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三、 Investment Tools: Quantitative Methods
1.A: Sampling and Estimation
a: Define simple random sampling.
Simple random sampling is a method of selecting a sample in such a way that each item or person in the population begin studied has the same (non-zero) likelihood of being included in the sample. This is the standard sampling design.
b: Define and interpret sampling error.
Sampling error is the difference between a sample statistic (the mean, variance, or standard deviation of the sample) and its corresponding population parameter (the mean, variance or standard deviation of the population).
The sampling error of the mean = sample mean - population mean = X bar -µ.
c: Define a sampling distribution
The sample statistic itself is a random variable, so it also has a probability distribution. The sampling distribution of the sample statistic is a probability distribution made up of all possible sample statistics computed from samples of the same size randomly drawn from the same population, along with their associated probabilities.
d: Distinguish between simple random and stratified random sampling.
Simple random sampling is where the observations are drawn randomly from the population. In a random sample each observation must have the same chance of being drawn from the population. This is the standard sampling design.
Stratified random sampling first divides the population into subgroups, called strata, and then a sample is randomly selected from each stratum. The sample drawn can be either a proportional or a non-proportional sample. A proportional sample requires that the number of items drawn from each stratum be in the same proportion as that found in the population.
e: Distinguish between time-series and cross-sectional data.
A time-series is a sample of observations taken at a specific and equally spaced points in time. The monthly returns on Microsoft stock from January 1990 to January 2000 are an example of time-series data.
Cross-sectional data is a sample of observations taken at a single point in time. The sample of reported earnings per share of all Nasdaq companies as of December 31, 2000 is an example of cross-sectional data.
f: State the central limit theorem and describe its importance.
The central limit theorem tells us that for a population with a mean µ and a finite variance σ2, the sampling distribution of the sample means of all possible samples of size n will be approximately normally distributed with a mean equal to µ and a variance equal to σ2/n.
The central limit theorem is extremely useful because the normal distribution is relatively easy to work with when doing hypothesis testing and forming confidence intervals. We can make very specific inferences about the population mean, using the sample mean, no matter the distribution of the population, as long as the sample size is "large."
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