1.6 Hypothesis Testing

Question 1

Hypothesis testing

Answer 1

used to determine whether a sample statistic is likely from a population with the hypothesized value of the population parameter

aims to provide an insight to this question by examining how a sample statistic describes a population parameter

Question 2

Hypothesis testing

Answer 2

used to determine whether a sample statistic is likely from a population with the hypothesized value of the population parameter

aims to provide an insight to this question by examining how a sample statistic describes a population parameter

Question 3

hypothesis

Answer 3

a statement about one or more populations tested using sample statistics

Question 4

The steps in the hypothesis testing process

Answer 4

1. State the hypotheses.
2. Identify the appropriate test statistic.
3. Specify the level of significance.
4. State the decision rule.
5. Collect data and calculate the test statistic.
6. Make a decision.

Question 5

The two hypotheses always stated:

Answer 5

Null hypothesis: H0

Alternative hypothesis: Ha

Question 6

Null hypothesis: H0

Answer 6

This is assumed true until the test proves otherwise.

Question 7

Alternative hypothesis: Ha

Answer 7

This is only accepted if there is sufficient evidence to reject the null hypothesis

Question 8

a two-sided hypothesis test

Answer 8

two-tailed

ex:

H0: μ = 10%

Ha: μ ≠ 10%

This is a two-sided hypothesis test because the null hypothesis will be rejected if the sample mean return is significantly different from 10%.

–> It could be a lot greater than or less than 10%, so it is a two-tailed test

Question 9

a one-sided hypothesis test

Answer 9

one-tailed

ex:

H0: μ ≤ 10%

Ha: μ > 10%

This is a one-sided hypothesis test because the null hypothesis will be rejected only if the sample mean return is significantly greater than 10%

It does not matter if the sample mean is a lot smaller than 10%

The analyst is only interested in whether the population mean is greater than 10% (instead of being different from 10%).
–> Therefore, it is a one-tailed test

Question 10

the two important rules in forming the hypotheses:

Answer 10

1. The null and alternative hypotheses should be mutually exclusive (i.e., do not overlap) and collectively exhaustive (i.e., cover all possibilities)
2. The null hypothesis includes the point of equality (i.e., H0 always contains an equal sign).

Question 11

The choice of null and alternative hypotheses should be based on what?

Answer 11

should be based on the hoped-for condition.

For example:

if an analyst is attempting to show that the mean annual return of a stock index has exceeded 10%, the null hypothesis (H0) should be that the mean return is less than or equal to 10%.

The alternative hypothesis (Ha) should only be accepted if statistical tests provide sufficient evidence that the mean return is not less than or equal to 10%

Question 12

The test statistic

Answer 12

the quantity calculated from the sample used to evaluate the hypothesis

can be calculated as follows:

z = (X¯ − μ0) / (σ/√nz)

X¯: Sample mean

μ0: Hypothesized mean

σ: Population standard deviation

n: Sample size

Question 13

The null hypothesis can be rejected or not rejected after the test statistic has been calculated.

The decision is based on what?

Answer 13

based on a comparison that assumes a specific significance level, which establishes how much evidence is required to reject the null hypothesis

Question 14

the four possible outcomes when we see whether a null hypothesis is to be rejected or not

Answer 14

Decision: Do not reject H0

–> H0 is True: Correct Decision

–> H0 is False: Type II Error (False negative)

Decision: reject H0

–> H0 is True: Type I Error (False positive)

–> H0 is False: Correct decision

Question 15

A Type I error

Answer 15

occurs if a true null hypothesis is mistakenly rejected

Question 16

A Type II error

Answer 16

occurs if a false null hypothesis is mistakenly accepted

Question 17

The probability of a Type I error

Answer 17

the level of significance of the test, which is denoted as α

Question 18

the confidence level

Answer 18

The complement of the level of significance of the test

1− α

The complement of the probability of a Type I error

Question 19

The probability of a Type II error

Answer 19

hard to quantify but is symbolized as β

Question 20

the power of a test

Answer 20

The complement of the probability of a Type II error

his is the probability of correctly rejecting the false null hypothesis

The power equals 1 − β

Question 21

explain the power of Test matrix

Answer 21

Decision:

Do not reject H0:

–> Ho is True: Confidence level (1 − α)
–> Ho is False:

reject H0:

–> Ho is True: Level of significance α
–> Ho is False: the power of test 1 − β

Question 22

It is common to set the level of significance to be at which levels

Answer 22

10%, 5%, and 1%.

Question 23

when must the decision rule be stated?

Answer 23

when comparing the test statistic’s calculated value to a given value based on the significance level of the test

Question 24

The critical value of the test statistic

Answer 24

the rejection point of the null hypothesis

Question 25

The critical value of the test statistic is based on what?

Answer 25

based on the level of significance and the probability distribution associated with the test statistic

Question 26

when is the null hypothesis rejected?

Answer 26

when the test statistic is calculated to be more extreme than the critical value(s)

Question 27

when is a result known to be statistically significant?

Answer 27

when the test statistic is calculated to be more extreme than the critical value(s)

when the null hypothesis is rejected

Question 28

For a two-tailed test, which are the two ways to reject the null hypothesis?

Answer 28

1. The sample/estimator is significantly smaller than the hypothesized value of the population parameter.
2. The sample/estimator is significantly larger than the hypothesized value of the population parameter

In order to capture these two rejection regions, two critical values are needed

Question 29

how do we find the two critical values needed to reject a two-tailed test?

Answer 29

the z-distribution (i.e., standard normal distribution) is used

–> the values would correspond to ±zα/2

Question 30

The confidence interval

Answer 30

Other than using critical values, which are based on the stated level of significance, the decision of a hypothesis test can be determined based on the confidence interval

The confidence interval explains what hypothesized values are acceptable so that the null hypothesis will not be rejected

Question 31

there are two ways to make a decision (whether to reject the null hypothesis) in a two-sided hypothesis test. Explain the following:

Metric: Critical Values

Explain the procedure and the decision

Answer 31

Procedure: Compare the test statistic with the critical values

Decision: Reject the null hypothesis if the test statistic is less than the lower critical value or greater than the upper critical value

Question 32

there are two ways to make a decision (whether to reject the null hypothesis) in a two-sided hypothesis test. Explain the following:

Metric: confidence interval

Explain the procedure and the decision

Answer 32

Procedure: Compare the hypothesized value of the population parameter with the confidence interval

Decision: Reject the null hypothesis if the hypothesized value falls outside of the confidence interval

Question 33

A statistical decision

Answer 33

can be made based on the conclusion of the hypothesis test

For example, an investor may perform a test on the null hypothesis (based on market consensus) that the mean return of a security is no greater than 5%.

–> If the test statistic exceeds the critical value, the investor makes the statistical decision to reject the null hypothesis. The mean return is expected to be greater than 5%.

Question 34

An economic decision

Answer 34

made based on the statistical decision

For example, if the investor rejects the null hypothesis that the mean return is no greater than 5% (in favor of the alternative hypothesis that the mean return is greater than 5%), the investor may consider investing in this security as it is currently undervalued.

Different economic factors such as the risk tolerance and the financial positions should be considered

Question 35

Statistically Significant but Not Economically Significant decision?

Answer 35

Statistical significance does not necessarily imply economic significance.

For example, if a security is only slightly undervalued (test statistic only slightly exceeds the critical value), transaction costs and other fees could prevent successful implementation.

Question 36

Which of the following statements is most accurate? A Type I error:

A
occurs when a false null hypothesis is not rejected.

B
is less likely to be committed if the level of significance is lowered.

C
is less likely to be committed if the likelihood of committing a Type II error is reduced.

Answer 36

B
is less likely to be committed if the level of significance is lowered.