Study Session 3: Hypothesis Testing
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 sample stats. 6. Make a decision regarding the Hypothesis 7. make a decision based on the results of the test
Situations that Call For Non Parametric Tests
1. The assumptions about the distribution of the random variables that support a parametric test are not met. (no variance and to small sample size). 2. When data are ranks (an ordinal measurement scale) rather than values. 3. The hypothesis does not involve the parameters of the distribution, such as testing whether a variable is normally distributed. Runs test will tell you if data is random.
Hypothesis Testing (Def)
A statistical assessment of a statement or idea regarding a population. Ex. "The mean return for U.S. stocks is greater than 0". Given the relevant returns data, hypothesis testing procedures can be employed to test the validity of this statement at a given significance level.
Test Statistic
A test statistic is calculated by comparing the point estimate of the population parameter with the hypothesized value of the parameter. It is calculated as the difference between the sample statistic and the hypothesized value, scaled by the standard error of the sample statistic. test statistic = (sample stat - hypo value) / standard error of the sample statistic.
Two Tailed Test
A two tailed test may be structured as: H₀: µ = µ₀ versus Ha: µ ≠ µ₀ Since the alt hypo allows for values above and below the hypothesized parameter, a two tailed test uses two critical values (or rejection points). General decision rule for a two tailed test is: Reject H₀ if: test statistic > upper critical value or test statistic < lower critical value
One and Two Tailed Tests of Hypotheses
Alt Hypos can be one sided or two sided. Whether the test is one or two sided depends on the proposition being tested. If a researcher wants to see if the mean is greater than zero, then you use a one sided. If you want to show that the mean is simply different than zero you use a two. Two sided tests allow for deviation on both sides of the hypothesized value.
One Tailed Test
For a one tailed hypothesis test of the population mean, the null and the alternative hypothesis are either: Upper Tail: H₀: µ ≤ µ₀ versus Ha: µ > µ₀, or Lower Tail: H₀: µ ≥ µ₀ versus Ha: µ < µ₀ The appropriate set of hypotheses depends on whether we believe the population mean, µ, to be greater than (upper tail) or less than (lower tail) the hypothesized value, µ₀.
F-Test Hypothesis
If we let δ₁² and δ₂² represent the variances of normal population 1 and 2, the hypothesis for the two-tailed F-test of differences in the variances can be constructed as: H₀: δ₁² = δ₂² versus Ha: δ₁² ≠ δ₂² One sided: H₀: δ₁² ≤ δ₂² versus Ha: δ₁² > δ₂² Or H₀: δ₁² ≥ δ₂² versus Ha: δ₁² < δ₂²
Hypothesis Testing Errors
Keep in mind that hypo testing is used to make inferences about the parameters of a given population. On the basis of stats computed for a sample that is drawn from that population. We must be aware that there is some probability that the sample does not represent the population and any conclusion based on the sample may be made in error.
Parametric Tests
Rely on assumptions regarding the distribution of the population and are specific to population parameters. For example the z test relies upon a mean and a standard deviation to define the normal distribution, and also requires that either the sample is large or that the populations is normally distributed.
t-test Statistic
The computed value for the test statistic based on the t-distribution is referred to as the t-statistic. For hypothesis tests of a population mean, a t-stat with n-1 degrees of freedom is calculated as: where xbar = sample mean µ₀ = hypothesized pop mean (i.e. the null) s = standard deviation of the sample n = sample size
F Distribution
The hypothesis concerned with the equality of the variances of two populations are tested with an F-distributed statistic. Hypothesis testing using a test stat that follows an F dist is referred to as the F test. The F test us used under the assumption that the populations from which the samples are drawn are normally distributed.
Null Hypothesis
The null Hypothesis, designated H₀, is the hypothesis that the researcher wants to reject. It is the hypothesis that is actually tested and is the basis for the selection of the test statistics. Typical statements are: H₀: µ = µ₀ H₀: µ ≤ µ₀ H₀: µ ≥ µ₀ Where µ is the population mean and µ₀ is the hypothesized value of the population mean.
F-Test Statistic
The test statistic for the F-test is the ratio of the sample variances. The F-statistic is computed as: F = s₁² / s₂² Where: s₁² = variance of the sample n₁ observ drawn from pop1 s₂² = variance of the sample n₂ observ drawn from pop2 Note that n₁-1 and n₂-1 are the degrees of freedom used to identify the appropriate critical value from the F-table.
t-Test
Use the t-test if the pop variance is unknown and either of the following conditions: - the sample is large (n≥30) - the sample is small (<30) but the dist of the pop is normal. If the sample is small and non-normal we have no reliable statistical test.
Chi-Square Test
Used for hypothesis tests concerning the variance of a normally distributed population. Letting δ² represent the true population variance and δ²₀ represent the hypothesized variance, the hypotheses for a two-tailed test of a single population variance are structured as: H₀: δ² = δ²₀ versus Ha: δ² ≠ δ²₀ For a one tailed test: H₀: δ² ≤ δ²₀ versus Ha: δ² > δ²₀ OR H₀: δ² ≥ δ²₀ versus Ha: δ² < δ²₀
Odd ball t-Tests
We have two oddball t-tests that involve t statistics and depend on the degrees of freedom. Note that when the samples are independent you can use the difference in means test, and when they are dependent the statistic is the average difference in (paired) observations divided by the standard error of the differences between observations.
Type I and Type II Errors
When drawing inferences from a hypo test, there are two types of errors: - Type I Errors: the rejection of the null hypothesis when it is actually true. - Type II Errors: the failure to reject the null hypothesis when it is actually false. The significance level is the probability of making a Type I error.
Power of a test
While the sig level is the prob of rejecting the null hypothesis when it is true, the power of a test is the prob of correctly rejecting the null hypo when it is false. The power of a test is actually 1 - P(Type II error).
Z Test Statistic
Z test is appropriate hypothesis test of the pop mean when the pop is normally distributed with known variance. Computed as follows where: xbar = sample mean µ₀ = hypothesized pop mean δ = standard deviation of the population n = sample size
Spearman Rank Correlation Test
can be used when data are not normally distributed. Consider the performance ranks of 20 mutual funds for 2 years. The ranks (1-20) are not normally distributed, so a standard t test is not appropriate. A large positive value of the spearman rank correlations such as .85 would indicate that a high (low) rank in one year is associated with a high (low) rank in the second year.
Chi-Squared T-Statistic
denoted as ℵ². The chi square distribution is asymmetrical and approaches the normal distribution in shape as the degrees of freedom increase. The test statistic ℵ² with n-1 degrees of freedom is calculated as: where: n= sample size s² = sample variance δ²₀ = hypothesized value for the pop variance.
Alternative Hypothesis
designated 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. Since you can never really prove anything with stats when the null hypothesis is discredited the implication is that the alternative is valid.
Nonparametric Tests
either do not consider a particular population parameter or have few assumptions about the population that is sampled. Non para tests are used when there is concern about quantities other than the parameters of a distribution or when the assumptions of parametric tests can't be supported.