed” Z score will be beyond ±1.96 (the “critical” Z score).
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Two-tailed Hypothesis Test: Z= -1.96 Z = +1.96 c cWhen α = .05, then .025 of the area is distributed on either side of the curve in area (C )The .95 in the middle section represents no significant difference between the population and the sample mean.The cut-off between the middle section and +/- .025 is represented by a Z-value of +/- 1.96.
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Testing Hypotheses:Using The Five Step Model…1. Make Assumptions and meet test requirements.2. State the null hypothesis.3. Select the sampling distribution and establish the critical region.4. Compute the test statistic.5. Make a decision and interpret results.
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Step 1: Make Assumptions and Meet Test Requirements Random sampling Hypothesis testing assumes samples were selected using random sampling. In this case, the sample of 117 cases was randomly selected from all education majors. Level of Measurement is Interval-Ratio GPA is I-R so the mean is an appropriate statistic. Sampling Distribution is normal in shape This is a “large” sample (N≥100).
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Step 2 State the Null Hypothesis H0: μ = 2.7 (in other words, H0: = μ) You can also state Ho: No difference between the sample mean and the population parameter (In other words, the sample mean of 3.0 really the same as the population mean of 2.7 – the difference is not real but is due to chance.) The sample of 117 comes from a population that has a GPA of 2.7. The difference between 2.7 and 3.0 is trivial and caused by random chance.
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Step 2 (cont.) State the Alternate Hypothesis H1: μ≠2.7 (or, H0: ≠ μ) Or H1: There is a difference between the sample mean and the population parameter The sample of 117 comes a population that does not have a GPA of 2.7. In reality, it comes from a different population. The difference between 2.7 and 3.0 reflects an actual difference between education majors and other students. Note that we are testing whether the population the sample comes from is from a different population or is the same as the general student population.
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Step 3 Select Sampling Distribution and Establish the Critical Region Sampling Distribution= Z Alpha (α) = .05 α is the indicator of “rare” events. Any difference with a probability less than α is rare and will cause us to reject the H 0.
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Step 3 (cont.) Select Sampling Distributionand Establish the Critical Region Critical Region begins at Z= ± 1.96 This is the critical Z score associated with α = .05, two-tailed test. If the obtained Z score falls in the Critical Region, or “the region of rejection,” then we would reject the H0.
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Step 4: Use Formula to Compute the TestStatistic (Z for large samples (≥ 100) Χ− µ Z= σ N
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When the Population σ is not known,use the following formula: Χ−µ Z= s N −1
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Test the Hypotheses 3.0 − 2.7 Z= = 4.62 .7 117 − 1 We can substitute the sample standard deviation S for σ (pop. s.d.) and correct for bias by substituting N-1 in the denominator. Substituting the values into the formula, we calculate a Z score of 4.62.
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Step 5 Make a Decision and Interpret Results The obtained Z score fell in the Critical Region, so we reject the H0. If the H0 were true, a sample outcome of 3.00 would be unlikely. Therefore, the H0 is false and must be rejected. Education majors have a GPA that is significantly different from the general student body (Z = 4.62, α = .05).* *Note: Always report significant statistics.
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Looking at the curve:(Area C = Critical Region when α=.05) Z= -1.96 Z = +1.96 c c z= +4.62 I
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Summary: The GPA of education majors is significantly different from the GPA of the general student body. In hypothesis testing, we try to identify statistically significant differences that did not occur by random chance. In this example, the difference between the parameter 2.70 and the statistic 3.00 was large and unlikely (p < .05) to have occurred by random chance.
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Summary (cont.) We rejected the H0 and concluded that the difference was significant. It is very likely that Education majors have GPAs higher than the general student body
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Special Tests of Significance for Large Samples: Proportions
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Special Tests of Significance for Large Samples: Proportions
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Special Tests of Significance for Large Samples: Difference of Means
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Special Tests of Significance for Large Samples: Difference of Means
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Special Tests of Significance for Large Samples: Difference of Means
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Special Tests of Significance for Large Samples: Difference of Means
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Special Tests of Significance for Large Samples: Difference of Means
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Special Tests of Significance for Large Samples: Difference of Means
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Special Tests of Significance for Large Samples: Difference of Proportions
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Special Tests of Significance for Large Samples: Difference of Proportions
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Special Tests of Significance for Large Samples: Difference of Proportions
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Special Tests of Significance for Small Samples: Means
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Special Tests of Significance for Small Samples: Means
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Special Tests of Significance for Small Samples: Means
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Special Tests of Significance for Small Samples: Means
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Using the Student’s t Distribution forSmall Samples (One Sample T-Test) When the sample size is small (approximately < 100) then the Student’s t distribution should be used (see Appendix B) The test statistic is known as “t”. The curve of the t distribution is flatter than that of the Z distribution but as the sample size increases, the t-curve starts to resemble the Z-curve (see text p. 230 for illustration)
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Degrees of Freedom The curve of the t distribution varies with sample size (the smaller the size, the flatter the curve) In using the t-table, we use “degrees of freedom” based on the sample size. For a one-sample test, df = N – 1. When looking at the table, find the t-value for the appropriate df = N-1. This will be the cutoff point for your critical region.
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Example A random sample of 26 sociology graduates scored 458 on the GRE advanced sociology test with a standard deviation of 20. Is this significantly different from the population average (µ = 440)?
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Solution (using five step model) Step 1: Make Assumptions and Meet Test Requirements: 1. Random sample 2. Level of measurement is interval-ratio 3. The sample is small (<100)
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Solution (cont.)Step 2: State the null and alternate hypotheses.H0: µ = 440 (or H0: = μ)H1: µ ≠ 440
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Solution (cont.) Step 3: Select Sampling Distribution and Establish the Critical Region1. Small sample, I-R level, so use t distribution.2. Alpha (α) = .053. Degrees of Freedom = N-1 = 26-1 = 254. Critical t = ±2.060
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Solution (cont.) Step 4: Use Formula to Compute the Test Statistic Χ−µ 458 − 440t= = = 4.5 S 20 N −1 26 − 1
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Looking at the curve for the t distributionAlpha (α) = .05 t= -2.060 t = +2.060 c c t= +4.50 I
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Step 5 Make a Decision and Interpret Results The obtained t score fell in the Critical Region, so we reject the H0 (t (obtained) > t (critical) If the H0 were true, a sample outcome of 458 would be unlikely. Therefore, the H0 is false and must be rejected. Sociology graduates have a GRE score that is significantly different from the general student body (t = 4.5, df = 25, α = .05).
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Testing Sample Proportions: When your variable is at the nominal (or ordinal) level the one sample z-test for proportions should be used. If the data are in % format, convert to a proportion first. The method is the same as the one sample Z-test for means (see above)
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Special Tests of Significance for Small Samples: Variance
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Special Tests of Significance for Small Samples: Variance
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Special Tests of Significance for Small Samples: Variance
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Special Tests of Significance for Small Samples: Difference of Means
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Special Tests of Significance for Small Samples: Difference of Means
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Special Tests of Significance for Small Samples: Ratios of Variances
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Special Tests of Significance for Small Samples: Ratios of Variances
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Special Tests of Significance for Small Samples: Ratios of Variances
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