stats ch. 5 and 6 flashcards

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what is randomness

ensures sample data are unbiased and accurate.

usually needs help of computer or other randomizing device

eg. drawing cards from a deck or names from a hat

what is probability

proportion or percent. measures how often random events occur. also known as likelihood, chance, or relative frequency

what is theoretical probability and when is it used?

relative frequency/long-run proportion based on theory. basically, probability in a population.

ex. theoretical probability of tails on a coin toss = ½ = 0.50

use when you can mathematically determine the probabilities (dice cards, genetics, etc.)

what is empirical probability and when is it used?

relative frequency/short-run proportion based on outcomes of an experiment or by observing the real-life process.

ex. empirical probability of tails on 50 coin tosses, landed on tails 22 times = 22/50 = 0.44

use when probabilities cannot be mathematically determined (weather, politics, business success, etc.)

obtained through sampling and simulations.

what are some common probability situations with equally likely outcomes?

tossing a coin: 2 outcomes (heads, tails) → ½ probability

rolling a die: 6 outcomes (1, 2, 3, 4, 5, 6) → 1/6 probability

drawing card from a deck: 52 outcomes (4 suits, 13 cards per suit) → 1/52 probability

what are useful facts about theoretical probabilities?

numbers between 0 and 1, inclusive

expressed as fractions, decimals, or percents

rule of complements: probability that an event will not occur is 1 minus probability that it will occur.

what are equally likely outcomes?

situations where all possible outcomes occur with same frequency

helpful to list all the possible outcomes, which is called the sample space, denoted by s.

eg. S = {1, 2, 3, 4, 5, 6}

→ an event is any collection of outcomes in the sample space

how do you calculate probability of an event?

number of outcomes in event / total number of possible outcomes

ONLY WORKS for equally likely outcomes within a sample space!!! (when you know the entire collection of outcomes)

what do you do when you see “AND” in theoretical probability

AND creates a new event that consists of only those outcomes in BOTH event A and event B. an “intersection.”

multiply the probabilities for the 2 events.

eg. 1/2 × 4/6 = 1/3

what do you do when you see “OR” in theoretical probability

OR creates a new event that consists of outcomes that are only in A, only in B, or in both A and B. basically at least 1 of the events must occur. a “union.”

add the probabilities for the 2 events.

eg. 1/6 + 1/6 = 2/6

how do you find empirical probabilities?

usually useful when you don’t know what assumptions to make to find a theoretical probability or when theoretical probability is too complex

so scientists use computer-based simulations

what is the law of large numbers

the more times you gacha the more likely you are to be close to the true probability

the more rare the thing is, the more pulls (trials) you need

generally at least 100 trials will work

what is probability distribution/model

table/graph/formula that gives all possible outcomes of a random experiment and the probability of each outcome

always between 0 and 1

all numbers in probability add to total of 1

difference between categorical and numerical data in probability distribution

both work for creating distributions

for categorical, discuss distribution in mode and variability. for numerical, discuss shape, center, and spread.

what is a discrete numerical variable?

numbers that can be listed or counted (1, 2, 3, numbers of classes taken, roll of a die)

if you graphed them they would skip along as happy dots

how do you make a table for discrete variables

table always has 2 columns

1st column is all possible outcomes. 2nd column is corresponding probabilities for all those outcomes.

CHECK THAT all probabilities are between 0 and 1, inclusive, and total adds to 1 or 100%.

how do you make a graph for discrete variables

x-axis shows variable, y-axis shows probability

CHECK THAT all probabilities are between 0 and 1, inclusive. the probability is the area under the curve, and totals/adds to 1.

what is a continuous numerical variable?

numbers that can’t be listed or counted because they occur over a range/smooth scale (time to finish an exam, exact weight)

if you graphed them it would be a straight line connecting all the dots

what are 3 characteristics of the normal model?

1. unimodal

2. symmetric

3. bell-shaped / mound-shaped

*also known as the Gaussian distribution!

important symbols for normal distribution?

μ = center or mean of a distribution

σ = standard deviation of a distribution

can be written in shorthand notation N(μ, σ) → (center, spread)

how do you calculate normal probabilities on the ti-84?

1. sketch a normal curve, label mean and the number you’re trying to find probability that it’s more/less than. shade to the appropriate side of that number (more than = right, less than = left)

2. 2nd → VARS → DISTR → 2: normalcdf

3. (where shading starts/lower, where shading ends/upper, mean, standard deviation)

   → if number goes to negative infinity, type -1E99. if positive infinity, type 1E99

how do you interpret area under the curve as probability

two ways:

1. probability that a random guy from the population has the characteristic described by the interval of values

2. proportion (%) of population with the characteristic described by the interval of values.

what is invnorm used for?

when you’re given a probability or percentile and want to find the value that corresponds to that percentile

eg. what is the weight of an adult giraffe that is at the 40th percentile?

how do you find values given a percentile on the ti-84?

1. sketch a normal curve, label mean and the percentile you want to find (starting from the left!) shade to the left.

2. 2nd → VARS → DISTR → 3: invNorm → (area to the LEFT of value of interest as a decimal, mean, standard deviation)

   → MAKE SURE it’s the left! even if question asks for top 10% you would still calculate for 90th percentile (0.90 to the left)