Flashcards in Exam #2 Deck (85):
What are Quartiles?
numbers which divide a set of data into four groups, with about 25% of the values in each group.
The first Quartile is the same as...
The second Quartile is the same as...
The third Quartile is the same as...
What are percentiles?
numbers which divide a set of data into 100 groups with about 1% of the values in each group.
a process with uncertain results that can be repeated
any collection of results or outcomes of a procedure
Define Sample Space
the set of all possible outcomes for a procedure
What does P(A) mean?
The probability of event A occurring
What is the rounding rule for probability?
a probability should always be expressed either as a simple fraction or as a decimal rounded to three significant digits.
What are the three methods for determining probability and which is the most commonly used?
1. Relative Frequency Approximation
2. Classical Approach (most common)
3. Subjective Approach
What is the formula for Relative Frequency Approximation?
P(A)=number of times event A occurred/number of times the procedure was repeated
What is the formula for the Classical Approach?
P(A)=number of ways A can occur (s)/number of possible outcomes (n)
*need a sample set to count the total possible outcomes and the number of outcomes containing the occurrence of A
What is the formula for the Subjective Approach?
no formula - this is an educated guess
(for example: what are the chances it will rain tomorrow?)
What is the Law of Large Numbers?
As a procedure is repeated again and again, the relative frequency probability tends to approach the actual probability of the event.
(example: the actual probability of flipping heads on a coin is 1/2 or 50%. The more times you flip the coin, the closer you will get to accurate results.)
What is the complement of an event?
The compliment of event A would be the event consisting of all the outcomes in which A does not occur.
What is the probability of an impossible event?
What is the probability of a certain event?
What is the probability of any event (shown as A)?
0 < P(A) < 1
Define an unlikely event.
an event that has a low probability of occurring. (usually less than or equal to 0.05 or 5%)
Define compound event.
any event combining two or more outcomes.
What does P(A or B)=
the probability of event A or event B occurring
(more likely; higher probability)
What does P(A and B)=
the probability of event A and event B both occurring
(more restrictive; lower probability)
In probability, "or" usually indicates...
in probability, "and" usually indicates...
What is the formal addition rule?
P(A or B) = P(A) + P(B) - P(A and B)
**P(A and B) are those that overlap so you don't want to add in twice.
What is the Intuitive addition rule?
To find P(A or B), find the sum of the number of ways that event A can occur and the number of ways event B can occur, adding in such a way that every outcome is counted only once, then divide by the total number of outcomes.
What are disjoint, or mutually exclusive, events?
two events are disjoint if they cannot both occur together; P(A and B) = 0
Are disjoint events compliments?
no because there are still other possibilities outside of A and B.
What is the rule of complementary events?
P(A) + P(complementary of A) = 1
therefore P(A) = 1-P(complementary of A)
What is the multiplication rule for independent events?
P(A and B) = P(A) x P(B)
What is the conditional probability notation?
P(B | A) = the probability of event B occurring after it is assumed that event A has already occurred.
(read "the probability of B given A" - changes the denominator)
What are independent events?
two events, A and B, are independent if the occurrence of one does not affect the probability of the other;
P(B | A) = P(B) and vice versa
What are dependent events?
The probability of one event occurring is dependent on the other event occurring. (example: A=study hard for exam, B=get an A on your exam)
What is the formal multiplication rule?
P(A and B) = P(A) x P(B | A)
What is the intuitive multiplication rule?
multiply the probability of event A by the probability of event B, taking into account the fact that event A has already occurred
What is sampling with replacement?
each time we randomly select an item, we record our result, then replace it in the group before selecting the next item. (independent)
What is sampling without replacement?
each time we select an item out of the larger group, we remove it from the group before randomly selecting the next object. (dependent)
How would you reword P(all 5 girls)?
P(1st is a girl and 2nd is a girl and 3rd....)
What is the sample size rule?
if a sample size is NO MORE than 5% of the size of the population, treat the selections as being independent - even if the selections are made without replacement.
How do you find the Probability of "at least one"?
1. if P(A) = the event of getting "at least one" of some result then P(complement of A) = the event of getting "none" of that result.
2. Find the probability of getting "none"
3. Solve for P(A) using the formula: P(A) = 1 - P(complement of A) ...or P(at least one) = 1 - P(none)
What is the Fundamental Counting Rule?
for a sequence of two events in which the first event can occur m ways and the second event can occur n ways, the events together can occur a total of (m)(n) ways... (m)(n)=mn NUMBERS ARE ALL INDEPENDENT
What is the factorial symbol?
if n > 0 (is a whole number) then the factorial symbol n! is defined as: n! = (n) x (n-1) x (n-2)...
(example: 4! = 4 x 3 x 2 x 1 = 24
What is the factorial rule?
any collection of n different items can be arranged in n! (n factorial) different ways.
How do you find a factorial of a number on the calculator?
1. type the number
3. arrow over to PRB
4. arrow down to !
6. ENTER again
When can the factorial rule be used? When can't it be used?
when you want to know all the different arrangements of a specific number of items. It cannot be used if you only want to know a specific number of arrangements of a specific number of items.
(example: can be used to find out all the different possibilities there are for the order of visiting a total of 30 stadiums. cannot be used if you want to find out the different possibilities of the order of only visiting 3 of the 30 stadiums.) NUMBERS ARE DEPENDENT
What is the permutations rule and formula?
The number of permutations (arrangements) of r items selected out of n items given. DEPENDENT
nPr = n!/(n-r)!
What is the combinations rule and formula?
The number of combinations (selections) of r items selected out of n items given. DEPENDENT
nCr = n!/(n-r)!r!
How does the combination rule relate to the permutations rule?
nPr = (r!)(nCr)
Once we determine how many ways to select r items out of n (combinations), we multiply by r! because there are r! different ways to arrange these items.
How do you use the calculator to find Permutations / Combinations?
1. type the value of n (total number of arrangements or combinations)
3. arrow over TO PRB
4. arrow down to nPr or nCr
5. type in the value of r (number of items selected)
When do we use the permutations rule?
when order matters - when we want to know how many ways there are to choose r items out of n AND ARRANGE them in different ways.
When do we use the combinations rule?
when order does not matter - when we are only interested in how many ways there are to select r items out of n.
What are the 4 counting techniques (rules)?
1. fundamental counting rule
2. factorial rule
3. permutations rule
4. combinations rule
What is a random variable?
a variable (usually represented by x) that has a single numerical value, determined by chance, for each outcome of a procedure.
(example: possible values of x for a roll of a die - X=1,2,3,4,5, or 6
What is a probability distribution?
a graph, table, or formula that gives the probability for each possible value of the random variable
What is a discrete random variable?
a random variable that takes on values consisting of discrete date.
what is a continuous random variable?
a random variable that takes on values over a continuous range of data
What are the requirements for a probability distribution?
The sum of all possible values for P(x) = 1
What is the rounding rule for probability distributions when calculating the mean, variance, and standard deviation?
round parameters to one more decimal than is present in the values of the random variable. (round to one more than the x value - not the P(x) value)
What is the formula to calculate the mean of a probability distribution?
meuw (mean) = SUM of all x times P(x)
What is the formula to calculate the variance of a probability distribution?
variance = SUM of all x squared times P(x) minus meow squared
What is the formula for standard deviation?
standard deviation = the square root of the variance
What probability is considered unusually high or low in trials?
1. a probability of 0.05 or less (P (x or more) < 0.05 or P(x or fewer) < 0.05)
2. Two standard deviation rule (mean plus 2 standard deviations or mean minus 2 standard deviations)
What is the expected value?
the expected value of a discrete random variable, denoted by E, is the same as the mean of the random variable and is calculated as E = SUM of all x times P(x)
What is a binomial probability distribution? (requirements)
a probability distribution of a random variable x, which meets the following requirements:
1. the procedure has a fixed number of trials
2. the trials ore independent
3. each trial has two possible outcomes (classified as success or failure)
4. the probability of each outcome remains constant for each trial.
5. the random variable, x, is defined to be the number of successes that occur.
What is the notation for binomial probability distributions?
S (success) and F (failure) - the two possible outcomes
n - the fixed number of trials
p - probability of success in any one trial
q - the probability of failure in any one trial
P(x) - the probability of getting exactly x successes in n trials
How are P(S) and P(F) related/
P(S) = p and P(F) = 1 - p = q
How do you use the graphing calculator to find binomial probabilities?
plug in: n,p, x (number of trials, probability of success, possible outcomes)
How do you simplify the formulas for the mean, variance, and standard deviation of a random variable with a binomial distribution?
mean = np
variance = npq
standard deviation = square root of npq
What is the formula to calculate a z-score?
z = x-mean/standard deviation
What is a density curve?
the graph of a continuous probability distribution satisfying the following properties:
1. the total area under the curve must be equal to 1
2. the curve may not fall below the x-axis
What is a uniform distribution?
a distribution in which all of the possible values of the variable have an equal probability of occurring; the density curve of a uniform distribution has a rectangular shape.
What shape is the density curve of a normal distribution?
Define a Standard Normal Distribution:
a normal probability distribution with a mean of 0 and a standard deviation of 1
What is the rounding rule for z-scores?
z-scores always round to 2 decimal places.
How do you use the calculator to find the probability that a random variable with a standard normal distribution takes on a value?
normalcdf (lower bound z-score, upper bound z-score)
**use 100 or -100 if there is no upper or lower given
What is the empirical rule for bell-shaped distributions?
1. about 68% of all data values fall within 1 s.d. of the mean
2. about 95% of all data values fall within 2 s.d. of the mean
3. about 99.7% of all data values fall within 3 s.d. of the mean
If I know a bottom portion percentage of a standard normal distribution, how do I find the z-score?
Change the percentage to a decimal then find that number on the z-score table OR
Change the percentage to a decimal then on the calculator do:
type in decimal
How do I apply the standard normal distribution chart to normal distributions with a mean and standard deviation other than 0 and 1?
P(x < ?)
Convert ? to a z-score then replace it in the formula:
P(z < z-score)
Find on the table
Define sampling variability
the variability of a sample statistic among different samples taken form the same population
Define sampling distribution of the mean
the probability distribution of sample means, with all samples having the same size, n.
What does it mean if a "statistic targets the population parameter"?
the mean of all values of a certain sample statistic for all samples of size n is equal to the population parameter
Which statistics target their corresponding parameters?
mean, variance, proportion
Which statistics do not target their corresponding parameters?
median, range, standard deviation