sample space

Examples of sample space in the following topics:

• The Multiplication Rule

  • If A and B are two events defined on a sample space, then: P(A AND B) = P(B) · P(A|B).

• Terminology

  • A sample space is a set of all possible outcomes.
  • The uppercase letter S is used to denote the sample space.
  • The sample space has four outcomes.
  • A conditional reduces the sample space.
  • The sample space S = { 1, 2, 3, 4, 5, 6 } .

• Complement of an event

  • This set of all possible outcomes is called the sample space (S) for rolling a die.
  • We often use the sample space to examine the scenario where an event does not occur.
  • Then the complement of D represents all outcomes in our sample space that are not in D, which is denoted by Dc = {1, 4, 5, 6}.
  • Figure 2.9 shows the relationship between D, Dc, and the sample space S.
  • S represents the sample space, which is the set of all possible events.

• Lab 2: Central Limit Theorem (Cookie Recipes)

  • Record your samples below.
  • Record them in the spaces provided.
  • Thistime, make the samples of size n = 10.
  • Record the samples below in Table 7.6.
  • Record them in the spaces provided.

• Estimation

  • Out of a random sample of 200 people, 106 say they support the proposition.
  • Thus in the sample, 0.53 ($\frac{106}{200}$) of the people supported the proposition.
  • It is rare that the actual population parameter would equal the sample statistic.
  • Bias leads to a sample mean that is either lower or higher than the true mean .
  • An estimate of expected error in the sample mean of variable $A$, sampled at $N$ locations in a parameter space $x$, can be expressed in terms of sample bias coefficient $\rho$ -- defined as the average auto-correlation coefficient over all sample point pairs.

• Standard Error

  • However, different samples drawn from that same population would in general have different values of the sample mean.
  • The standard error of the mean (i.e., of using the sample mean as a method of estimating the population mean) is the standard deviation of those sample means over all possible samples (of a given size) drawn from the population.
  • SEM is usually estimated by the sample estimate of the population standard deviation (sample standard deviation) divided by the square root of the sample size (assuming statistical independence of the values in the sample):
  • $s$ is the sample standard deviation (i.e., the sample-based estimate of the standard deviation of the population), and
  • If values of the measured quantity $A$ are not statistically independent but have been obtained from known locations in parameter space $x$, an unbiased estimate of the true standard error of the mean may be obtained by multiplying the calculated standard error of the sample by the factor $f$:

• Mann-Whitney U-Test

  • For small samples a direct method is recommended.
  • Call this "sample 1," and call the other sample "sample 2. "
  • where $n_1$ is the sample size for sample 1, and $R_1$ is the sum of the ranks in sample 1.
  • Note that it doesn't matter which of the two samples is considered sample 1.
  • $U$ remains the logical choice when the data are ordinal but not interval scaled, so that the spacing between adjacent values cannot be assumed to be constant.

• Degrees of Freedom

  • Degrees of freedom can be seen as linking sample size to explanatory power.
  • In fitting statistical models to data, the random vectors of residuals are constrained to lie in a space of smaller dimension than the number of components in the vector.
  • The sample mean could serve as a good estimator of the population mean.
  • The difference between the height of each man in the sample and the observable sample mean is a residual.
  • That means they are constrained to lie in a space of dimension n − 1, and we say that "there are n − 1 degrees of freedom for error. "

• Samples

  • This process of collecting information from a sample is referred to as sampling.
  • The best way to avoid a biased or unrepresentative sample is to select a random sample, also known as a probability sample.
  • Several types of random samples are simple random samples, systematic samples, stratified random samples, and cluster random samples.
  • A sample that is not random is called a non-random sample, or a non-probability sampling.
  • Some examples of nonrandom samples are convenience samples, judgment samples, and quota samples.

• Which Average: Mean, Mode, or Median?

  • For example, the mode of the sample $[1, 3, 6, 6, 6, 6, 7, 7, 12, 12, 17]$ is 6.
  • For samples, if it is known that they are drawn from a symmetric distribution, the sample mean can be used as an estimate of the population mode.
  • For example, consider the data sample $\{1,2,3,4\}$.
  • Unlike median, the concept of mean makes sense for any random variable assuming values from a vector space.
  • Then "Kim" would be the mode of the sample.