Probability Distributions
We may represent a probability distribution in any of these three ways.
What is a formula, a graph, or a table?
We may recognize a binomial probability distribution according to the fact that these are the only two well-defined outcomes.
What are success and failure?
We may recognize the standard normal distribution according to these three properties regarding its shape and center.
What is a bell-shaped graph, mean \mu = 0, and standard deviation \sigma = 1?
It is the Excel command for the left probability of an observation x in any normal distribution of mean \mu and standard deviation \sigma.
What is NORM.DIST(x, \mu, \sigma, 1)?
It is the least sample size in a simple random sample for which we may invoke the Central Limit Theorem.
What is n = 31?
Every random variable in a probability distribution encodes this type of data.
What is quantitative (or numerical)?
Given that the success probability in a binomial distribution is p, the failure probability q satisfies this equation.
What is q = 1 - p?
We may compute the area to the left of the observation x = \alpha in any continuous probability distribution as this quantity.
What is the probability P(x \leq \alpha) that x is less than or equal to (or at most) \alpha?
It is the bell-shaped and symmetric distribution with mean \mu = 0 and standard deviation \sigma = 1 that measures the z-score of any observation.
What is the standard normal distribution?
By the Central Limit Theorem, in any simple random sample of at least 31 individuals, this statistic is approximately normally distributed.
What is the sample mean \bar x?
Binomial distributions are an example of this type of distribution that is encoded by a random variable that counts something.
What is discrete?
Given a binomial distribution with n = 64 independent trials of success probability p = 0.5, it is the mean of the binomial distribution.
What is \mu = 32?
Given any uniform distribution defined for 0 \leq x \leq 10, it is the probability that x lies between the values 2 and 8.
What is P(2 \leq x \leq 8) = \frac 1 {10} (8 - 2) = 0.6.
It is the distribution of the sample mean \bar x of any simple random sample taken from any normally distributed population.
What is the normal distribution?
Under these two conditions, we cannot apply the Central Limit Theorem.
What is a sample size of 30 or fewer individuals taken from a population that is not normally distributed?
Uniform distributions and normal distributions are examples of this type of distribution that is encoded by a random variable that cannot be counted.
What is continuous?
Given a binomial distribution with n = 64 independent trials of success probability p = 0.5, it is the standard deviation of the distribution.
What is \sigma = 4?
We may compute the probability P(x \geq \alpha) in any normal probability distribution with mean \mu and st'd deviation \sigma according to this Excel formula.
What is =1 - NORM.DIST(\alpha, \mu, \sigma, 1)?
Given a probability distribution in a random variable x, if P(x or \text{more}) \leq 0.05 or P(x or \text{less}) \leq 0.05, then we refer to the observation x as this alliterative term.
What is statistically significant?
Given any simple random sample of size n taken from a normally distributed population with mean \mu and standard deviation \sigma, it is the mean \mu_{\bar x} of the sample mean \bar x.
What is \mu_{\bar x} = mu?
Given a probability distribution in a random variable x, it is the parameter obtained by summing the products of all square deviations of a value x from the mean and the probability of x.
What is the variance?
Given a binomial distribution with n = 64 independent trials of success probability p = 0.5, it is the least significantly high number of successes.
What is x = 40?
We may compute the probability P(\alpha \leq x \leq \beta) in any normal probability distribution with mean \mu and st'd deviation \sigma according to this Excel formula.
What is =NORM.DIST(\beta, \mu, \sigma, 1) - NORM.DIST(\alpha, \mu, \sigma, 1)?
Given a normally distributed population with mean \mu = 3.14 and standard deviation \sigma = 0.369, it is the percentage of observations that are less than \alpha = 3.61.
What is P(x \leq 3.61) = 0.90?
Given any simple random sample of size n taken from a normally distributed population with mean \mu and standard deviation \sigma, it is the standard deviation \sigma_{\bar x} of the sample mean \bar x.
What is \sigma_{\bar x} = \frac \sigma {\sqrt n}?
Given a probability distribution in a random variable x with P(x = 0) = \frac 1 3 and P(x = 1) = \frac 2 3, it is the expected value of x.
What is E(x) = \frac 2 3?
Given a binomial distribution with n independent trials, it is the number of outcomes with exactly x successes among the n independent trials.
What is n choose x or \frac{n!}{(n - x)! x!}?
We may compute the z-score corresponding to the confidence level \alpha in any normal probability distribution with mean \mu and st'd deviation \sigma according to this Excel formula.
What is =NORM.INV(1 - \alpha, \mu, \sigma)?
Given a normally distributed population with mean \mu = 3.14 and standard deviation \sigma = 0.369, it is the probability that the mean value of 100 observations is at least \alpha = 3.61.
What is P(\bar x \geq 3.61) = 0?
Given a simple random sample of 100 individuals from a population with mean \mu = 64 and standard deviation \sigma = 6, it is the z-score corresponding to a sample mean \bar x = 82.
What is z = \frac{\bar x - \mu_{\bar x}}{\sigma_{\bar x}} = \frac{82 - 64}{\frac 6 {\sqrt{100}}} = 30?