READING 19: HYPOTHESIS TESTING AND CONFIDENCE INTERVALS

Population And Sample Statistics

        Population Mean

        Population Variance

        Sample Mean

        Sample Variance

        The standard error of the sample mean is the standard deviation of the distribution of the sample means.

        where:
         σ_x= standard error of the sample mean
         σ = standard deviation of the population
         n = size of the sample

Confidence Interval

        For a normally distributed population, a confidence interval for its mean can be constructed using a z-statistic when variance is known, and a t-statistic when the variance is unknown. The z-statistic is acceptable in the case of a normal population with an unknown variance if the sample size is large (30+).
In general, we have:

Hypothesis Testing

        The hypothesis testing process requires a statement of a null and an alternative hypothesis, the selection of the appropriate test statistic, specification of the significance level, a decision rule, the calculation of a sample statistic, a decision regarding the hypotheses based on the test, and a decision based on the test results.
         The test statistic is the value that a decision about a hypothesis will be based on. For a test about the value of the mean of a distribution:

        With unknown population variance, the t-statistic is used for tests about the mean of a normally distributed population:

. If the population variance is known, the appropriate test statistic is

for tests about the mean of a population.

One-Tailed and Two-Tailed Tests of Hypotheses

        A two-tailed test results from a two-sided alternative hypothesis (e.g., HA: μ ≠ μ0). A one-tailed test results from a one-sided alternative hypothesis (e.g., HA: μ > μ0, or HA: μ < μ0).

Type I and Type II Errors

        Type I error: the rejection of the null hypothesis when it is actually true.
         Type II error: the failure to reject the null hypothesis when it is actually false.

Hypothesis Testing Results

        Hypothesis testing compares a computed test statistic to a critical value at a stated level of significance, which is the decision rule for the test.
         A hypothesis about a population parameter is rejected when the sample statistic lies outside a confidence interval around the hypothesized value for the chosen level of significance.

The p-Value

        The 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.

The t-Test

        The t-test is a widely used hypothesis test that employs a test statistic that is distributed according to a t-distribution. Following are the rules for when it is appropriate to use the t-test for hypothesis tests of the population mean.
        Use the t-test if the population variance is unknown and either of the following conditions exist:

• The sample is large (n ≥ 30).
  
• The sample is small (n < 30), but the distribution of the population is normal or approximately normal.

        If the sample is small and the distribution is non-normal, we have no reliable statistical test.
        For hypothesis tests of a population mean, a t-statistic with n − 1 degrees of freedom is computed as:

        where:
         x = sample mean
         μ0 = hypothesized population mean (i.e., the null)
         s = standard deviation of the sample
         n = sample size

The z-Test

        The z-test is the appropriate hypothesis test of the population mean when the population is normally distributed with known variance. The computed test statistic used with the z-test is referred to as the z-statistic. The z-statistic for a hypothesis test for a population mean is computed as follows:

        where:
         x = sample mean
         μ0 = hypothesized population mean
         σ = standard deviation of the population
         n = sample size

Back-testing Value at Risk

        Back-testing is the process of comparing losses predicted by the value at risk (VaR) model to those actually experienced over the sample testing period. If a model were completely accurate, we would expect VaR to be exceeded with the same frequency predicted by the confidence level used in the VaR model. In other words, the probability of observing a loss amount greater than VaR should be equal to the level of significance.