Quantitative Methods

Question 1

Central Limit Theorem

Answer 1

The central limit theorem states that for simple random samples of size n from a population with a mean µ and a finite variance σ^2, the sampling distribution of the sample mean x approaches a normal probability distribution with mean µ and a variance equal to (σ^2)/n as the sample size becomes large.

Question 2

Confidence interval (variance unknown) =

Answer 2

Question 3

Confidence interval (variance known) =

Answer 3

Question 4

Normal Distribution:

68% of observations fall within how many σ.

90% fall within how many σ.

95% fall within how many σ.

99% fall within how many σ.

Answer 4

68% of observations fall within ± 1σ.

90% fall within ± 1.65σ.

95% fall within ± 1.96σ.

99% fall within ± 2.58σ.

Question 5

Standard normal distribution reliability factors

90% confidence intervals

95% confidence intervals

99% confidence intervals

Answer 5

z_α/2 = 1.645 for 90% confidence intervals (the significance level is 10%, 5% in each tail).

z_α/2 = 1.960 for 95% confidence intervals (the significance level is 5%, 2.5% in each tail).

z_α/2 = 2.575 for 99% confidence intervals (the significance level is 1%, 0.5% in each tail).

Question 6

Chi-Square Test Statistic

Answer 6

The test of a hypothesis about the population variance for a normally distributed population uses a chi-square test statistic

Question 7

F-distributed test statistics

Answer 7

The test comparing two variances based on independent samples from two normally distributed populations uses an F-distributed test statistic: F=(S₁^{^2})/(S₂^{^2}), where (S₁^{^2}) is the variance of the first sample and (S₂^{^2}) is the (smaller) variance of the second sample.

Question 8

Effective annual yield

Answer 8

effective annual yield = EAY = (1 + HPY)^365/t – 1

Question 9

Bank discount yield

Answer 9

Question 10

Coefficient of Variation (CV)

Answer 10

Coefficient of variation expresses how much dispersion exists relative to the mean of a distribution and allows for direct comparison of the degree of dispersion across different data sets. It measures risk per unit of expected return.

CV = standard deviation / mean return

When comparing two investments using the CV criterion, the one with the lower CV is the better choice.

Question 11

Z-score

Answer 11

“Standardizes” observation from normal distributions; represents number of standard deviations a given observation is from population mean.

z = (Observation - population mean) / Standard deviation

Question 12

Standard Error

Answer 12

Question 13

Type I error

Type II error

Answer 13

• Type I error: rejection of null hypothesis when it is actually true.
• Type II error: failure to reject null hypothesis when it is actually false.

Question 14

Criteria for Selecting the Appropriate Test Statistic

Answer 14

Question 15

Leptokurtic, mesokurtic, and platykurtic

Answer 15

• A distribution that is more peaked than normal is called leptokurtic. Will have fatter tails
• A distribution that is neither more peaked nor less peaked than normal is called mesokurtic.
• A distribution that is less peaked than normal is called platykurtic.

Question 16

Steps in Hypothesis Testing

Answer 16

• State the hypothesis.
• Select a test statistic.
• Specify the level of significance.
• State the decision rule for the hypothesis.
• Collect the sample and calculate statistics.
• Make a decision about the hypothesis.
• Make a decision based on the test results.

Question 17

Calculate test statistic

Answer 17

Test statistic = ( Sample statistic − Value of the population parameter under H₀) / Standard error of the sample statistic

Question 18

Binomial Distribution

Answer 18

Question 19

A Priori Probability vs. Empirical Probability

Answer 19

• A Priori probability is one that is estimated ahead of time based on what you already know
• Empirical probability is based on experiments or historical data.

Question 20

Independent event vs. mutually exclusive event

Answer 20

• Independent event: P(X|Y) = P(X)
• Mutually exclusive event:
  
  • P(XY) = 0
  • P(X) + P(Y) = P(X or Y)

Question 21

Labeling problem

Answer 21

Labeling refers to the situation where there are n items that can each receive one of k different labels. The number of items that receives label 1 is n₁ and the number that receive label 2 is n₂, and so on, such that n₁ + n₂ + n₃ + … + n_k = n. The total number of ways that the labels can be assigned is:

n! / [(n₁!)×(n₂!)×…×(n_k!)]

Question 22

Parametric vs. Non-parametric test

Answer 22

• Parametric test: to estimate population parameters or based on assumptions about parameters
• Non-parametric test: other than for a parameter. eg: test of correlation between ranked observations

Question 23

Chebyshev’s inequality

Answer 23

• Minimum probability of an outcome within “k” standard deviations of the mean = 1 - (1 / k²)
• Applies to any distribution

Question 24

Addition Rule: P (X or Y)

Multiplication Rule: P (XY)

Answer 24

• P (X or Y) = P (X) + P (Y) - P (XY)
• P (XY) = P (X|Y) x P (Y)
  
  • = P (Y|X) x P (X)

Question 25

Measurement scale: NOIR

Answer 25

• Nominal: classified by characteristic
• Ordinal: can be arranged in order
• Interval: equal scale difference
• Ratio: zero indicates absence