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Statistics

Key Terms

StatisticsKey Terms
binomial distribution
a discrete random variable (RV) that arises from Bernoulli trials; there are a fixed number, n, of independent trials
Independent means that the result of any trial (for example, trial 1) does not affect the results of the following trials, and all trials are conducted under the same conditions. Under these circumstances the binomial RV Χ is defined as the number of successes in n trials. The notation is: X ~ B(n, p) μ = np and the standard deviation is σ=  npq σ=  npq . The probability of exactly x successes in n trials is P(X=x)=( n x ) p x q n−x P(X=x)=( n x ) p x q n−x .
confidence interval (CI)
an interval estimate for an unknown population parameter
 This depends on the following:
  • The desired confidence level.
  • Information that is known about the distribution (for example, known standard deviation).
  • The sample and its size.
hypothesis
a statement about the value of a population parameter; in the case of two hypotheses, the statement assumed to be true is called the null hypothesis (notation H0) and the contradictory statement is called the alternative hypothesis (notation Ha)
hypothesis testing
based on sample evidence, a procedure for determining whether the hypothesis stated is a reasonable statement and should not be rejected, or is unreasonable and should be rejected
level of significance of the test
probability of a Type I error (reject the null hypothesis when it is true)
 Notation: α. In hypothesis testing, the level of significance is called the preconceived α or the preset α.
normal distribution

a bell-shaped continuous random variable X, with center at the mean value (μ) and distance from the center to the inflection points of the bell curve given by the standard deviation (σ)
We write X~N( μ,σ ) X~N( μ,σ ) . If the mean value is 0 and the standard deviation is 1, the random variable is called the standard normal distribution, and it is denoted with the letter Z.

p-value
the probability that an event will happen purely by chance assuming the null hypothesis is true; the smaller the p-value, the stronger the evidence is against the null hypothesis
standard deviation
a number that is equal to the square root of the variance and measures how far data values are from their mean; notation: s for sample standard deviation and σ for population standard deviation
Student's t-distribution
investigated and reported by William S. Gosset in 1908 and published under the pseudonym Student
 The major characteristics of the random variable (RV) are as follows
  • It is continuous and assumes any real values.
  • The pdf is symmetrical about its mean of zero. However, it is more spread out and flatter at the apex than the normal distribution.
  • It approaches the standard normal distribution as n gets larger.
  • There is a family of t-distributions: every representative of the family is completely defined by the number of degrees of freedom, which is one less than the number of data items.
Type 1 error
the decision is to reject the null hypothesis when, in fact, the null hypothesis is true
Type 2 error
the decision is not to reject the null hypothesis when, in fact, the null hypothesis is false
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