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statistics Facts

statistics I

What are the characteristics of random variables? They are numerical, They are random, They are associated with the outcome of an experiment
How do outcomes associate to random variables? ● Each outcome of an experiment can be mapped to only 1 random variable. ● More than one outcome can be mapped to a single random variable
What are the two types of random variables? ● Discrete ● Continuous
If a random variable can take on any value between 0 and 1 with two digits of precision to the right of the decimal (e.g. .01, .02, .03, … , .97, .98, .99), is it a discrete or continuous random variable? Discrete, because there are gaps between each of the those decimals (e.g. .015 is between .01 and .02)
Is it possible for a discrete random variable (X) to have an infinite number of possible x values? Yes. i.e. of this is the prob dist of how many flips it will take until the first H is hit. It is possible for a H never to come up. a fair coin & the flips are independent, then the prob of x = 1 is ½, the prob of x = 2 is ¼, the prob of x = 3 is 1/8.
The probability of each value of x must be within what range? 0 ≤ P(x) ≤ 1 for all x
What is ∑p(x) equal to (where the sum is across all values of x) ∑P(x) = 1
For a random variable X, E(x) (i.e. the expected value of x) is equal to what population parameter? The mean (μ)
1. What are the characteristics of a Binomial Random Variable? All trials are identical & indep of each other. Each trial has only 2 outcomes. Each trial has the same prob of a success (and therefore the same probability of a failure) The bi random variable (x) is the number of successes in the n trials
For a Binomial Random Variable, what are the two possible outcomes for a single trial and what symbols do we typically use to represent the probability of each outcome? ● Success: p(success) = p ● Failure: p(failure) = 1 - p
For a continuous random variable, what do the density functions represent? The density function defines the height of the curve for any particular x value in that distribution. It is used to calculate the cumulative function.
For a continuous random variable, what do the cumulative functions represent? The cumulative function is the area under the curve of the density function for a range of x values and determines the probability of that range of x values for that distribution.
Can a continuous random variable X, ever be bounded on the maximum and minimum values that x can assume? Yes. For example, all uniform distributions are bounded.
Can a continuous random variable X, ever have a finite number of possible x values? No​. While the x values may be bounded, there are an infinite number of possible x values between any two x values within the domain of the function.
For any continuous random variable, what does the sum of all the area under the density function curve (and above the x axis) equal? It equals 1
Can a density function curve ever dip below the x-axis? No. That would result in negative probabilities, which is not allowed.
Can the cumulative function for a continuous random variable ever have a greater value for x1 than x2 , if x1 < x2? No, for the same reason as the prior question. For the cumulative function to reduce, then the density function must be negative.
8. Can the density function ever have a value > 1 for any particular value of x? Yes. However, the range of x values that have a P(x) > 1, must be narrow enough that the total area under the curve still = 1.
Does a continuous random density function have to be symmetrical about its mean? No. The normal curve is symmetrical, but many other continuous random density functions are not symmetrical
What are the maximum and minimum x values for any normal distribution? − ∞ and ∞
What population parameter determines the width (i.e. the region that includes the bulk of the probability) of a normal distribution curve? σ
For a normal distribution curve, if the width gets greater, then what must happen to the height? It must get lower, so that the total area under the curve = 1.
Can the normal density function ( ), ever be = 0? No. It is always > 0.
For any particular normal density function (i.e. for whatever combination of µ and σ), what value of x has the greatest density function value? When x = μ
What is the z-value formula for a particular x coming from a normal distribution? z = x−μ/σ
What distribution does the z-value for a normal distribution always follow? The standard normal distribution.
Under what circumstances does the normal distribution closely approximate the binomial distribution? When n · p ≥ 10 and n · (1 – p) ≥ 10.
1. If x is a random variable, is x ​for a sample of 10 also a random variable? Yes.
What are the formulas to determine the mean and standard deviation for x (i.e. the mean of the sampling distribution) based on the mean and standard deviation of the underling distribution for x? μx = μ; σx =σ/√n
The central limit theorem says what? As the sample size (n) approaches ∞ , the distribution for the sample average (x ) approaches the norm dist For practical purposes, any sample greater than or equal to 30 will have asample average ( x ) that is approximated by the normally distribution
What sample size cut off do we use as a ‘rule of thumb’ to ensure the normal distribution is a close enough approximation of x ’s distribution for practical purposes? 30
. If the underlying x population is normally distributed, then what size of a sample is necessary to ensure that the distribution for x is also normally distributed? 1 is fine.
If a 99% conf. level is used to build a sym conf interval for x to est. the true value of the underlying popul. µ, what % of the time will µ be larger than the entire conf interval? What % of the time will µ be smaller than the entire conf interval? .5% (.005) of the time µ will be outside the confidence interval on the high side .5% (.005) of the time µ will be outside the confidence interval on the low side
If we wish for a confidence interval to get smaller, but to keep the same confidence level, what can be done to accomplish that? increase sample size.
For samples sizes ≤ 30, what are the two required conditions for a confidence interval for µ to be valid if σ is unknown? ● The sample is a random sample from the target population ● The distribution of x has to be near normally distributed.
As the degrees of freedom gets smaller for the same α, what happens to the t-values? They get larger.
Can a t-value ever be smaller than the corresponding z-value? Never
For a binomial random variable, what is the mean and standard deviation of the x values in terms of n and p? What is the standard deviation of pˆ ? σ = √npq (this is the standard deviation of the x values – the count of the # of successes out of n trials)
. For a given sample size n, what value of pˆ generates the largest standard deviation for σp ? .5
Can pˆ vary more than x? no
What rule do we use for n and p to put a valid confidence interval around pˆ to estimate p using z-values? np ≥ 10 and n(1 – p) ≥ 10
When we put a confidence interval around pˆ to estimate p using z-values, we are using what distribution instead of the binomial to create that confidence interval? The normal distribution
Created by: aiur100
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