Hypothesis Testing (Reading 11)

Learning Outcome Statements (LOS)

a

Define a hypothesis, describe the steps of hypothesis testing, describe and interpret the choice of the null and alternative hypotheses, and distinguish between one-tailed and two-tailed tests of hypotheses:

b

Explain a test statistic, Type I and Type II errors, a significance level, and how significance levels are used in hypothesis testing:

c     

Explain a decision rule, the power of a test, and the relation between confidence intervals and hypothesis tests:

d    

Distinguish between a statistical result and an economically meaningful result:     

e  

Explain and interpret the p-value as it relates to hypothesis testing:

f 

Identify the appropriate test statistic and interpret the results for a hypothesis test concerning the population mean of both large and small samples when the population is normally or approximately distributed and the variance is 1) known or 2) unknown:

g

Identify the appropriate test statistic and interpret the results for a hypothesis test concerning the equality of the population means of two at least approximately normally distributed populations, based on independent random sample with 1) equal or 2) unequal assumed variances:

h

Identify the appropriate test statistic and interpret the results for a hypothesis test concerning the mean difference of two normally distributed populations:

i

Identify the appropriate test statistic and interpret the results for a hypothesis test concerning 1) the variance of a normally distributed population, and 2) the equality of the variance of two normally distributed populations based on two independent random samples:

j

Distinguish between parametric and nonparametric tests and describe the situations in which the use of nonparametric tests may be appropriate:

Formulas:

Exercise Problems:

1.      When an investigator wants to test whether a particular parameter is larger than a specific value, the null and alternative hypothesis are best defined as:

A.    H₀: θ=θ₀ versus H_a: θ≠θ₀

B.     H₀: θ≤θ₀ versus H_a: θ>θ₀

C.     H₀: θ≥θ₀ versus H_a: θ<θ₀

 Ans: B; the null hypothesis is the hypothesis to be tested. It is a proposition that is considered true unless the sample we use to conduct the hypothesis test gives convincing evidence that the null hypothesis is false.

The alternative hypothesis is the hypothesis accepted when the null hypothesis is rejected.

In this problem, the proposition is that the particular parameter is larger than a specific value, so the null hypothesis, as opposite, is θ≤θ₀, and the alternative hypothesis is θ>θ₀.

2.      Which of the following steps in hypothesis testing most likely follows collecting the data and calculating the test statistic?   

A.    Stating the decision rule.

B.     Making the statistical decision.

C.     Specifying the significance level.

Ans: B; the steps in testing a hypothesis are as follows:

1.       Stating the hypothesis

2.       Identifying the appropriate test statistic and its probability distribution

3.       Specifying the significance level

4.       Stating the decision rule

5.       Collecting the data and calculating the test statistic

6.       Making the statistical decision

7.       Making the economic or investment decision

3.      A hypothesis test fails to reject a false null hypothesis. This is best described as a:

A.    Type ? error

B.     Type II error

C.     Test with little power

Ans: B; type I error is that we reject a true null hypothesis; while type II error is that we do not reject a false null hypothesis.

The power of a test is the probability  of correctly rejecting the null—that is, the probability of rejecting the null when it is false.

4.      A test statistic is best defined as the difference between the sample statistic and the value of the population parameter under H₀ divided by the:

A.  Sample standard deviation.

B.  Standard error of the sample statistic.

C. appropriate value from the t-distribution.

Ans: B; the formula of test statistic is