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# statistics Facts

### statistics I

Question | Answer |
---|---|

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 |