CFA 2018 SS 03 reading 12 hypothesis testing

Hypothesis Testing INTRODUCTION Statistical inference refers to a process of making judgments regarding a population on the basis of information obtained from a sample Two branches of Statistical inference include: 1) Hypothesis testing: It involves making statement(s) regarding unknown population parameter values based on sample data In a hypothesis testing, we have a hypothesis about a parameter's value and seek to test that hypothesis e.g we test the hypothesis “the population mean = 0” • Hypothesis: Hypothesis is a statement about one or more populations 2) Estimation: In estimation, we estimate the value of unknown population parameter using information obtained from a sample HYPOTHESIS TESTING Steps in Hypothesis Testing: Stating the hypotheses: It involves formulating the null hypothesis (H0) and the alternative hypothesis (Ha) Determining the appropriate test statistic and its probability distribution: It involves defining the test statistic and identifying its probability distribution Specifying the significance level: The significance level should be specified before calculating the test statistic Stating the decision rule: It involves identifying the rejection/critical region of the test statistic and the rejection points (critical values) for the test • Critical Region is the set of all values of the test statistic that may lead to a rejection of the null hypothesis • Critical value of the test statistic is the value for which the null is rejected in favor of the alternative hypothesis • Acceptance region is the set of values of the test statistic for which the null hypothesis is not rejected Collecting the data and calculating the test statistic: The data collected should be free from measurement errors, selection bias and time period bias Making the statistical decision: It involves comparing the calculated test statistic to a specified possible value or values and testing whether the calculated value of the test statistic falls within the acceptance region Making the economic or investment decision: The hypothesized values should be both statistically significant and economically meaningful Null Hypothesis: The null hypothesis (H0) is the claim that is initially assumed to be true and is to be tested e.g it is hypothesized that the population mean risk premium for Canadian equities ≤ • The null hypothesis will always contain equality Alternative Hypothesis: The alternative hypothesis (Ha) is the claim that is contrary to H0 It is accepted when the null hypothesis is rejected e.g the alternative hypothesis is that the population mean risk premium for Canadian equities > • The alternative hypothesis will always contain an inequality Formulations of Hypotheses: The null and alternative hypotheses can be formulated in three different ways: H0: θ = θ0 versus Ha: θ ≠ θ0 • It is a two-sided or two-tailed hypothesis test • In this case, the H0 is rejected in favor of Ha if the population parameter is either < or > θ0 –––––––––––––––––––––––––––––––––––––– Copyright © FinQuiz.com All rights reserved –––––––––––––––––––––––––––––––––––––– FinQuiz Notes – Reading 12 Reading 12 Hypothesis Testing FinQuiz.com ܵ௑ത = ܵ √݊ Thus, Test statistic = ‫ି܋ܑܜܛܑܜ܉ܜ܁ ܍ܔܘܕ܉܁‬۶‫ܚ܍ܜ܍ܕ܉ܚ܉۾ ܖܗܑܜ܉ܔܝܘܗ۾ ܍ܐܜ ܎ܗ ܍ܝܔ܉܄ ܌܍ܢܑܛ܍ܐܜܗܘܡ‬ ‫ܛ‬ √‫ܖ‬ H0: θ ≤ θ0 versus Ha: θ>θ0 • It is a one-sided right tailed hypothesis test • In this case, the H0 is rejected in favor of Ha if the population parameter is > θ0 H0: θ ≥ θ0 versus Ha: θ < θ0 • It is a one-sided left tailed hypothesis test • In this case, the H0 is rejected in favor of Ha if the population parameter is < θ0 When a null hypothesis is tested, it may result in four possible outcomes i.e A false null hypothesis is rejected → this is a correct decision and is referred to as the power of the test Power of a test = – Probability of a Type-II error When more than one test statistic is available to conduct a hypothesis test, then the most powerful test statistic should be selected A true null hypothesis is rejected → this is an incorrect decision and is referred to as a Type-I error A false null hypothesis is not rejected → this is an incorrect decision and is referred to as a Type-II error A true null hypothesis is not rejected → this is a correct decision Type I and Type II Errors in Hypothesis Testing True Situation Decision H0 True H0 False Do not reject H0 Correct Decision Type II Error Reject H0 (Accept Ha) Type I Error Correct Decision Source: Table 1, CFA® Program Curriculum, Volume 1, Reading 12 where, θ = Value of population parameter θ0 = Hypothesized value of population parameter NOTE: Ha: θ > θ0 and Ha: θ < θ0 more strongly reflect the beliefs of the researcher Test Statistic: A test statistic is a quantity that is calculated using the information obtained from a sample and is used to decide whether or not to reject the null hypothesis Test statistic = ‫ି܋ܑܜܛܑܜ܉ܜܛ ܍ܔܘܕ܉܁‬۶‫ܚ܍ܜ܍ܕ܉ܚ܉ܘ ܖܗܑܜ܉ܔܝܘܗܘ ܍ܐܜ ܎ܗ ܍ܝܔ܉܄ ܌܍ܢܑܛ܍ܐܜܗܘܡ‬ ‫∗܋ܑܜܛܑܜ܉ܜܛ ܍ܔܘܕ܉ܛ ܍ܐܜ ܎ܗ ܚܗܚܚ܍ ܌ܚ܉܌ܖ܉ܜ܁‬ • The smaller the standard error of the sample statistic, the larger the value of the test statistic and the greater the probability of rejecting the null hypothesis (all else equal) • As the sample size (n) increases, the standard error decreases (all else equal) *When the population S.D is unknown, the standard error of the sample statistic is given by: • Type-I and Type-II errors are mutually exclusive errors • The probability of a Type-I error is referred to as a level of significance and is denoted by alpha, α o The lower the level of significance at which the null hypothesis is rejected, the stronger the evidence that the null hypothesis is false • The probability of a Type-II error is denoted by beta, β The probability of type-II error is difficult to quantify • All else equal, the smaller the significance level, the smaller the probability of making a type-I error and the greater the probability of making a type-II error • Type I and II errors probabilities can be simultaneously reduced by increasing the sample size (n) • Type-I error is more serious than Type-II error Rejection Points Approach to Hypothesis Testing: Critical region for two-tailed test at 5% level of significance (i.e α = 0.05): • Null hypothesis: H0: θ = θ0 • Alternative hypothesis: Ha: θ ≠ θ0 Reading 12 Hypothesis Testing The two critical/rejection points are Z 0.025 = 1.96 and – Z0.025 = –1.96 FinQuiz.com Confidence Interval Approach to Hypothesis Testing: The 95% confidence interval for the population mean is stated as: X ± 1.96s x • It implies that there is 95% probability that the interval X ± 1.96s x contains the population mean's value Lower limit Acceptance region • The Null hypothesis is rejected when Z < -1.96 or Z > 1.96; otherwise, it is not rejected Critical region for one-tailed test at 5% level of significance (i.e α = 0.05): • Null hypothesis: H0: θ ≤ θ0 • Alternative hypothesis: Ha: θ>θ0 The critical/rejection point is Z0.05 = 1.645 µ < X −1.96sx µ > X + 1.96s x Upper limit • When the hypothesized population mean (µ0) < the lower limit, H0 is rejected When the hypothesized population mean (à0) > the upper limit, H0 is rejected • When the hypothesized population mean (µ0) lies between the lower and upper limit, H0 is not rejected P-value Approach to hypothesis testing: The p-value is also known as the marginal significance level The pvalue is the smallest level of significance at which the null hypothesis can be rejected • The smaller the p-value, the greater the probability of rejecting the null hypothesis • The p-value approach is considered more efficient relative to rejection points approach Decision Rule: • When p-value < α • When p-value ≥ α • The Null hypothesis is rejected when Z > 1.645; otherwise, it is not rejected Critical region for one-tailed test at 5% level of significance (i.e α = 0.05): • Null hypothesis: H0: θ ≥ θ0 • Alternative hypothesis: Ha: θ 1.96 or if z < -1.96 H0: θ ≤ θ0 versus Ha: θ>θ0 The rejection points are z0.05 = 1.645 Decision Rule: Reject the null hypothesis if z > 1.645 H0: θ ≥ θ0 versus Ha: θ 2.575 or if z < -2.575 H0: θ ≤ θ0 versus Ha: θ > θ0 The rejection points are z0.01 = 2.33 Decision Rule: Reject the null hypothesis if z > 2.33 H0: θ ≥ θ0 versus Ha: θ < θ0 The rejection points are z0.01 = -2.33 Decision Rule: Reject the null hypothesis if z < -2.33 The test statistic is calculated as follows: t n −1 = X −µ s n where, tn–1 = t-statistic with n – degrees of freedom (n is the sample size) ܺത = sample mean ߤ଴ = hypothesized value of the population mean s = sample standard deviation NOTE: As the sample size increases, the difference between the rejection points for the t-test and z-test decreases Test Concerning the Population Mean (Population Variance Unknown) Large Sample (n ≥ 30 Small Sample (n +1.96 or 1.645 or if z < –1.645 H0: θ ≤ θ0 versus Ha: θ > θ0 The rejection points are z0.10 = 1.28 Decision Rule: Reject the null hypothesis if z > 1.28 H0: θ ≥ θ0 versus Ha: θ 1.96 nor < 1.96, we not reject H0 at 5% level of significance Example: Suppose, n H0: µ Ha: µ α = 5% = 25 ≤ 368 > 368 = 0.05 Reading 12 X σ Hypothesis Testing FinQuiz.com • Since hypothesized value of mean return i.e 2.10% falls within this confidence interval, H0 is not rejected = 372.5 = 15 Since, it is a one-tailed test, critical value is 1645 Decision Rule: Reject H0 when calculated value of Z >1.645 Z= ଷ଻ଶ.ହିଷ଺଼ భఱ √మఱ Practice: Example & 3, Volume 1, Reading 12 = 1.50 3.2 Tests Concerning Differences between Means H0: µ1 – µ2 = versus Ha: µ1 – µ2 ≠ or µ1 ≠ µ2 H0: µ1 – µ2 ≤ versus Ha: µ1 – µ2> or µ1>µ2 H0: µ1 – µ2 ≥ versus Ha: µ1 – µ2< orµ1 1.645, we not reject H0 at 5% level of significance Example: Suppose, an equity fund has been in existence for 25 months It has achieved a mean monthly return of 2.50% with sample S.D of 3.00% It was expected to earn a 2.10% mean monthly return during that time period H0: Underlying mean return on equity fund (µ) = 2.10% Ha: Underlying mean return on equity fund (à) 2.10% Level of significance = α = 10% • Since, population variance is not known, a t-test is used with degree of freedom = n -1 = 25 – = 24 • The rejection points or critical values = t α/2, n-1 = t 0.10/2, t-value from t-table are 1.711 and 25-1 = t 0.05, 24 1.711 Decision Rule: Reject the null hypothesis when t > 1.711 or t < –1.711 t-statistic = ଶ.ହ଴ିଶ.ଵ଴ య.బబ √మఱ = 0.667 • Since, calculated t-value is neither > 1.711 nor < 1.711, we not reject the null hypothesis at 10% significance level Using Confidence interval approach: ܺത − ‫ݐ‬ഀ ‫ݏ‬௑ത ‫ܺ ݋ݐ‬ത + ‫ݐ‬ఈ/ଶ ‫ݏ‬௑ത మ where, tα/2→α/2 of the probability remains in the right tail t -α/2→ -α/2 of the probability remains in the left tail 90% confidence interval is: 2.5 – (1.711) (0.60) = 1.473 AND 2.5 + (1.711) (0.60) = 3.5266 [1.473, 3.5266] Test Statistic for a Test of the Difference between Two Population Means (Normally Distributed Populations, Population Variances Unknown but Assumed Equal) based on Independent samples: A t-test based on independent random samples is given by: ‫=ݐ‬ ሺܺതଵ − ܺതଶ ሻ − (ߤଵ − ߤଶ ) ቀ ௌ೛మ ௡భ + ௌ೛మ ௡మ ଵ/ଶ ቁ where, Sp2= Pooled estimator of the Common variance ܵ௣ଶ = ሺ݊ଵ − 1ሻܵଵଶ + ሺ݊ଶ − 1ሻܵଶଶ ݊ଵ + ݊ଶ − The number of degrees of freedom is n1 + n2– Test Statistic for a Test of the Difference between Two Population Means (Normally Distributed Populations, Unequal and Unknown Population Variances) based on independent samples: In this case, an approximate t-test based on independent random samples is given by: ‫=ݐ‬ ሺܺതଵ − ܺതଶ ሻ − (ߤଵ − ߤଶ ) ቀ ௌభమ ௡భ + ௌమమ ௡మ ଵ/ଶ ቁ In this case, modified degrees of freedom is used It is calculated as follows: ݂݀ = ቀ ௌభమ ௡భ మ ௌమ ൬ భ ൗ௡భ ൰ ௡భ Practice: Example & 5, Volume 1, Reading 12 + ௌమమ ௡మ + ቁ ଶ మ ௌమ ൬ మ ൗ௡మ ൰ ௡మ Reading 12 3.3 Hypothesis Testing Tests Concerning Mean Differences When samples are dependent, the test concerning mean differences is referred to as paired comparisons test and is conducted as follows H0: µd= µd0 versus Ha: µd≠ µd0 H0: µd≤ µd0 versus Ha: µd>µd0 H0: µd≥ µd0 versus Ha: µd 1.714 or if t < –1.714 Calculated test statistic = t = ି૙.૟૙ ି ૙ ૚.૜૛૟ૡ૙ૠ = –0.452213 • Since, calculated t statistic is not < -1.714, we fail to reject the null hypothesis at 10% significance level Thus, we conclude that the difference in mean quarterly returns is not statistically significant at 10% significance level Practice: Example 6, Volume 1, Reading 12 ෍ ݀௜ ݊ ௜ୀଵ ଶ ∑௡௜ୀଵ൫݀௜ − ݀̅൯ = ݊−1 s2d n = number of pairs of observations ࡿ࢚ࢇ࢔ࢊࢇ࢘ࢊ ࢋ࢘࢘࢕࢘ ࢕ࢌ ࢚ࢎࢋ ࢙ࢇ࢓࢖࢒ࢋ ࢓ࢋࢇ࢔ ࢊ࢏ࢌࢌࢋ࢘ࢋ࢔ࢉࢋ = ‫̅݀ݏ‬ ܵௗ = √݊ Example: • H0: The mean quarterly return on Portfolio A = Mean quarterly return on Portfolio B from 2000 to 2005 • Ha: The mean quarterly return on Portfolio A ≠ Mean quarterly return on Portfolio B from 2000 to 2005 The two portfolios share the same set of risk factors; thus, their returns are dependent (not independent) Hence, a paired comparisons test should be used The following test is conducted: H0: µd = versus Ha: µd ≠ at a 10% significance level where, µd = population mean value of difference between the returns on the two portfolios 2000 to 2005 4.1 Tests Concerning a Single Variance We can formulate hypotheses as follows: H0: σ2 = σ20 versus Ha: σ2 ≠ σ20 H0: σ2 ≤ σ20 versus Ha: σ2 > σ20 H0: σ2 ≥ σ20 versus Ha: σ2 < σ20 where, σ20 = hypothesized value of σ20 Test Statistic for Tests Concerning the Value of a Population Variance (Normal Population): If we have n independent observations from a normally distributed population, the appropriate test statistic is chi-square test statistic, denoted χ2 χଶ = ሺ݊ − 1ሻܵ ଶ ߪ଴ଶ where, n– = degrees of freedom s2 = sample variance, calculated as follows ∑௡௜ୀଵሺܺ௜ − ܺതሻଶ ݊−1 Assumptions of the chi-square distribution: ܵଶ = • The sample is a random sample or • The sample is taken from a normally distributed Reading 12 Hypothesis Testing FinQuiz.com population Properties of the chi-square distribution: • Unlike the normal and t-distributions, the chi-square distribution is asymmetrical • Unlike the t-distribution, the chi-square distribution is bounded below by i.e χ2 values cannot be negative • Unlike the t-distribution, the chi-square distribution is affected by violations of its assumptions and give incorrect results when assumptions not hold • Like the t-distribution, the shape of the chi-square distribution depends upon the degrees of freedom i.e as the number of degrees of freedom increases, the chi-square distribution becomes more symmetric Rejection Points for Hypothesis Tests on the Population Variance: Example: Suppose, H0: The variance, σ2 ≤ 0.25 Ha: The variance, σ2 > 0.25 It is a right-tailed test with level of significance (α) = 0.05 and d.f = 41 – = 40 degrees Using the chi-square table, the critical value is 55.758 Decision rule: Reject H0 if χ2 > 55.758 Using the X2–test, the standardized test statistic is: Two-tailed test: H0: σ2 = σ20 versus Ha: σ2≠ σ20 Decision Rule: Reject H0 if i The test statistic > upper α/2 point (χ2α/2) of the chisquare distribution with df = n – or ii The test statistic < lower α/2 point (χ21-α/2) of the chisquare distribution with df = n – χ2 = (n − 1) s σ = (41 − 1)(0.27) = 43.2 0.25 • Since, χ2 is not > 55.758, we fail to reject the H0 Right-tailed test: H0: σ2≤ σ20 versus Ha: σ2 > σ20 Decision Rule: Reject H0 if the test statistic > upper α point of the chi-square distribution with df = n -1 Left-tailed test: H0: σ2 ≥ σ20 versus Ha: σ2< σ20 Decision Rule: Reject H0 if the test statistic < lower α point of the chi-square distribution with df = n -1 Finding the critical values for the chi-square distribution from a table: • For a right-tailed test, use the value corresponding to d.f and α • For a left-tailed test, use the value corresponding to d.f and - α • For a two-tailed test, use the values corresponding to d.f.& ½ α and d.f.& –½ α Chi-square confidence intervals for variance: Unlike confidence intervals based on z or t-statistics, chi-square confidence intervals for variance are asymmetric A two-sided confidence interval for population variance, based on a sample of size n is as follows: • Lower limit = L = (n-1) s2 / χ2α/2 • Upper limit = U = (n -1) s2 / χ2 1-α/2 When the hypothesized value of the population variance lies within these two limits, we fail to reject the null hypothesis Practice: Example 7, Volume 1, Reading 12 Reading 12 4.2 Hypothesis Testing Tests Concerning the Equality (Inequality) of Two Variances H0: σ2 = σ22 versus Ha: σ21 ≠ σ22 σ2 = σ22 implies that σ2 / σ22 = H0: σ21 ≤σ22 versus Ha: σ2 >σ22 H0: σ21 ≥σ22 versus Ha: σ2 upper α / point of the F-distribution with the specified numerator and denominator degrees of freedom Right-tailed test: H0: σ21 ≤σ22 versus Ha: σ21> σ22 Decision Rule: Reject H0 at the “α significance level” if the test statistic > upper α point of the Fdistribution with the specified numerator and denominator degrees of freedom Relationship between the chi-square and F-distribution: F = (χ12 / m) ÷ (χ22 / n) • It follows an F-distribution with m numerator and n denominator degrees of freedom where, χ12 is one chi-square random variable with m degrees of freedom χ22 is another chi-square random variable with n degrees of freedom Test Statistic for Tests Concerning Differences between the Variances of Two Populations (Normally Distributed Populations): Assumption: The samples are random and independent and taken from normally distributed populations S 21 F= S where, s21 = sample variance of the first sample with nl observations Left-tailed test: H0: σ21 ≥ σ22 versus Ha: σ21 < σ22 Decision Rule: Reject H0 at the “α significance level” if the test statistic > upper α point of the Fdistribution with the specified numerator and denominator degrees of freedom B When the convention of using the larger of the two ratios s21 / s22 or s22 / s21 is NOT followed: In this case if the calculated value of F < 1, F-table can still be used by using a reciprocal property of F-statistics i.e., F n, m = 1/ Fm, n Important to Note: • For a two-tailed test at the α level of significance, the rejection points in F-table are found at α / significance level • For a one-tailed test at the α level of significance, the rejection points in F-table are found at α significance level Reading 12 Hypothesis Testing Example: Suppose, H0: σ21 ≤ σ22 Ha: σ21>σ22 • • • • • n1 = 16 n2 = 16 S21 = 5.8 S22 =1.7 df1=df2 = 15 From F table with 15 and 15 df and α = 0.05, the critical value of F = 2.40 (from the table below) Decision Rule: Reject H0 if calculated F-statistic > critical value of F Since S21 > S22, we will use convention F = s21 / s22 F= s12 5.8 = = 3.41 s22 1.7 • Since calculated F-statistic (3.41) > 2.40, we reject H0 at 5% significance level F-values for α = 0.05 Practice: Example & 9, Volume 1, Reading 12 FinQuiz.com Reading 12 Hypothesis Testing FinQuiz.com OTHER ISSUES: NONPARAMETRIC INFERENCE Source: Table 9, CFA® Program Curriculum, Volume 1, Reading 12 Parametric test: A parametric test is a hypothesis test regarding a parameter or a hypothesis test that is based on specific distributional assumptions • Parametric tests are robust i.e they are relatively unaffected by violations of the assumptions • Parametric tests have greater statistical power relative to corresponding non-parametric tests Non parametric test: A non parametric test is a test that is either not regarding a parameter or is based on minimal assumptions about the population • Nonparametric tests are considered distribution-free methods because they not rely on any underlying distributional assumption • Nonparametric statistics are useful when the data are not normally distributed A non parametric test is mainly used in three situations: 1) When data not meet distributional assumptions 2) When data are given in ranks 3) When the hypothesis is not related to a parameter In a nonparametric test, generally, observations (or a function of observations) are converted into ranks according to their magnitude Thus, the null hypothesis is stated as a thesis regarding ranks or signs The nonparametric test can also be used when the original data are already ranked Important to Note: Non-parametric test is less powerful i.e the probability of correctly rejecting the null hypothesis is lower So when the data meets the assumptions, parametric tests should be used Example: If we want to test whether a sample is random or not, we will use the appropriate nonparametric test (a so-called runs test) Parametric Nonparametric Tests concerning a single mean t-test z-test Wilcoxon signedrank test Tests concerning differences between means t-test Approximate ttest Mann-Whitney U test Tests concerning mean differences (Paired comparisons tests) t-test 5.1 Tests Concerning Correlation: The Spearman Rank Correlation Coefficient When the population under consideration does not meet the assumptions, a test based on the Spearman rank correlation coefficient rS can be used Steps of Calculating rS: Rank the observations on X in descending order i.e from largest to smallest • The observation with the largest value is assigned number • The observation with second-largest value is assigned number 2, and so on • If two observations have equal values, each tied observation is assigned the average of the ranks that they jointly occupy e.g if the 4th and 5th-largest values are tied, both observations are assigned the rank of 4.5 (the average of and 5) Calculate the difference, di, between the ranks of each pair of observations on X and Y The Spearman rank correlation is calculated as: ‫ݎ‬ௌ = − ∑௡௜ୀଵ ݀ଵଶ ݊(݊ଶ − 1) a) For small samples, the rejection points for the test based on rS are found using Table 11 below b) For large samples (i.e n> 30), t-test can be used to test the hypothesis i.e ‫=ݐ‬ (݊ − 2)ଵ/ଶ ‫ݎ‬ௌ (1 − ‫ݎ‬ௌଶ )ଵ/ଶ With degrees of freedom = n – Example: Suppose, H 0: ρ = Ha: ρ ≠ where, Wilcoxon signedrank test Sign test ρ = Population correlation of X and Y after ranking Reading 12 Hypothesis Testing FinQuiz.com Spearman Rank Correlation Distribution Approximate Upper-Tail Rejection Points Portfolio Managers Sharpe Ratio (X) –1.50 –1.00 –0.90 –1.00 –0.95 Management Fee (Y) 1.25 0.95 0.90 0.98 0.90 X Rank 3.5 3.5 Y Rank 4.5 4.5 di( X – Y) 0.5 –3.5 1.5 –2.5 d 2i 16 0.25 12.25 2.25 6.25 Sum of d2i = 37 • The first two rows in the table above contain the original data • In the row of X Rank, the Sharpe ratios are converted into ranks • In the row of Y Rank, the management fees are converted into ranks It is a two-tailed test with a 0.05 significance level and sample size (n) = NOTE: Both variables X and Y are not normally distributed; the ttest assumptions are not met rS = – [(6 ∑d2i) / n (n2 – 1)] rS = – (6 × 37) / (25 – 1) = -0.85 Important to Note: Since the sample size is small i.e (n < 30), the rejection points for the test must be looked up in Table 11 • Upper-tail rejection point for n = and α/2 = 0.05/ = 0.025 from table 11 is 0.9000 Decision Rule: Reject H0 if rS> 0.900 or rS 0.900, we not reject the null hypothesis Sample Size: n α = 0.05 α = 0.025 α = 0.01 0.8000 0.9000 0.9000 0.7714 0.8286 0.8857 0.6786 0.7450 0.8571 0.6190 0.7143 0.8095 0.5833 0.6833 0.7667 10 0.5515 0.6364 0.7333 11 0.5273 0.6091 0.7000 12 0.4965 0.5804 0.6713 13 0.4780 0.5549 0.6429 14 0.45930 0.5341 0.6220 15 0.4429 0.5179 0.6000 16 0.4265 0.5000 0.5824 17 0.4118 0.4853 0.5637 18 0.3994 0.4716 0.5480 19 0.3895 0.4579 0.5333 20 0.3789 0.4451 0.5203 21 0.3688 0.4351 0.5078 22 0.3597 0.4241 0.4963 23 0.3518 0.4150 0.4852 24 0.3435 0.4061 0.4748 25 0.3362 0.3977 0.4654 26 0.3299 0.3894 0.4564 27 0.3236 0.3822 0.4481 28 0.3175 0.3749 0.4401 29 0.3113 0.3685 0.4320 30 0.3059 0.3620 0.4251 NOTE: The corresponding lower tail critical value is obtained by changing the sign of the upper-tail critical value Source: Table 11, CFA® Program Curriculum, Volume 1, Reading 12 Practice: Example before Table 10, Volume 1, Reading 12 & End of Chapter Practice Problems for Reading 12  ... to Hypothesis Testing: Critical region for two-tailed test at 5% level of significance (i.e α = 0.05): • Null hypothesis: H0: θ = θ0 • Alternative hypothesis: Ha: θ ≠ θ0 Reading 12 Hypothesis Testing. .. Reading 12 Hypothesis Testing FinQuiz.com OTHER ISSUES: NONPARAMETRIC INFERENCE Source: Table 9, CFA Program Curriculum, Volume 1, Reading 12 Parametric test: A parametric test is a hypothesis. .. lies within these two limits, we fail to reject the null hypothesis Practice: Example 7, Volume 1, Reading 12 Reading 12 4.2 Hypothesis Testing Tests Concerning the Equality (Inequality) of Two