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An overview of statistical concepts related to sampling distributions and hypothesis testing. It covers topics such as empirical and theoretical probability distributions, discrete and continuous distributions, population parameters, expected value and mean, variance and standard deviation, chebyshev's inequality, normal distribution, hypothesis testing with known and unknown standard errors, and the t distribution. The document also discusses the central limit theorem and the sampling distribution of sample means.
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Lecture
POLI 7962:
Seminar in Quantitative Political Analysis
August 2-6, 2007
I. Sample Estimation of Population Parameters
A. Introduction
a. Random sampling
a. Inference: the process of making generalizations or drawing conclusions about the attributes of a population based on evidence contained in a sample.
b. More broadly, an inference is the process of drawing conclusions about that which is not observed directly.
B. Probability Distributions
a. Empirical probability distribution : a probability distribution for a set of empirical observations.
b. Theoretical probability distribution : a probability distribution for a set of theoretical observations.
C. Describing Discrete Probability Distributions
a. The single outcome that best describes a probability distribution is its expected value, which is also the mean of the probability distribution:
E(Y) = Σ Y (^) ip(Y (^) i)
μ = E(Y)
b. If one is looking for a single number that (1) characterizes a given distribution, (2) minimizes the sum of deviations from the mean, and (3) represents the best “guess” of a randomly-selected number from a given distribution, the mean (i.e., expected value) is that number.
D. Chebycheff's Inequality Theorem
E. Normal Distributions
a. The shape of any given normal curve can be determined by two values: the population mean and variance.
b. The values of any normal distribution can easily be converted to Z-scores using the formula:
Z = (Y - μY ) / σY
F. The Central Limit Theorem
a. The central limit guarantees that a given sample mean can be made to come close to the population mean in value by simply choosing an N large enough, since the variance of the sample distribution of means becomes small as N gets larger.
b. Based on information pertaining to normal distributions, one should expect that 95% of all sample means will fall within 1.96 standard errors of the population mean.
G. Sample Point Estimates and Confidence Intervals
a. In general:
Y ± (Z (^) α/2)(σY )
b. For 95% confidence interval:
Y ± (1.96)(σY )
(1) The first step in hypothesis testing is to specify the null hypothesis and the alternate hypothesis. In testing hypotheses about μ, the null hypothesis (H0) is a hypothesized value of μ, usually associated with a null effect. The alternative hypothesis (H (^) 1) represents the research hypothesis.
(2) The second step is to choose a significance level, or a level of uncertainty that one is willing to accept in falsely rejecting the null hypothesis. Normally one should assume the .05 level is chosen.
(3) The third step is to establish the critical test statistic—in this case, a Z value--associated with the level of significance from step #2. This is going to be the compared to a test statistic that will determine whether the observed mean is sufficiently different from the null hypothesized mean justify rejection of the null hypothesis.
(4) The fourth step is to calculate the test Z score, using the following formula:
Z = (M - μH0 ) / σM
(5) In the fifth step, one compares the critical Z and test Z. If the test statistic is further away from 0 than the critical Z, then one rejects the null hypothesis and “accepts” the working hypothesis. Otherwise one is unable to reject the null hypothesis.
K. Hypothesis testing when the standard error is unknown: The t distribution
a. t distribution: a test statistic used with small samples selected from a normally distributed population or, for large samples, drawn from a population with any shape.
For hypothesis tests involving the mean, the following equation is used for calculating the critical t statistic:
t = (M - μH0 ) / s (^) M
a. For small samples (N < 100), the use of the t distribution to test hypotheses assumes that the sample is drawn from a normally distributed population. While this assumption appears to be restrictive, research has demonstrated that violations of this assumption have only minor effects on the computations of the test statistic. Therefore, unless there is evidence that the underlying population distribution is grossly non-normal, one can use the t test to test hypotheses even when N is small.
b. A t distribution for a given sample size has a larger variance than a normal Z distribution. Therefore, the standard error of a t distribution is larger than that of a normal Z distribution. However, as N becomes large (i.e. in the range of 100), a t distribution becomes increasingly similar to a normal Z distribution in shape. Therefore, as N gets large, the standard error of the t distribution approaches that of a normal Z distribution. This means that for N = 100, the probabilities associated with outcomes in the two distributions are virtually identical.
L. One- and two-tailed hypothesis tests