Reading 12 - Hypothesis Testing

Question 1

Hypothesis testing

Answer 1

Hypothesis testing is the process of evaluating the accuracy of a statement regarding a population parameter (e.g., the population mean) given sample information (e.g., the sample mean).

Question 2

Hypothesis

Answer 2

A hypothesis is a statement about the value of a population parameter developed for the purpose of testing a theory. Let’s assume that we think (hypothesize) that the average points scored in each game by a basketball player throughout his career is greater than 30. First, we would need to get some sample information. Then we would conduct a hypothesis test on the sample information (average of his scores in, let’s say, 49 randomly selected games) in order to be able to comment on the accuracy of the statement pertaining to the population parameter (his average score across all the games that he played in his entire career).

Question 3

Null Hypothesis (H₀)

Answer 3

The null hypothesis (H0) generally represents the status quo, and is the hypothesis that we are interested in rejecting. This hypothesis will not be rejected unless the sample data provides sufficient evidence to reject it. Null hypotheses regarding the mean of the population can be stated in the following ways:

H0 : μ <= μ 0

H0 : μ ³ μ 0

H0 : μ = μ 0

Where:


μ = population mean


μ0 = hypothesized value of the population mean

Question 4

Alternative hypothesis (H_a)

Answer 4

The alternate hypothesis (Ha) is essentially the statement whose validity we are trying to evaluate. The alternate hypothesis is the statement that will only be accepted if the sample data provides convincing evidence of its truth. It is the conclusion of the test if the null hypothesis is rejected. Alternate hypotheses can be stated as:

Ha : μ > μ 0

Ha : μ < μ 0

Ha : μ Does not equal μ 0

Question 5

Essentially, a hypothesis test involves the comparison of a sample’s test statistic to a critical value. The test statistic is calculated as:

Answer 5

Test statistic = (Sample statistic – Hypothesized statistic)/(Standard error of sample statistic)

The critical value depends on the relevant distribution, sample size, and level of significance used to test the hypothesis. 


Question 6

Hypothesis tests can be either one‐tailed or two‐tailed. Under one‐tailed tests:

Answer 6

we assess whether the value of a population parameter is either greater than or less than a given hypothesized value. Hypotheses for one‐tailed tests can be stated as: 


1. 0 : μ <= μ 0 versus Ha : μ > μ 0 


When we are testing whether the population mean is greater than a given hypothesized value.

1. 0 : μ ³ μ 0 versus Ha : μ < μ 0 


When we are testing whether the population mean is less than a given hypothesized value.

Question 7

The following rejection rules apply when trying to determine whether a population mean is greater than the hypothesized value.

Answer 7

Reject H0 when:


Test statistic > positive critical value 


Fail to reject H0 when:


Test statistic ≤ positive critical value 


Question 8

Under two‐tailed tests, we assess whether the value of the population parameter is simply different from a given hypothesized value. The hypotheses for two‐tailed tests are stated as: 


Answer 8

H0 : μ = μ0

Ha : μ ≠ μ0

Two‐tailed hypotheses tests have 2 rejection regions.

Question 9

Rejection Rule for a Two-Tailed Hypothesis test

Answer 9

Reject H₀ when:


Test statistic < Lower critical value

Test statistic > Upper critical value 


Fail to reject H0 when:


Lower critical value ≤ test statistic ≤ Upper critical value 


Question 10

Hypothesis tests are used to make inferences about population parameters using sample statistics. There is always a possibility that the sample may not be perfectly representative of the population, and that the conclusions drawn from the test may be wrong. There are two types of errors that can be made when conducting a hypothesis test: 


Answer 10

Type I error: Rejecting the null hypothesis when it is actually true. 


Type II error: Failing to reject the null hypothesis when it is actually false. 


Question 11

Power of a test

Answer 11

The power of a test is the probability of correctly rejecting the null hypothesis when it is false.

Power of a test = 1 − P(Type II error)

Question 12

Power of the test – 4 Important notes

Answer 12

1) The higher the power of the test, the better it is for purposes of hypothesis testing. Given a choice of tests, the one with the highest power should be preferred.
This statement is fairly straightforward—the test with the highest probability of rejecting the null hypothesis when it is false should be preferred. 

2) Decreasing the significance level reduces the probability of Type I error. However, reducing the significance level means shrinking the rejection region, and inflating the “fail to reject the null region.” This increases the probability of failing to reject a false null hypothesis (Type II error) and reduces the power of the test. 

3) The power of the test can only be increased by reducing the probability of a Type II error. This can only be accomplished by reducing the “fail to reject the null region,” which is equivalent to increasing the size of the rejection region and increasing the probability of a Type I error. Basically, an increase in the power of a test comes at the cost of increasing the probability of a Type I error. 

4) The only way to decrease the probability of a Type II error given the significance level (probability of Type I error) is to increase the sample size. 


Question 13

Confidence interval vs. Hypothesis interval

Answer 13

• In a confidence interval, we aim to determine whether the hypothesized value of the population mean (μ ₀), lies within a computed interval with a particular degree of confidence (1‐α). Here the interval represents the “fail‐to‐reject‐the‐null region” and is based around, or centered on the sample mean, x. 

• In a hypothesis test, we examine whether the sample mean, x lies in the rejection region (i.e., outside the interval) or in the fail‐to‐reject‐the‐null region (i.e., within
the interval) at a particular level of significance (α). Here the interval is based around, or centered on the hypothesized value of the population mean (μ 0). 


Question 14

p-value

Answer 14

The p‐value is the smallest level of significance at which the null hypothesis can be rejected. It represents the probability of obtaining a critical value that would lead to rejection of the null hypothesis.

Question 15

Hypothesis testing: Distinguishing between statistical results and economically meaningful results

Answer 15

Sometimes differences between a variable and its hypothesized value are statistically significant but not practically or economically meaningful. Suppose we are testing a hypothesis that the returns from a technical trading strategy are greater than zero. If we use a large sample (n) when conducting the test, our standard error will be small, the “fail‐to‐reject region” narrower, and the greater the chance that the null will be rejected. The sample error decreases as sample size increases, and as sample size increases we can have situations where the null is rejected even when the sample mean deviates only slightly from the hypothesized value. Even though

a trading strategy might provide a statistically significant return of greater than zero (based
on a hypothesis test) it does not mean that we can guarantee that trading on this strategy
would result in economically meaningful positive returns. The returns may not be economically significant after accounting for taxes, transaction costs and risks inherent in the strategy.

Even if we conclude that a strategy’s results are economically significant, we should examine whether there is a logical reason to explain the apparently‐significant returns offered by the strategy before actually implementing it.

Question 16

Hypothesis tests concerning a single mean

Answer 16

In the process of hypothesis testing, the decision whether to use critical values based on the z‐distribution or the t‐distribution depends on sample size, the distribution of the population and whether the variance of the population is known.

Question 17

When is the t-test used?

Answer 17

The t‐test is used when the variance of the population is unknown and either of the conditions below holds:

• The sample size is large. 

• The sample size is small, but the underlying population is normally distributed or 
approximately normally distributed. 


Question 18

The test statistic (t‐statistic) for hypothesis tests concerning the mean of a single population is: Used when variance is unknown 


Answer 18

Question 19

Why is the t-test popular?

Answer 19

In a t‐test, the sample’s t‐statistic is compared to the critical t‐value with n‐1 degrees of freedom, at the desired level of significance. Practically speaking, the variance of the population is rarely ever known, so the t‐test is very popular. 


Question 20

The z‐test: When & What

Answer 20

Can be used to conduct hypothesis tests of the population mean when the population is normally distributed and its variance is known. 


Question 21

The z‐test can also be used when the population’s variance is unknown, but the sample size is large.

Answer 21

Question 22

Hypotheses describing the tests of means of two populations can be structured as: 


Answer 22

• H0: μ1 – μ2 = 0 versus Ha: μ1 – μ2 ≠ 0 when we want to test if the two populations’ means are not equal. 

• H0: μ1 – μ2 ≥ 0 versus Ha: μ1 – μ2 < 0 when we want to test if the mean of Population 1 is less than the mean of Population 2. 

• H0: μ1 – μ2 ≤ 0 versus Ha: μ1 – μ2 > 0 when we want to test if the mean of Population 1 is greater than the mean of Population 2. 


Question 23

In tests where it is assumed that the variances of the two populations are equal, we use the pooled variance (s2p) in the calculation of the t‐stat. The test statistic, the pooled variance, and the degrees of freedom for the t‐test are calculated as follows:

Answer 23

Question 24

In hypothesis tests where it is assumed that the variances of the two populations are unequal, the test statistic and the degrees of freedom for the t‐test are calculated as follows:

Answer 24

Question 25

The table given below summarizes the important concepts that you should bear in
mind from the examination perspective. We highly doubt that any question testing the ability of financial analysts would entail memorizing the complicated formulas above and performing tedious calculations. However, questions related to recognizing the test appropriate to verify the hypotheses, the test statistic, and drawing conclusions given the critical values and the test statistic are fair game. Hypothesis tests concerning the mean of two populations.

Answer 25

Question 26

Sometimes we may need to perform tests on the variance of normally distributed populations. We use σ2to represent the population variance andσ02to denote the hypothesized value of the population variance.

Tests relating to the variance of normally distributed populations can be one‐tailed or two-tailed:

Answer 26

One-tailed tests:

H₀ : σ² ≤ σ²₀ versus H_a : σ² > σ²₀

When testing whether the population variance is greater than the hypothesized variance

H0 : σ² ≥ σ²₀ versus Ha : σ² < σ²₀

When testing whether the population variance is lower than the hypothesized variance

Two-tailed tests:

H0 : σ² = σ²₀ versus Ha : σ² ≠ σ²₀

When testing whether the population variance is different from the hypothesized variance

Question 27

Hypothesis tests for testing the variance of a normally distributed population involve the use of the chi‐square distribution, where the test statistic is denoted as χ2. Three important features of the chi‐square distribution are:

Answer 27

• It is asymmetrical. 

• It is bounded by zero. Chi‐square values cannot be negative. 

• It approaches the normal distribution in shape as the degrees of freedom increase. 


Question 28

The chi‐square test statistic is calculated as:

Answer 28

Question 29

When is the F-Test used?

Answer 29

Hypotheses related to the equality of the variance of two populations are tested with an F‐ test. This test is used under the assumptions that:

• The populations from which samples are drawn are normally distributed. 

• The samples are independent. 


Question 30

F-test: Hypothesis tests concerning the variance of two populations can be structured as one‐tailed or two‐tailed tests: 


Answer 30

One__‐__tailed tests: 


H₀ : σ₁² ≤ σ₂ 2 versus H_a : σ₁² > σ₂²

H₀ : σ₁² ≤ σ₂² versus H_a : σ₁² < σ₂² 


Two__‐__tailed tests: 


H₀ : σ₁² = σ₂² versus H_a : σ₁² =/ σ₂² 


σ₁² = variance of Population 1

σ₂² = variance of Population 2

Question 31

The test statistic for the F‐test is given by:

Answer 31

Question 32

Features of the F‐distribution:

Answer 32

• It is skewed to the right. 

• It is bounded by zero on the left. 

• It is defined by two separate degrees of freedom. 


Question 33

Which hypothesis tests do you use concerning the variance?

Answer 33

Variance of a single, normally distributed population

Chi‐square stat

Equality of variance of two independent, normally distributed populations

F‐stat

Question 34

A parametric test has at least one of the following two characteristics:

Answer 34

It is concerned with parameters, or defining features of a distribution. 


It makes a definite set of assumptions. 


Question 35

Non-parametric test:

Answer 35

A non‐parametric test is not concerned with a parameter, and makes only a minimal set of assumptions regarding the population. 


Question 36

non-parametric tests are used when: 


Answer 36

• The researcher is concerned about quantities other than the parameters of the distribution. 

• The assumptions made by parametric tests cannot be supported.
• When the data available is ranked (ordinal measurement scale). For example, 
non‐parametric methods are widely used for studying populations such as movie reviews, which receive one to five stars based on people’s preferences. 


Question 37

Spearman rank correlation

Answer 37

The Spearman rank correlation coefficient is a non‐parametric test that is calculated based on the ranks of two variables within their respective data sets. It lies between –1 and +1 where –1 (+1) denotes a perfectly inverse (positive) relationship between the ranks of the two variables, and 0 represents no correlation. 


Question 38

Nonparametric Alternatives to Parametric Tests



Answer 38

Question 39

Estimation

Answer 39

With reference to statistical inference, the subdivision dealing with estimating the value of a population parameter.

Question 40

Hypothesis testing

Answer 40

With reference to statistical inference, the subdivision dealing with the testing of hypotheses about one or more populations.

Question 41

Hypothesis

Answer 41

With reference to statistical inference, a statement about one or more populations.

Question 42

Steps in Hypothesis Testing. The steps in testing a hypothesis are as follows:

Answer 42

1) Stating the hypotheses.
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.

Question 43

Null Hypothesis

Answer 43

Definition of Null Hypothesis. The null hypothesis is the hypothesis to be tested. For example, we could hypothesize that the population mean risk premium for Canadian equities is less than or equal to zero.

Question 44

Alternative hypothesis

Answer 44

Definition of Alternative Hypothesis. The alternative hypothesis is the hypothesis accepted when the null hypothesis is rejected. Our alternative hypothesis is that the population mean risk premium for Canadian equities is greater than zero.

Question 45

Two-sided hypothesis test

Answer 45

A test in which the null hypothesis is rejected in favor of the alternative hypothesis if the evidence indicates that the population parameter is either smaller or larger than a hypothesized value.

Question 46

Two-tailed hypothesis test

Answer 46

A test in which the null hypothesis is rejected in favor of the alternative hypothesis if the evidence indicates that the population parameter is either smaller or larger than a hypothesized value.

Question 47

One-sided hypothesis test

Answer 47

A test in which the null hypothesis is rejected only if the evidence indicates that the population parameter is greater than (smaller than) θ₀. The alternative hypothesis also has one side

Question 48

One-tailed hypothesis test

Answer 48

A test in which the null hypothesis is rejected only if the evidence indicates that the population parameter is greater than (smaller than) θ₀. The alternative hypothesis also has one side.

Question 49

Risk premium

Answer 49

An extra return expected by investors for bearing some specified risk.

Question 50

Test statistic

Answer 50

A test statistic is a quantity, calculated based on a sample, whose value is the basis for deciding whether or not to reject the null hypothesis.

Question 51

Type I Error

Answer 51

The error of rejecting a true null hypothesis.

Question 52

Type II Error

Answer 52

The error of not rejecting a false null hypothesis.

Question 53

Level of significance

Answer 53

The probability of a Type I error in testing a hypothesis.

For example, a level of significance of 0.05 for a test means that there is a 5 percent probability of rejecting a true null hypothesis. The probability of a Type II error is denoted by the Greek letter beta, β.

Question 54

Power of the test

Answer 54

The probability of correctly rejecting the null—that is, rejecting the null hypothesis when it is false.

Question 55

Statistically significant

Answer 55

A result indicating that the null hypothesis can be rejected; with reference to an estimated regression coefficient, frequently understood to mean a result indicating that the corresponding population regression coefficient is different from 0.

Question 56

Rejection Point (Critical Value) for the Test Statistic

Answer 56

A rejection point (critical value) for a test statistic is a value with which the computed test statistic is compared to decide whether to reject or not reject the null hypothesis.

Question 57

Definition of p-Value

Answer 57

The p-value is the smallest level of significance at which the null hypothesis can be rejected.

Question 58

Argument against the merits of p-value approach

Answer 58

The P value approach does not necessarily force us to make a decision about the null hypothesis. If we obtain a P value of, say, 0.000001, we will almost certainly want to reject the null. But if we obtain a P value of, say, 0.04, or even 0.004, we are not obliged to reject it. We may simply file the result away as information that casts some doubt on the null hypothesis, but that is not, by itself, conclusive. We believe that this somewhat agnostic attitude toward test statistics, in which they are merely regarded as pieces of information that we may or may not want to act upon, is usually the most sensible one to take.”

Question 59

The formula for the variance of a t-distribution is:

Answer 59

df/(df − 2)

Question 60

How can a test be robust?

Answer 60

The quality of being relatively unaffected by a violation of assumptions.

Question 61

Paired observations

Answer 61

Observations that are dependent on each other.

Question 62

Paired comparison test

Answer 62

A statistical test for differences based on paired observations drawn from samples that are dependent on each other.

Question 63

Parametric test

Answer 63

Any test (or procedure) concerned with parameters or whose validity depends on assumptions concerning the population generating the sample.

Question 64

Nonparametric test

Answer 64

A test that is not concerned with a parameter, or that makes minimal assumptions about the population from which a sample comes.

Question 65

Spearman rank correlation

Answer 65

A measure of correlation applied to ranked data.

Question 66

Hypothesis

Answer 66

A hypothesis is a statement about one or more populations.

Question 67

The steps in testing a hypothesis are as follows:

Answer 67

1) Stating the hypotheses.
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.

Question 68

What are the two hypotheses we state?

Answer 68

We state two hypotheses: The null hypothesis is the hypothesis to be tested; the alternative hypothesis is the hypothesis accepted when the null hypothesis is rejected.

Question 69

There are three ways to formulate hypotheses:

Answer 69

• 1) H*₀: θ = θ₀ versus H_a: θ ≠ θ₀
• 2) H*₀: θ ≤ θ₀ versus H_a: θ > θ₀
• 3) H*₀: θ ≥ θ₀ versus H_a: θ < θ₀

where θ₀ is a hypothesized value of the population parameter and θ is the true value of the population parameter. In the above, Formulation 1 is a two-sided test and Formulations 2 and 3 are one-sided tests.

Question 70

A “suspected” or “hoped for” condition vs neutral attitude when proving a hypothesis. Explain.

Answer 70

When we have a “suspected” or “hoped for” condition for which we want to find supportive evidence, we frequently set up that condition as the alternative hypothesis and use a one-sided test. To emphasize a neutral attitude, however, the researcher may select a “not equal to” alternative hypothesis and conduct a two-sided test.

Question 71

Test statistic

Answer 71

A test statistic is a quantity, calculated on the basis of a sample, whose value is the basis for deciding whether to reject or not reject the null hypothesis. To decide whether to reject, or not to reject, the null hypothesis, we compare the computed value of the test statistic to a critical value (rejection point) for the same test statistic.

Question 72

In reaching a statistical decision, we can make two possible errors:

Answer 72

We may reject a true null hypothesis (a Type I error), or we may fail to reject a false null hypothesis (a Type II error).

Question 73

Level of significance of a test (Hypothesis Testing)

Answer 73

The level of significance of a test is the probability of a Type I error that we accept in conducting a hypothesis test. The probability of a Type I error is denoted by the Greek letter alpha, α. The standard approach to hypothesis testing involves specifying a level of significance (probability of Type I error) only.

Question 74

Power of a test

Answer 74

The power of a test is the probability of correctly rejecting the null (rejecting the null when it is false).

Question 75

A decision rule consists of what?

Answer 75

A decision rule consists of determining the rejection points (critical values) with which to compare the test statistic to decide whether to reject or not to reject the null hypothesis. When we reject the null hypothesis, the result is said to be statistically significant.

Question 76

What does the (1 − α) confidence interval represent?

Answer 76

The (1 − α) confidence interval represents the range of values of the test statistic for which the null hypothesis will not be rejected at an α significance level.

Question 77

Statistical vs economic decision

Answer 77

The statistical decision consists of rejecting or not rejecting the null hypothesis. The economic decision takes into consideration all economic issues pertinent to the decision.

Question 78

P-value

Answer 78

The p-value is the smallest level of significance at which the null hypothesis can be rejected. The smaller the p-value, the stronger the evidence against the null hypothesis and in favor of the alternative hypothesis. The p-value approach to hypothesis testing does not involve setting a significance level; rather it involves computing a p-value for the test statistic and allowing the consumer of the research to interpret its significance.

Question 79

For hypothesis tests, when do we use the t-statistic and when do we use the z-statistic?

Answer 79

For hypothesis tests concerning the population mean of a normally distributed population with unknown (known) variance, the theoretically correct test statistic is the t-statistic (z-statistic). In the unknown variance case, given large samples (generally, samples of 30 or more observations), the z-statistic may be used in place of the t-statistic because of the force of the central limit theorem.

Question 80

Describe the t-distribution and compare it to the standard normal distribution

Answer 80

The t-distribution is a symmetrical distribution defined by a single parameter: degrees of freedom. Compared to the standard normal distribution, the t-distribution has fatter tails.

Question 81

Hypothesis Testing: When we want to test whether the observed difference between two means is statistically significant, what do we do?

Answer 81

When we want to test whether the observed difference between two means is statistically significant, we must first decide whether the samples are independent or dependent (related). If the samples are independent, we conduct tests concerning differences between means. If the samples are dependent, we conduct tests of mean differences (paired comparisons tests).

Question 82

How do we conduct a test of the difference between two population means from normally distributed populations with unknown variances and we CAN assume the variances are equal?

Answer 82

When we conduct a test of the difference between two population means from normally distributed populations with unknown variances, if we can assume the variances are equal, we use a t-test based on pooling the observations of the two samples to estimate the common (but unknown) variance. This test is based on an assumption of independent samples.

Question 83

How do we conduct a test of the difference between two population means from normally distributed populations with unknown variances and we CANNOT assume the variance are equal?

Answer 83

When we conduct a test of the difference between two population means from normally distributed populations with unknown variances, if we cannot assume that the variances are equal, we use an approximate t-test using modified degrees of freedom given by a formula. This test is based on an assumption of independent samples.

Question 84

In tests concerning two means based on two samples that are not independent, what can be done?

Answer 84

In tests concerning two means based on two samples that are not independent, we often can arrange the data in paired observations and conduct a test of mean differences (a paired comparisons test). When the samples are from normally distributed populations with unknown variances, the appropriate test statistic is a t-statistic. The denominator of the t-statistic, the standard error of the mean differences, takes account of correlation between the samples.

Question 85

Chi-square

Answer 85

In tests concerning the variance of a single, normally distributed population, the test statistic is chi-square (χ²) with n − 1 degrees of freedom, where n is sample size.

Question 86

When do we use the F-test?

Answer 86

For tests concerning differences between the variances of two normally distributed populations based on two random, independent samples, the appropriate test statistic is based on an F-test (the ratio of the sample variances).

Question 87

F-statistic

Answer 87

The F-statistic is defined by the numerator and denominator degrees of freedom. The numerator degrees of freedom (number of observations in the sample minus 1) is the divisor used in calculating the sample variance in the numerator. The denominator degrees of freedom (number of observations in the sample minus 1) is the divisor used in calculating the sample variance in the denominator. In forming an F-test, a convention is to use the larger of the two ratios, S₁²/S₂² or S₂²/S₁², as the actual test statistic.

Question 88

Parametric test

Answer 88

A parametric test is a hypothesis test concerning a parameter or a hypothesis test based on specific distributional assumptions. In contrast, a nonparametric test either is not concerned with a parameter or makes minimal assumptions about the population from which the sample comes.

Question 89

Nonparametric test

Answer 89

A nonparametric test is primarily used in three situations: when data do not meet distributional assumptions, when data are given in ranks, or when the hypothesis we are addressing does not concern a parameter.

Question 90

Spearman rank

Answer 90

The Spearman rank correlation coefficient is calculated on the ranks of two variables within their respective samples.