Reading 6
For a one-tailed hypothesis test of the population mean, the null and alternative hypotheses are either:
Upper tail: H0: µ ≤ µ0 versus Ha: µ > µ0, or Lower tail: H0: µ ≥ µ0 versus Ha: µ < µ0
The critical value for the appropriate test statistic—the value against which the computed test statistic is compared—
depends on its distribution.
A _________________ variance is used with the t-test for testing the hypothesis that the means of two normally distributed populations are equal, when the variances of the populations are unknown but assumed to be equal.
pooled
If the sample means are very close together, the numerator of the t-statistic (and the t-statistic itself) are small, and ...
we do not reject equality.
a t-statistic with n - 1 degrees of freedom is computed as:
(tn−1)=(¯x−μ0) / (s/√n) where: ¯x= sample mean µ0= hypothesized population mean (i.e., the null) s= standard deviation of the sample n= sample size
Let's use the upper tail test structure where H0: µ ≤ µ0 and Ha: µ > µ0.
-If the calculated test statistic is greater than 1.645, we conclude that the sample statistic is sufficiently greater than the hypothesized value. In other words, we reject the null hypothesis. -If the calculated test statistic is less than 1.645, we conclude that the sample statistic is not sufficiently different from the hypothesized value, and we fail to reject the null hypothesis.
Keep in mind that we have been describing two distinct hypothesis tests, one about the significance of the difference between the means of two populations and one about the significance of the mean of the differences between pairs of observations. Here are rules for when these tests may be applied:
-The test of the differences in means is used when there are two independent samples. -A test of the significance of the mean of the differences between paired observations is used when the samples are not independent.
This is the range within which we fail to reject the null for a two-tailed hypothesis test at a given level of significance.
-critical value ≤ test statistic ≤ +critical value
A confidence interval is a range of values within which the researcher believes the true population parameter may lie.
A confidence interval is determined as: {[samplestatistic-(criticalvalue)(standarderror)]≤ populationparameter≤[samplestatistic+(criticalvalue)(standarderror)]}
F-test:
Always put the larger variance in the numerator (s12 ). Following this convention means we only have to consider the critical value for the right-hand tail.
f the observations in the two samples both depend on some other factor, we can construct a "paired comparisons" test of whether the means of the differences between observations for the two samples are different.
Dependence may result from an event that affects both sets of observations for a number of companies or because observations for two firms over time are both influenced by market returns or economic conditions.
A two-tailed test for the population mean may be structured as:
H0: µ = µ0 versus Ha: µ ≠ µ0
the hypotheses for a two-tailed test of a single population variance are structured as:
H0: σ2 = σ02 versus Ha: σ2 ≠ σ02
The hypotheses for one-tailed tests are structured as:
H0: σ2 ≤ σ02versus Ha : σ2 > σ02 or H0: σ2 ≥ σ02 versus Ha σ2 < σ02
Hypothesis testing procedures, 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 is an unreasonable statement and should be rejected. The process of hypothesis testing consists of a series of steps shown in _______ ______ ____
Hypothesis Testing Procedure*
Memorize this chart!
Level of Significance. Two-Tailed Test One-Tailed Test 0.10 = 10%. ±1.65. +1.28 or -1.28 0.05 = 5%. ±1.96. +1.65 or -1.65 0.01 = 1%. ±2.58. +2.33 or -2.33
_______________ tests either do not consider a particular population 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 or when the assumptions of parametric tests can't be supported. They are also used when the data are not suitable for parametric tests (e.g., ranked observations).
Nonparametric
______________ 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. The z-test also requires that either the sample is large, relying on the central limit theorem to assure a normal sampling distribution, or that the population is normally distributed.
Parametric
The general decision rule for a two-tailed test is:
Reject H0 if: test statistic > upper critical value or test statistic < lower critical value
Example: Difference between means - equal variances Sue Smith is investigating whether the abnormal returns for acquiring firms during merger announcement periods differ for horizontal and vertical mergers. She estimates the abnormal returns for a sample of acquiring firms associated with horizontal mergers and a sample of acquiring firms involved in vertical mergers. Smith finds that abnormal returns from horizontal mergers have a mean of 1.0% and a standard deviation of 1.0%, while abnormal returns from vertical mergers have a mean of 2.5% and a standard deviation of 2.0%. Smith assumes that the samples are independent, the population means are normally distributed, and the population variances are equal. Smith calculates the t-statistic as -5.474 and the degrees of freedom as 120. Using a 5% significance level, should Smith reject or fail to reject the null hypothesis that the abnormal returns to acquiring firms during the announcement period are the same for horizontal and vertical mergers?
Since the test statistic, -5.474, falls to the left of the lower critical t-value, Smith can reject the null hypothesis and conclude that mean abnormal returns are different for horizontal and vertical mergers.
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.
The sample is large (n ≥ 30). The sample is small (less than 30), but the distribution of the population is normal or approximately normal. If the sample is small and the distribution is nonnormal, we have no reliable statistical test. 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-statistic with n - 1 degrees of freedom is computed as:
Example: Paired comparisons test Joe paired comparisons test Andrews is examining changes in estimated betas for the common stock of companies in the telecommunications industry before and after deregulation. Andrews believes that the betas may decline because of deregulation since companies are no longer subject to the uncertainties of rate regulation or that they may increase because there is more uncertainty regarding competition in the industry. Andrews calculates a t-statistic of 10.26 for this hypothesis test, based on a sample size of 39. Using a 5% significance level, determine whether there is a change in betas.
The test statistic, 10.26, is greater than the critical t-value, 2.024—it falls in the rejection region to the right of 2.024 in the previous figure. Thus, we reject the null hypothesis of no difference, concluding that there is a statistically significant difference between mean firm betas before and after deregulation.
Note that when the sample variances are equal, the value of the test statistic is 1.
The upper critical value is always greater than one (the numerator is significantly greater than the denominator), and the lower critical value is always less than one (the numerator is significantly smaller than the denominator). In fact, the lower critical value is the reciprocal of the upper critical value. For this reason, in practice we put the larger sample variance in the numerator and consider only the upper critical value.
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:
Z-statistic = (¯x−μ0) / *σ/√n)
The _______________________ 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. Why? Because you can never really prove anything with statistics, when the null hypothesis is discredited, the implication is that the alternative hypothesis is valid.
alternative
the test statistic is the
difference between the sample statistic and the hypothesized value, divided by the standard error of the sample statistic.
Statisticians refer to these incorrect rejections of the null hypothesis as _____ positives.
false
p value equation
if your level of significance >= p value ... then reject null hypothesis
the power of a test
is the probability of correctly rejecting the null hypothesis when it is false
The ____________ 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. The null is generally stated as a simple statement about a population parameter. Typical statements of the null hypothesis for the population mean include H0: µ = µ0, H0: µ ≤ µ0, and H0: µ ≥ µ0, where µ is the population mean and µ0 is the hypothesized value of the population mean.
null
The ___-_____ 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 one-tailed tests, the p-value is the probability that lies above the computed test statistic for upper tail tests or below the computed test statistic for lower tail tests. 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.
p-value
The appropriate test statistic for the hypothesis that the population correlation equals zero, when the two variables are normally distributed, is:
r(√n−2)/(√1−r^2), where r = sample correlation and n = sample size
Type II error:
the failure to reject the null hypothesis when it is actually false.
Type I error:
the rejection of the null hypothesis when it is actually true.
Hypothesis testing involves two statistics:
the test statistic calculated from the sample data and 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.
Sample size and the choice of significance level (Type I error probability) will together determine the probability of a Type II error. The relation is not simple, however, and calculating the probability of a Type II error in practice is quite difficult. Decreasing the significance level (probability of a Type I error) from 5% to 1%, for example, will increase the probability of failing to reject a false null (Type II error) and therefore reduce the power of the test. Conversely, for a given sample size, we can increase the power of a test only with the cost that the probability of rejecting a true null (Type I error) increases. For a given significance level, we can decrease the probability of a Type II error and increase the power of a test, only by increasing the sample size.
true!
The LOS here says, "Describe how to interpret the significance of a test ..." It does not indicate that calculations will be required. Perhaps if you just remember that we compare the reported p-values (ranked from lowest to highest) to the adjusted significance levels (significance level times rank / number of tests), and then count only those tests for which the p-values are less than their adjusted significance levels as rejections, you'll be able to handle any questions based on this LOS.
true!
You should instead focus on the fact that both of these tests involve t-statistics and depend on the degrees of freedom. Also note that when samples are independent, you can use the difference in means test, and when they are dependent, we must use the paired comparison (mean differences) test. In that case, with a null hypothesis that there is no difference in means, the test statistic is simply the mean of the differences between each pair of observations, divided by the standard error of those differences. This is just a straightforward t-test of whether the mean of a sample is zero, which might be considered "fair game" for the exam.
true!
Since the alternative hypothesis allows for values above and below the hypothesized parameter, a _____-_____ test uses two critical values (or rejection points).
two-tailed
If the sample means are far apart, the numerator of the t-statistic (and the t-statistic itself) are large, and ..... Perhaps not as easy to remember is the fact that this test is only valid for two populations that are independent and normally distributed.
we reject equality.
Adjusted significance =
α×(Rank of p-value/Number of tests)
The chi-square test statistic, χ2, with n - 1 degrees of freedom, is computed as:
χ2(n−1)=(n−1)s2/σ20 where: n= sample size s2= sample variance σ02 = hypothesized value for the population variance.