Unit 2: Probability, Random Variables, and Probability Distributions

21 terms. Say what each one means, then open the card to check.

• Two-way table

  A table of counts that sorts each individual by two categorical variables at once, one along the rows and one along the columns, with row and column totals.

  Example. A poll of 150 shoppers sorted by age group (under 40, 40 and over) in the rows and by how they pay (card, cash, phone) in the columns.

• Conditional relative frequency

  A count in one cell divided by the total of its row or its column, so it describes only the individuals in that one group.

  Example. If 45 of the 150 ninth graders in a survey play a sport, the conditional relative frequency of playing a sport among ninth graders is 45/150 = 0.30.

• Association (categorical)

  Two categorical variables are associated when the conditional distribution of one variable changes from one category of the other variable to the next.

  Example. If 45 percent of club members but 30 percent of non-members go to the game, membership and attendance are associated in that data.

• Complement rule

  The probability that an event does not happen is 1 minus the probability that it does happen.

  Example. If the chance that a flight is late is 0.15, the chance that it is not late is 1 - 0.15 = 0.85.

• Mutually exclusive events

  Events that cannot happen on the same trial, so the probability that both happen is 0.

  Example. On one roll of a die, getting a 2 and getting a 5 are mutually exclusive, since a single roll shows only one face.

• General addition rule

  P(A or B) equals P(A) plus P(B) minus P(A and B). The subtraction removes the overlap that was counted twice.

  Example. If 40 percent of shoppers use a coupon, 20 percent pay by phone and 10 percent do both, then 40 + 20 - 10 = 50 percent do at least one.

• Conditional probability

  The probability of an event given that another event has happened: P(A given B) is P(A and B) divided by P(B).

  Example. If 10 percent of shoppers use a coupon and pay by phone, and 20 percent pay by phone, the chance of a coupon given phone payment is 0.10/0.20 = 0.5.

• Independent events

  Two events are independent when knowing that one happened does not change the probability of the other, so P(A given B) equals P(A).

  Example. Two flips of a fair coin are independent: the chance of heads on the second flip is 0.5 whatever the first flip showed.

• Discrete random variable

  A numerical outcome of a chance process that takes a countable set of values, each with a probability, and those probabilities add to 1.

  Example. The number of siblings a randomly chosen student has, which can be 0, 1, 2 or 3, each with its own probability.

• Expected value

  The mean of a random variable: the sum of each value times its probability. It is the long-run average over many repetitions.

  Example. A game with an expected net gain of 0.35 dollars per play wins about 35 cents per play on average over many plays.

• Standard deviation of a random variable

  The typical distance of the random variable's values from its mean, found as the square root of the sum of each squared deviation times its probability.

  Example. If the number of calls a help desk gets in an hour has standard deviation 2.1, a given hour typically differs from the mean by about 2 calls.

• Variances add

  For independent random variables, the variance of a sum and the variance of a difference both equal the sum of the variances. Standard deviations do not add.

  Example. Independent times with standard deviations 6 and 8 minutes give a total with variance 36 + 64 = 100, so a standard deviation of 10 minutes, not 14.

• Binomial setting

  A fixed number of trials, each a success or a failure, with the trials independent and the same probability of success on every trial.

  Example. Counting how many of 15 randomly dialed calls are answered, when each call is answered with probability 0.4, independently of the others.

• Binomial mean and standard deviation

  For a binomial count with n trials and success probability p, the mean is np and the standard deviation is the square root of np(1 - p).

  Example. For 150 seeds that each fail to sprout with probability 0.08, the mean number that fail is 12 and the standard deviation is about 3.32.

• Normal distribution

  A symmetric, single-peaked, bell-shaped model set by its mean and standard deviation. Areas under the curve give proportions.

  Example. The lengths of a fictional fish species modeled as normal with mean 52 cm and standard deviation 7 cm.

• z-score

  How many standard deviations a value is above or below the mean: the value minus the mean, divided by the standard deviation.

  Example. A runner finishes in 21 minutes in a race where the mean time is 25 minutes and the standard deviation is 2, so z = 21 - 25/2 = -2.

• Empirical rule

  In a normal distribution, about 68 percent of values fall within 1 standard deviation of the mean, 95 percent within 2 and 99.7 percent within 3.

  Example. Commute times that are normal with mean 40 minutes and standard deviation 6 minutes put about 68 percent of commutes between 34 and 46 minutes.

• Sampling distribution

  The distribution of the values a statistic takes over all possible random samples of the same size from the same population.

  Example. The pattern of sample means of battery life you would get from every possible random sample of 25 batteries.

• Standard deviation of a sample proportion

  The spread of the sampling distribution of a sample proportion: the square root of p(1 - p)/n, when the sample is less than 10 percent of the population.

  Example. With p = 0.30 and n = 250, the standard deviation is the square root of 0.30(0.70)/250, about 0.029.

• Central limit theorem

  For large random samples, the sampling distribution of a sample mean is approximately normal even when the population is not normal.

  Example. Household water use is skewed right, but the means of random samples of 40 households pile up in a roughly normal shape.

• Unbiased estimator

  A statistic whose sampling distribution is centered at the parameter it estimates, so it does not tend to run high or low.

  Example. A sample proportion from a random sample is unbiased: its sampling distribution has mean equal to the population proportion p.