CFA: Quantitative Methods: Hypothesis Testing
What is similar about the means of two populations vs the significance of the mean of the differences between pairs of observations?
- both involve t-statistics - depend on degrees of freedom -
Using the z statistic, when the sample size is large and the population variance is unknown, the z statistic is?
= x̄−μ₀ / s⁄√n
Hypothesis
A statement about the value of a population parameter developed for the purpose of testing a theory or belief. Tested in terms of the population parameter to be tested.
When you reject Hₒ when Hₒ is true what is this called and the relation to the significance level?
Incorrect decision, called a Type 1 error, Significant level a = P(Type 1 error)
Difference between statistical significance and economic significance
Statistical significance does not necessarily imply economic significance. One few considerations to think about: - transaction costs - taxes - risk Any of these factors could make committing funds to a strategy unattractive, even though the statistical evidence of positive returns is highly significant. By nature of statistical tests, a very large sample size can result in highly significant results that are quite small in absolute terms.
T Test - explain the t-statistic, the use
T-Statistic: t(n-1)= x̄−μ₀ / s⁄√n Use the t-test if the population variance is unknown and either of the following conditions exist: - The sample is large (n≥30) - If the sample size is n<30, but the distribtuion of the population is normal To test the hypothesis, calculate the critical t-value at the level of significance with the appropriate degrees of freedom. Compare to t-statistic.
When you do not reject Hₒ when Hₒ is false what is that called?
Type II error, and this is an incorrect decision
One tailed hypothesis test of the population mean, the null and alternative hypotheses are either:
Upper tail: Hₒ: µ≤µₒ, versus Hₐ : µ>µₒ Lower tail: Hₒ: µ≥µₒ, versus Hₐ : µ<µₒ
Z Test- explain the z-statistic, the use, what are the critical z values, if population variance is unknown?
Z Statistic: = x̄−μ₀ /σ⁄√n Use the z-test if the population variance is known and the population is normally distributed. To test the hypothesis, the z statistic is compared to the zvalue corresponding to the significance of the test. What are the critical z values: - Level of significance .10=10%, Two tailed test- ±1.65, One Tailed Test +1.28, or -1.28 - Level of significance .05=5%, Two tailed test ±1.96, One tailed test, +1.65 or - 1.65 - Level of significance .01=1%, Two tailed test ±2.58, One tailed test, + 2.33 or -2.33 Unknown variance: Z Statistic: = x̄−μ₀ /s⁄√n
What is the appropriate hypothesis test of the population mean when the population is normally distributed with known variance.
Z- test or Z Statistic z statistic= x̄−μ₀ / σ⁄√n Where: x̄= sample mean, μ₀ = hypothesied population mean (null), σ- standard deviation of the population, n= sample size
What is the spearman rank correlation test
can be used when data is not normally distributed.
How is a test statistic calculated?
comparing the point estimate of the population parameter with the hypothesized value of the parameter (ie the value specified in the null hypothesis). So as an example we are concerned with the difference between the mean return of the sample x=.0001 and the hypothesized mean return (ie µₒ=0) Test statistic= (sample statistic-hypothesized value)/standard error of the sample statistic
When Hₒ is true and your decision is "do not reject Hₒ" is that corect?
correct decision
When you reject Hₒ and Hₒ is false , is that a correct decision and relation to power of test?
correct decision, power of test=1-P(Type II error)
Non parametric tests
either do not consider a particular parameter or have few assumptions about the population that is sampled. Nonparametric tests are used when - There is concern about quantities other than the parameters of a distribution - When the assumptions of parametric tests can't be supported. - When the data are not suitable for parametric tests.
When the null is less than or equal to, the alternative is framed as...
greater than, and a one tail test is appropriate... i.e. when we are trying to demonstrate that a return is greater than the risk free rate. We will have set up the null and alternative hypothesis so that the rejection of the null will lead to the acceptance of the alternative which was the GOAL IN PERFORMING THE TEST. The null for one tailed tests will include the equal to sign (greater than or less than equal to). the alternative will include the "less than" or "greater than"
When is it appropriate to use the t-test ?
if the population variance is unknown AND either of the following conditions exist - the same is large (n≥30) - The sample is small (less than 30) but the distribution of the population is normal or approximately normal.
According to the hypothesis testing, if the sample size is small and the distribution is non normal, do we have a reliable statistical test?
no
If the sample is small and the distribution is nonnormal,
no reliable statistical test
Parametric Tests
rely on assumptions regarding distribution of the population and are specific to population parameters.
Hypothesis Testing
statistical assessment of a statement or idea regarding a population. Hypothesis testing procedures can be employed to test the validity of the statement at a given significance level. "the mean return for the US equity market is greater than zero"
(Difference Between The Means) Appropriate test statistic for a hypothesis test concerning the equality of population means of two at least approximately normally distributed populations, based on independent random samples are EQUAL. Variance is unknown (independent and normally distributed)
t statistic= (x̄₁−x̄₂)/ [(sᵨ²/n₁)+(sᵨ²/n₂)]½ sᵨ²= [(n₁-1)s₁² + (n₂-1)s₂²)/(n₁+n₂-2)] s₁²= variance of first sample s₂²= variance of second sample n₁= # of observations in first sample n₂= # of observations in second sample Df: n₁+n₂-2
Type 2 error
the failure to reject the null hypothesis when it is actually false
What is the alternative (vs. the null) ?
the most hoped for hypothesis
Power of a test
the probability of correctly rejecting the null hypothesis when it is false. the power of a test is one minus the probability of making a Type II error, or 1-P (Type II error)
What is the relationship between the significance level and the probability of making a Type 1 error?
the significance level is the probability of making a Type 1 error (rejecting the null hypothesis when it is true). For example a significance level of 5% (a=.05) means there is a 5% chance of rejecting a true null hypothesis.
How many statistics does hypothesis testing have?
two 1. the test statistic calculated from the sample data 2. the critical value of the test statistic The value of the computed test statistic relative to the critical value is a key step in assessing the validity of a hypothesis.
Two tailed test(two sided alternative hypothesis)
two sided test. a two tail test should be used if the research question is whether the return on options is simply different than zero. Two sided tests allow for deviation on both sides of the hypothesized value (zero). In practice most hypothesis tests are constructed as two tailed tests.
When there is a equal to and not equal to alternative what kind of test do you use?
two tailed test, where the null will include the equal to sign and the alternative will include the not equal to sign.
What is a "runs test"
we can use a non parametric test, called a runs test, to determine whether data are random. A runs test provides an estimate of the probability that a series of changes are random.
can the alternative hypothesis be one or two sided?
yes
Standard error of the sample statistic
σx̄= σ/√n- when the population standard deviation σ is known sx̄= s/√n- when the population standard deviation σ is not known. (estimated using the standard deviation of the sample s)
What are the steps to hypothesis testing or the hypothesis testing procedure?
1. State the hypothesis 2. Select the appropriate test statistic 3. Specify the level of significance 4. State the decision rule regarding the hypothesis 5. Collect the sample and calculate the sample statistics 6. make a decision regarding the hypothesis 7. Make a decision based on the results of the test.
Chi Square Test- single population variance
Chi-square test statistic= χ² ₙ₋₁ = (n-1)s² / σ₀² σ₀²= hypothesized value for the population variance s²= sample variance = Chi square distribution is bounded below by zero, chi-square values can't be negative Don't forget if they may give you a standard deviation so square it for the chi square test
F Test- Statistic- Equality of the variances of two normally distributed populations
Hypothesis can be created as: H₀ : σ₁²=σ₂² vs Hₐ: σ₁²≠σ₂² One sided test structures can be H₀ : σ₁²≤σ₂² vs Hₐ: σ₁²>σ₂² or H₀ : σ₁²≥σ₂² vs Hₐ: σ₁²<σ₂² F Test or Statistic: = s₁²/s₂² always put the variance of the larger number on teh - both samples need to be independent & normally distributed - Upper critical value is always greater than 1
Alternative Hypothesis
Hₐ , is what is concluded if there is sufficient evidence to reject the null hypothesis. It is usually the alternative hypothesis that you are really trying to assess. When the null hypothesis is discredited, the implication is that the alternative hypothesis is valid.
What is the general decision rule for a two tailed test?
Reject Hₒ if : test statistic > upper critical value or test statistic< lower critical value for example .. for a two tailed test using z test statistic at a 5% level of significance , a = .05 - the computed test statistic is compared with critical z values of ± 1.96. -if the computed test statistic falls outside the range of critical z values (ie test statistic>1.96, or test statistic < -1.96..... we want to REJECT the null and conclude that the sample statistic is sufficiently different from the hypothesized value. - if the computed statistic falls between ± 1.96 we conclude that the sample statistic is not sufficiently different from the hypothesized value (µ=µₒ in this case) and we fail to reject the null hypothesis. THE DECISION RULE (rejection rule)CAN BE STATED AS.. Reject Hₒ if: test statistic < -1.96 or test statistic > 1.96
(Difference Between the Means) Appropriate test statistic for a hypothesis test concerning the equality of population means of two at least approximately normally distributed populations, based on independent random samples are UNEQUAL. Variance is unknown (independent and normally distributed)
Uses sample variances for both populations. With no assumption of equal variances, the denominator or standard error is based on the individual sample variances for each sample T Statistic used. The numerator of the t statistic and the t statistic are small, and we do not reject equality. If the sample means are far apart, the numerator of the t statistic and the t-statistic are rather large, and we reject equality. REMEMBER THAT THIS TEST IS ONLY VALID FOR TWO POPULATIONS THAT ARE INDEPENDENT AND NORMALLY DISTRIBUTED
P-value
is the probability of obtaining a test statistic that would lead to a rejection of the null hypothesis, assuming the null hypothesis is true. It is the smallest level of significance for which the null hypothesis can be rejected. for two tailed tests, the p value is the probability that lies above the positive value of the computed test statistic plus the probability that lies below the negative value of the computed test statistic.
What is the computed value for the test statistic based on the t-distribution?
it is called the t statistic. With n-1 degrees of freedom, it is computed.... t(n-1)= x̄−μ₀ / s⁄√n Where: x̄= sample mean μ₀=hypothesized population mean (the null) s= standard deviation of the sample n=sample size
one tailed test (one sided alternative hypothesis)
one sided test. For example if a researcher wants to test whether the return on stock options is greater than zero, a one tailed test should be used.
Confidence Interval
range of values within which the researcher believes the true population parameter may lie. {[sample statistic-(critical value)(standard error)≤ population parameter≤[sample statistic + (critical value)(standard error)]} the interpretation of a confidence level is that for a given level of confidence of 95%, for example, there is 95% probability that the true population parameter is contained in the inteval.
Type 1 error
the rejection of the null hypothesis when it is actually true
What are situations where a non parametric test is called for:
- The assumptions about the distribution of the random variable that support a parametric test are not met. (mean value comes from non normal distribution and of small size so z or t tests don't work - When data are ranks (ordinal) rather than values - They hypothesis does not involve the parameters of the distribution, such as testing whether a variable is normally distributed.
what's distinct between the means of two populations vs the significance of the mean of the differences between pairs of observations?
- the test of the difference in means is used when there are two independent samples - the test of the significance of the mean of the differences between paired observations is used when the samples are not independent.
Hypothesis testing procedures and purpose
Based on sample statistics and probability theory, are used to determine whether a hypothesis is a reasonable statement and should not be rejected or if it an unreasonable statement and should be rejected.
Mean Difference of two normally distributed populations - find the appropriate test statistic and interpret results (Dependent based on some factor)
Dependent samples- construct a "paired comparison" test of whether the means of the differences between observations for the two samples are different. The paired comparison test is just a test of whether the average difference between monthly returns is significantly different from zero, based on the standard error of the differences in monthly returns Paired comparison test requires the sample data be normally distributed. t= sample mean difference- μdz / (Sd/√n)
Null Hypothesis
Designated, Hₒ , is the hypothesis the researcher wants to reject. It is the hypothesis that is actually tested and is the basis for the selection of the test statistics. Generally stated as a simple statement about a population parameter. Typical statements of the null hypothesis include Hₒ: µ=µₒ, Hₒ ; µ≤µₒ , and Hₒ ; µ≥µₒ where µ is the population mean and µₒ is the hypothesized value of the population mean. Null hypothesis always includes the "equal to" condition
How is a two tailed test for the population mean constructed or structured?
Hₒ: µ=µₒ, versus Hₐ : µ≠µₒ In this case, the alternative hypothesis allows for values above and below the hypothesized parameter, a two tailed test uses two critical values or rejection points.