AP Statistics Chapter 6 Review

What is a random variable?

Variable that takes numerical values that describes the outcomes of some chance process.

What is probability distribution?

Tracks a random variable and gives its possible values and their probabilities.

EX: (Let X = Number of Tails of a coin flipped 3 times)

X: 0 1 2 3
P: 1/8 3/8 3/8 1/8

A good way to find this is by creating a sample space of all possible outcomes, then counting.

What is a discrete random variable?

When X takes a fixed set of possible values with gaps between.

How do you know if you have a discrete random variable?

If the number is countable, like a 6 sided dice.

What do you say when describing a distribution?

Shape, center, variability, outliers.

How do you find the mean (expected value) of a discrete random variable?

(x1*p1)+(x2*p2)+(x3*p3)+…

How do you find variance? (formula)

(x1-ux [mean])²*p1+(x2-ux [mean])²*p2+…

How do you find standard deviation?

sqrt (x-mean)²*pi

just square the variance :)

How do you interpret the mean?

If a (explanatory variable) is chosen many, many times at random and the number of (response variable) is recorded at random for each (explanatory variable), the average is _____

EX: If a high school student is chosen at random many times and a the number of languages is recorded for each studying, the average is 1.457.

How do you interpret the standard deviation?

Over many times, A randomly chosen (explanatory variable) and the recorded (response variable) typically differs from the mean of ___ by ___.

EX: A randomly chosen students number of languages spoken typically differs from the mean of 1.457 by 0.671.

How do you interpret variance?

The weighted average of the squared deviations of the (explanatory variable) to (response variable) from their mean is _____.

What is a continuous random variable?

When X takes all values in an interval of numbers. EX: weight

NOTE: This means the probability to a whole number is 0, since it could be 1.49583895893485.

How do you find the mean of two different variables (they are independent)

You add them.

How do you find the difference of two means that are independent?

You subtract them.

How do you find the standard deviation of two independent events when adding them together?

Find the variances (Square the standard deviation), add them together, then take the square root of that.

How do you find the standard deviation of two independent events when trying to find their difference?

Find the variances (square the standard deviation), add them together, then take the square root of that.

NOTE: Variability stays the same when adding or substract.

When you add or subtract random variables, know that variation (increase/decreases)

increases (ALWAYS INCREASES)

Remember to include units on the ….

mean, standard deviation! NOT the variance!

What happens when we ADD a constant?

Only the center changes (mean). Add the same constant to every value besides standard deviation.

What happens when we MULTIPLY by a constant?

The center changes (mean) and the variability. Remember to multiple every value.

What is a binomial setting?

When several independent trials of the same chance process and the number of times that an outcome occurs are recorded.

What are the four conditions for a binomial setting?

Binary (successes/failures), Independent, Number (fixed set of trials), Success (same probability)

What is the Binomial probability formula?

P(X=k) = (n) * p^k * (1-p)^n-k
(k)

k = Number of successes.
n-k= Number of failures
n = Number of trials

What does binomPdf find?

The probability of one outcome for a binomial setting.

What does binomCdf find?

The probability of a range of outcomes.

How do you find the mean of a binomial variable?

n*p

How do you find the standard deviation of a binomial variable?

sqrt(n*p*(1-p))

What does it mean when the 10% condition is met?

You can use this to determine if using a binomial setting is acceptable if the sample is no more than 10% of the population.

What is the 10% condition formula?

n< N/10
-

n= number of trials
N= population size

If a scenario violates the 10% plan and you still need to find the probability of something, what do you use?

Conditional probability formula:

EX: 6/10 × 5/9

What is the Large Counts Condition? What can you assume if the condition is true?

np>10
-

You can assume the distribution is approximately normal with a mean of n*p and standard deviation of sqrt(np(1-p))

What is a geometric setting?

When we perform independent trials of the same chance process and record the number of trials it takes to get one success.

What is a geometric random variable?

The number of trials Y that it takes to get a success in a geometric setting.

What is a geometric distribution?

The probability distribution of Y with a parameter pm possible values of Y are 1, 2, 3, …

What is the geometric probability formula?

P(Y=k) =(1-P)^k-1 * P

When you are asked for more than, always …

Go above the number one time. For example, if it asks above than 4, plug in 5!

Find the probability that the difference in the lengths of the two pregnancies is greater than 13 days. (Approximately Normal Distribution)

U= 2
O = 22.63

normCDF(LB: -13, UB: 13, U: 2, O, 22.63)=0.434343

Remember to pay attention to the size of things like toothpaste (they’re probably useful in the normCDF command)!!!!

EX: Jimmy’s 0.85oz of toothpaste should last him the entire 6-day trip. Mean: 0.13, Std. Dev: 0.02 (for one day)

What is the probability that he’ll use all the toothpaste in the 6 days?

EXPLANATION:

First, find the new mean and deviation after he brushes 6 times. Then, take that and do normCDF(LB: 0.85, UB: infinity, U: 0.78, O: 0.049) = 0.077