Statistical Distributions, Monte Carlo Simulation, and Hypothesis Testing, Exams of Nursing

Download Statistical Distributions, Monte Carlo Simulation, and Hypothesis Testing and more Exams Nursing in PDF only on Docsity! CFA Level 1- Quantitative Methods Part 2 Questions and Answers Exam. Cumulative Distribution Function (CDF) - Correct Answer distribution function, defines the probability that a random variable, X, takes on a value equal to or less than a specific value, x. Represents the sum, or cumulative value, of the probabilities of the outcomes up to and including a specific outcome. F(x) = P(X ≤ x) Return on equity for a firm is defined as a continuous distribution over the range -20% to 30% and had a cumulative distribution function of F(x)=(x+20)/50. Calculate the probability ROE is between 0 and 15% - Correct Answer P(0 ≤ x ≤ 15) = F(15) - F(0) F(15) = (15 + 20) / 50 = 0.70 F(0) = (0 + 20) / 50 = 0.40 F(15) - F(0) = 0.70 - 0.40 = 0.30 = 30% Discrete Uniform Random Variable - Correct Answer the probabilities of all possible outcomes for a discrete random variable are equal Determine p(6), F(6), and P(2 ≤ X ≤ 8) for the discrete uniform distribution function defined as: X = {2, 4, 6, 8, 10}, p(x) = 0.2 - Correct Answer p(6) = 0.2, since p(x) = 0.2 for all x. F(6) = P(X ≤ 6) = np(x) = 3(0.2) = 0.6. Note that there are 3 possible values for x that are less than or equal to 6 P(2 ≤ X ≤ 8) = 4(0.2) = 0.8. Note that k = 4, since there are four outcomes in the range 2 ≤ X ≤ 8. continuous uniform distribution - Correct Answer defined over a range that spans between some lower limit, a, and some upper limit b, which serves as the parameters of the distribution. properties of continuous uniform distribution - Correct Answer For all a ≤ x1 < x2 ≤ b (i.e., for all x1 and x2 between the boundaries a and b). P(X < a or X > b) = 0 (i.e., the probability of X outside the boundaries is zero). P(x1 ≤ X ≤ x2) = (x2 − x1)/(b − a). This defines the probability of outcomes between x1 and x2. X is uniformly distributed between 2 and 12. Calculate the probability that X will be between 4 and 8. - Correct Answer (8-4)/(12-2)=40% Binomial Random Variable - Correct Answer Variable may be defined as the number of successes in a given number of trials where the outcome can be either a success or failure; Bernoilli random variable - Correct Answer A binomial random variable for which the number of trials is 1 Binomial Probability Formula - Correct Answer P(x)= (nCx) (p^x) (1-p)^n-x assuming a binomial distribution, compute the probability of drawing three black beans from a bowl of black and white beans when p(black bean)=.6. You will draw 5 black beans from the bowl - Correct Answer p(x=3)= 3Cr5*(.6)^3*(.4)^(5-3) Expected Value of a binomial random variable Variance of a binomial random variable - Correct Answer E(x) = np Var(x)=np(1-p) Based on empirical data, the probability that the Dow Jones Industrial Average (DJIA) will increase on any given day has been determined to equal 0.67. Assuming that the only other outcome is that it decreases, we can state p(UP) = 0.67 and p(DOWN) = 0.33. Further, assume that movements in the DJIA are independent (i.e., an increase in one day is independent of what happened on another day). Using the information provided, compute the expected value of the number of up days in a 5-day period. - Correct Answer E(X | n = 5, p = 0.67) = (5)(0.67) = 3.35 key properties of a normal distribution - Correct Answer It is completely described by its mean, μ, and variance, Skewness = 0, mean = median = mode Kurtosis = 3; this is a measure of how flat the distribution is. Recall that excess kurtosis is measured relative to 3, the kurtosis of the normal distribution. A linear combination of normally distributed random variables is also normally distributed. The probabilities of outcomes further above and below the mean get smaller and smaller but do not go to zero (the tails get very thin but extend infinitely). step 2: (Rcc) is 20% / 2 = 10%. t distribution - Correct Answer A bell shaped distribution symmetrical about its median used to make confidence intervals with small samples (<30) and unknown population variance; Degrees of freedom = n - 1 t distribution characteristics - Correct Answer symmetric, defined by a single parameter (df) where df=n-1, centered at zero; more probability in the tails (fatter tails); as degrees of freedom increase, the t distribution approaches normal (becomes more spiked with thinner tails) t distribution is ______ than a normal distribution - Correct Answer flatter ticker tails= - Correct Answer more outliers chi-square distribution - Correct Answer a family of distributions each based on degrees of freedom. It is the distribution of the sum of squared values of n random variables and k where df=n-1 chi-square distribution characteristics - Correct Answer bounded from below by 0, asymmetric, symmetry increases with df, as fd gets largers the shape approaches normal distribution chi-square distribution is most often used in tests of - Correct Answer the value of the variance of a normally distributed population F distribution - Correct Answer distribution of the quotient of two appropriately scaled independent chi square variables with df of m and n. F=(x^2/m)/(x^2/n) F distribution is used to determine - Correct Answer the probability that the variances of two independent normal distributions are equal F-distribution characteristics - Correct Answer Bounded from below by 0, asymmetric, as df increases the shape approaches normal Monte Carlo Simulation - Correct Answer a technique based on the repeated generation of one or more risk factors that affect security values, in order to generate a distribution of security values. Monte Carlo Simulation is used for - Correct Answer -Value complex securities. -Simulate the profits/losses from a trading strategy. -Calculate estimates of value at risk (VaR) to determine the riskiness of a portfolio of assets and liabilities. -Simulate pension fund assets and liabilities over time to examine the variability of the difference between the two. -Value portfolios of assets that have nonnormal returns distributions. limitations of monte carlo simulation - Correct Answer it is fairly complex and will provide answers that are no better than the assumptions about the distributions of the risk factors and the pricing/valuation model that is used. Also, simulation is not an analytic method, but a statistical one, and cannot provide the insights that analytic methods can probability sampling - Correct Answer refers to selecting a sample when we know the probability of each sample member in the overall population random sampling - Correct Answer each item is assumed to have the same probability of being selected simple random sampling - Correct Answer If we have a population of data and select our sample by using a computer to randomly select a number of observations from the population, each data point has an equal probability of being selected simple random sampling is useful for - Correct Answer If we want to estimate the mean profitability for a population of firms, this may be an appropriate method. Non-probability sampling is based on - Correct Answer low cost and easy access to some data items, or on using the judgment of the researcher in selecting specific data items. Less randomness in selection may lead to greater sampling error. sampling error of the mean - Correct Answer = sample mean - population mean = x - µ sampling distribution of the sample statistic is: - Correct Answer a probability distribution of all possible sample statistics computed from a set of equal-size samples that were randomly drawn from the same population. systematic sampling - Correct Answer selecting every nth member from a population Stratified random sampling - Correct Answer uses a classification system to separate the population into smaller groups based on one or more distinguishing characteristics. From each subgroup, or stratum, a random sample is taken and the results are pooled. The size of the samples from each stratum is based on the size of the stratum relative to the population. Stratified sampling is often used in - Correct Answer bond indexing Cluster sampling - Correct Answer based on subsets of a population, but in this case we are assuming that each subset (cluster) is representative of the overall population with respect to the item we are sampling. For example, we may have data on personal incomes for a state's residents by county. one-stage cluster sampling two-stage cluster sampling - Correct Answer -a random sample of clusters is selected and all the data in those clusters comprise the sample. -random samples from each of the selected clusters comprise the sample. Two-stage cluster sampling can be expected to have _____ sampling error than one- stage cluster sampling. - Correct Answer greater Convenience sampling - Correct Answer refers to selecting sample data based on its ease of access, using data that are readily available. Because such a sample is typically not random, sampling error will be greater. It is most appropriate for an initial look at the data prior to adopting a sampling method with less sampling error. Judgmental sampling - Correct Answer refers to samples for which each observation is selected from a larger data set by the researcher, based on her experience and judgment central limit theorem - Correct Answer The theory that, as sample size increases, the distribution of sample means of size n, randomly selected, approaches a normal distribution. Important properties of the central limit theorem - Correct Answer If the sample size n is sufficiently large (n ≥ 30), the sampling distribution of the sample means will be approximately normal. The mean of the population, µ, and the mean of the distribution of all possible sample means are equal. The variance of the distribution of sample means is the population variance divided by the sample size. standard error of the sample mean - Correct Answer the standard deviation of the distribution of the sample means. When the standard deviation of the population, σ, is known, the standard error of the sample mean is calculated as: - Correct Answer standard deviation of the population/ (square root n) The mean hourly wage for Iowa farm workers is $13.50 with a population standard deviation of $2.90. Calculate and interpret the standard error of the sample mean for a sample size of 30. - Correct Answer 2.9/(√30) Survivorship bias - Correct Answer - most common form of sample selection bias - only include data currently in existence "the survivors" and doesn't include data that as ceased to exist - solution is to use a sample of funds that all start at the same time and not drop funds that have been dropped from sample Look-ahead bias - Correct Answer occurs when a study tests a relationship using sample data that was not available on the test date. Time-period bias - Correct Answer time period over which data is gathered is too short or too long Hypothesis - Correct Answer statement about the value of a population parameter developed for the purpose of testing a theory or belief. hypothesis steps - Correct Answer 1.) state the hypothesis 2.) select appropriate test statistic 3.) specify the level of significance 4.) state the decision rule 5.) collect the sample and sample statistics 6.) make a decision on the hypothesis 7.) make a decision on the results of the test Null Hypothesis (H0) - Correct Answer the hypothesis that the researcher wants to reject include H0: µ = µ0, H0: µ ≤ µ0, and H0: µ ≥ µ0 the null hypothesis always includes what condition - Correct Answer the "equal to" Alternative Hypothesis (Ha) - Correct Answer what is concluded if there is sufficient evidence to reject the null hypothesis. two-tailed test - Correct Answer H0: µ = µ0 versus Ha: µ ≠ µ0 Reject H0 if: test statistic > upper critical value or test statistic < lower critical value one-tailed test - Correct Answer Upper tail: H0: µ ≤ µ0 versus Ha: µ > µ0, or Lower tail: H0: µ ≥ µ0 versus Ha: µ < µ0 test statistic for a hypothesis test - Correct Answer (sample statistic- hypothesized value)/standard error of the sample statistic standard error of the sample statistic for hypothesis testing - Correct Answer σ/√n when population st dev is known s/√n when population st dev is unknown Type 1 error (alpha) - Correct Answer Rejecting null hypothesis when it is true type 2 error - Correct Answer failing to reject a false null hypothesis probability of making a type I error - Correct Answer the significance level, α power of a test - Correct Answer probability of correctly rejecting the null hypothesis when it is false (1-P(type II error)) what determines the probability of a type II error? - Correct Answer sample size and choice of significance level decreasing the significance level will ____ the probability of a type II error - Correct Answer increase For a given significance level, we can decrease the probability of a Type II error and increase the power of a test, only by - Correct Answer increasing the sample size A researcher has gathered data on the daily returns on a portfolio of call options over a recent 250-day period. The mean daily return has been 0.1%, and the sample standard deviation of daily portfolio returns is 0.25%. The researcher believes that the mean daily portfolio return is not equal to zero. Construct a 95% confidence interval for the population mean daily return over the 250- day sample period. Construct a hypothesis test of the researcher's belief. - Correct Answer s/√n=0.25√250=0.0158% At the 5% level of significance, the critical z-values for the confidence interval are z = +- 1.96 0.1 - 1.96(0.0158) ≤ µ ≤ 0.1 + 1.96(0.0158), or 0.069% ≤ µ ≤ 0.131% Reject H0 if test statistic < -1.96 or test statistic > +1.96 test statistic = (sample statistic- hypothesized value)/standard error of the sample statistic = .001-0/(.25/√250)= 6.33 6.33 lies outside of range so reject null p-value - Correct Answer the probability of obtaining a test statistic that would lead to a rejection of the null hypothesis, assuming the null hypothesis is true. It is the smallest level of significance for which the null hypothesis can be rejected. Reject null hypothesis when... - Correct Answer the p-value is less than the significance level. incorrect rejections of the null hypothesis are known as - Correct Answer false positives (= to the significance level) t- test statistic for hypothesis testing when population variance is unknown - Correct Answer T(n-1)= (x−μ)/ (s/√n) z-test statistic for hypothesis testing when population variance is known - Correct Answer z= (x−μ)/ (σ/√n) A pooled variance is used when - Correct Answer testing the hypothesis that the means of two normally distributed populations are equal t- statistic formula (for a difference between the means of two populations) - Correct Answer t= (x1−x2)/√(s^2p/n1+ s^2p/n2) where s^2p= [(n1−1)s1^2*(n2−1)s2^2]/ (n1+n2−2) note df=n-2 If the sample means are very close together: - Correct Answer the t-statistic is small, and we do not reject equality. For the paired comparisons test, the t-statistic with n - 1 degrees of freedom is computed as: - Correct Answer t= (d−μ) / (sd) where d is sample mean difference chi square test statistic - Correct Answer Χ² = (n-1)s² / σ² F statistic is computed as - Correct Answer s1²/ s2² Parametric tests rely on - Correct Answer assumptions regarding the distribution of the population and are specific to population parameters. Nonparametric tests - Correct Answer Do not make any assumptions about the population and are used when parametric tests cannot be supported Situations where a nonparametric test is called for are the following: - Correct Answer a hypothesis test of the mean value for a variable that comes from a distribution that is not normal and is of small size so that neither the t-test nor the z-test is appropriate.