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Statistics
Chapter 7 & 8 Vocabulary // Statistics
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Gravity
Terms in this set (20)
Average
a number that describes the central tendency of the data; there are a number of specialized averages, including the arithmetic mean, weighted mean, median, mode, and geometric mean.
Central Limit Theorem
Given a random variable (RV) with known mean μ and known standard deviation, σ, we are sampling with size n, and we are interested in two new RVs: the sample mean, X¯¯¯, and the sample sum, ΣΧ. If the size (n) of the sample is sufficiently large, then X¯¯¯ ~ N(μ, σn√)
Exponential Distribution
a continuous random variable (RV) that appears when we are interested in the intervals of time between some random events, for example, the length of time between emergency arrivals at a hospital
Mean
a number that measures the central tendency; a common name for mean is "average." The term "mean" is a shortened form of "arithmetic mean." By definition, the mean for a sample (denoted by x¯) is x¯ = Sum of all values in the sampleNumber of values in the sample, and the mean for a population (denoted by μ) is μ = Sum of all values in the population Number of values in the population.
Normal Distribution
a continuous random variable (RV) with pdf f(x) = 1σ2π√ e-(x - μ)22σ2, where μ is the mean of the distribution and σ is the standard deviation.; notation: X ~ N(μ, σ). If μ = 0 and σ = 1, the RV is called the standard normal distribution.
Sampling Distribution
Given simple random samples of size n from a given population with a measured characteristic such as mean, proportion, or standard deviation for each sample, the probability distribution of all the measured characteristics is called a sampling distribution.
Standard Error of the Mean
the standard deviation of the distribution of the sample means, or σn√.
Uniform Distribution
a continuous random variable (RV) that has equally likely outcomes over the domain, a < x < b; often referred as the Rectangular Distribution because the graph of the pdf has the form of a rectangle.
Binomial Distribution
a discrete random variable (RV) which 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.
Confidence Interval (CI)
an interval estimate for an unknown population parameter.
This depends on:
the desired confidence level,
information that is known about the distribution (for example, known standard deviation),
the sample and its size.
Confidence Level (CL)
the percent expression for the probability that the confidence interval contains the true population parameter; for example, if the CL = 90%, then in 90 out of 100 samples the interval estimate will enclose the true population parameter.
Degrees of Freedom (df)
the number of objects in a sample that are free to vary
Error Bound for a Population Mean (EBM)
the margin of error; depends on the confidence level, sample size, and known or estimated population standard deviation.
Error Bound for a Population Proportion (EBP)
the margin of error; depends on the confidence level, the sample size, and the estimated (from the sample) proportion of successes.
Inferential Statistics
also called statistical inference or inductive statistics; this facet of statistics deals with estimating a population parameter based on a sample statistic. For example, if four out of the 100 calculators sampled are defective we might infer that four percent of the production is defective.
Normal Distribution
a continuous random variable (RV) with pdf f(x)=1σ2π√e-(x-μ)2/2σ2, where μ is the mean of the distribution and σ is the standard deviation, notation: X ~ N(μ,σ). If μ = 0 and σ = 1, the RV is called the standard normal distribution
Parameter
a numerical characteristic of a population
Point Estimate
a single number computed from a sample and used to estimate a population parameter
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. Gossett in 1908 and published under the pseudonym Student; the major characteristics of the random variable (RV) are:
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 get larger.
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Verified questions
QUESTION
A company says its premium mixture of nuts contains 10% Brazil nuts, 20% cashews, 20% almonds, and 10% hazelnuts, and the rest are peanuts. You buy a large can and separate the various kinds of nuts. Upon weighing them, you find there are 112 grams of Brazil nuts, 183 grams of cashews, 207 grams of almonds, 71 grams of hazelnuts, and 446 grams of peanuts. You wonder whether your mix is significantly different from what the company advertises. What might you do instead of weighing the nuts in order to use a $$ \chi ^ { 2 } $$ test?
STATISTICS
Suppose that $$ Y _ { 1 } , Y _ { 2 } , \ldots , Y _ { n } $$ constitute a random sample from the density function $$ f ( y | \theta ) = \left\{ \begin{array} { l l } { e ^ { - ( y - \theta ) } , } & { y > \theta } \\ { 0 , } & { \text { elsewhere } } \end{array} \right. $$ where $$ \theta $$ is an unknown, positive constant. Find an estimator $$ \hat { \theta } _ { 2 } \text { for } \theta $$ by the method of maximum likelihood.
STATISTICS
Let X and Y have the joint pdf f(x, y) = 1, −x < y < x, 0 < x < 1, zero elsewhere. Show that, on the set of positive probability density, the graph of E(Y |x) is a straight line, whereas that of E(X|y) is not a straight line.
PROBABILITY
Calculate upper and lower bounds for the perimeter of the rectangle shown (below), whose dimensions are accurate to 2 d.p. width = 4.86 m, length = 2.00 m.