MODULE 8.2: Types of Hypothesis Tests

Question 1

What are the null and alternative hypotheses for testing if a mean daily portfolio return is not equal to zero?

Answer 1

Answer:* H0: µ0 = 0 versus Ha: µ0 ≠ 0

Question 2

When do we use a t-distribution versus a z-distribution for testing a population mean?

Answer 2

Use a t-distribution for smaller sample sizes or when the population standard deviation is unknown. For large sample sizes, either t or z can be used.

Question 3

Question: How do you calculate the test statistic for a population mean?

Answer 3

• Answer: Test statistic = (Sample Mean - Hypothesized Mean) / (Standard Error)

Question 4

Question: What is the decision rule for a two-tailed test with a 5% significance level?

Answer 4

• Answer: Reject H0 if the test statistic < – 1.96 or > +1.96.

Question 5

Question: Calculate the standard error for a sample with a standard deviation of 0.25% and n = 250.

Answer 5

• Answer: SE = 0.25% / √250 = 0.0158%

Question 6

Question: What are the assumptions for a t-test for differences between means?

Answer 6

• Answer: Samples are independent, populations are normally distributed, and variances are equal.

Question 7

Question: What is the null hypothesis for comparing means of two populations?

Answer 7

• Answer: H0: µ1 – µ2 = 0 versus Ha: µ1 – µ2 ≠ 0

Question 8

Question: When should you reject the null hypothesis for a t-test comparing means at a 5% significance level?

Answer 8

• Answer: Reject H0 if the t-statistic < – 1.980 or > +1.980.

Question 9

Question: What is the interpretation if the t-statistic = –5.474 and the critical value = 1.980?

Answer 9

• Answer: Reject H0. There is a statistically significant difference between the two means.

Question 10

Question: What test is used for dependent samples?

Answer 10

• Answer: A paired comparisons t test OR paired T test

Question 11

Question: What is the null hypothesis for a paired comparisons test?

Answer 11

• Answer: H0: µd = µdz (commonly µdz = 0)

the mean difference between the paired values is zero

Question 12

Question: How is the test statistic for paired differences calculated?

Answer 12

• Answer: Test statistic = (Mean of Differences) / (Standard Error of Differences)

Question 13

Question: If the t-statistic = 10.26 and the critical value = 2.024, what is the conclusion?

Answer 13

• Answer: Reject H0. There is a statistically significant difference between the means.

Question 14

Question: What test is used for hypothesis testing of a single population variance?

Answer 14

• Answer: Chi-squared test.

Question 15

Question: What are the hypotheses for a two-tailed test of a single population variance?

Answer 15

• Null: This states that the population variance is equal to the specified value.
• Alternate: The population variance is not equal to the specified value
• Chi squared distribution
• Answer: H0: σ2 = σ0² versus Ha: σ2 ≠ σ0²

Question 16

Question: If the test statistic = 20.76 and the critical value is not exceeded, what is the conclusion?

Answer 16

• Answer: Fail to reject H0. There is no significant difference between the variances.

Question 17

Question: What test is used for comparing variances of two populations?

Answer 17

• Answer: F-test.

Question 18

Question: What are the hypotheses for an F-test?

Answer 18

• Answer:
• H0: σ1² = σ2² versus Ha: σ1² ≠ σ2²
• H0: σ1² ≤ σ2² versus Ha: σ1² > σ2²

Question 19

Question: How is the F-statistic calculated?

Answer 19

• Answer: F = S1² / S2² (larger variance on top).

Question 20

Question: If F = 1.28 and the critical value = 1.94, what is the conclusion?

Answer 20

• Answer: Fail to reject H0. The variances are not significantly different.

Question 21

Question: What is the key difference between parametric and nonparametric tests?

Answer 21

• Answer: Parametric tests rely on distribution assumptions and population parameters. Nonparametric tests do not require these assumptions.

Question 22

Z-Test
- meaning
- null / alternate hypothesis
- when are they used

Answer 22

Zealously Known Variance

Meaning: Tests whether the population mean (𝜇) is equal to a hypothesized value.

Key Points:
Used when the population variance is known and the sample size is large (𝑛>30).

Hypothesis:
Null: 𝜇 = 𝜇0
alternate 𝜇 NOT = 𝜇0

Question 23

t-Test
- meaning
- null / alternate hypothesis
- when are they used

Answer 23

Tiny Sample or Unknown Variance

Meaning: Tests whether the population mean (𝜇) is equal to a hypothesized value when the population variance is unknown.

Key Points:
Used when the sample size is small (n<30).

Hypothesis:
Null: 𝜇 = 𝜇0
alternate 𝜇 NOT = 𝜇0

Question 24

Chi-Squared Test
- meaning
- null / alternate hypothesis
- when are they used

Answer 24

Checking Variance

Meaning: Tests whether the population variance (𝜎squared) is equal to a hypothesized value

Use the chi-squared distribution with n−1 degrees of freedom.

Question 25

F-Test
- meaning
- null / alternate hypothesis
- when are they used

Answer 25

Fair Fight Between Variances

Question 26

t-Test for Equality of Means (Independent Samples)

• meaning
• null / alternate hypothesis
• when are they used

Answer 26

Two Groups, Two Means

df = n1 + n2 -2

Question 27

6. Paired t-Test (Dependent Samples)

• meaning
• null / alternate hypothesis
• when are they used

Answer 27

Pairs of t’s Stay Together. deliberate link between first / second observations (like before / after people took the drug or studied for a test). df is still n-1

Question 28

7. ANOVA (Analysis of Variance)
   - meaning
   - null / alternate hypothesis
   - when are they used

Answer 28

All Groups Over Variance

Question 29

8. Z-Test for Proportions
   - meaning
   - null / alternate hypothesis
   - when are they used

Answer 29

Zeroing in on Proportions

Question 30

9. Chi-Squared Test for Independence
   - meaning
   - null / alternate hypothesis
   - when are they used

Answer 30

Cross-Tab Independence

Question 31

define and interpret the P value

Answer 31

P value of a test is the probability of getting the test statistic (or a result more extreme) if the null were true

P value < significance level = REJECT NULL

Question 32

How are significance and p value related
what if the p .0213 and the significance levels are 5%, 3% , and 1%

Answer 32

p value is the smallest level of significance at which the null can be rejected

5% > 2.13% so reject the null
3% > 2.13% so reject the null
1% < 2.13% so fail to reject the null

Question 33

What does a chi squared distribution look like

Answer 33

all the values are positive and they start after zero, and it’s skewed to the right

Question 34

Chi squared test statistic calculation

Answer 34

(n-1)var^2 / hyp variance