psych stats 240 final exam unit 3

which of the following numbers below is not a possible value for the probability of an outcome?

-0.2 (can't be less than 0 or greater than 1)

a normal distribution has a mean of 80 and a SD of 10. what is the probability of a score below 60?

0.025 (in this example 0.60 is 2 SD below the mean. the probability of an observation falling within 2 SDs of the mean is 0.95, so the probability of an observation falling outside 2 SDs is 0.05. this is divided between the two tails with 0.025 on each side)

You have two bags of marbles. Bag A contains 4 red marbles and 6 black marbles. Bag B contains 3 blue marbles and 7 yellow marbles. You draw a marble from each bag. What is the probability that you draw a red marble and a blue marble?

0.12 (by multiplication rule; p(red and blue) = p(red)*p(blue); p(red=4) and p(blue=.3) so p(red and blue) = .3*.4 = .12)

in the probability density function which of the following is the best estimate of the probability of a value in the shaded region?

0.2 (the proportion of the area shaded in this region is much too small to be 1 or 0.5 and too big to be 0.01, you make an educated guess based on the graph and the options given for answers and eliminate)

shown a graph, what is the approximate difference in the probability of sleeping more than 8 hours if you have had coffee before bed rather than water?

0.25 (area under the curve is 1, looking at 8 hours+ so you guess how much it is then subtract them since there are two curves)

negative kurtosis

when a distribution looks too humpy

positive kurtosis

when a distribution looks too peaky

discrete probability distribution

when we can identify specific events, we call the corresponding probability distribution this

if you do the z transform on a variable that is normally distributed, the z scores themselves will always follow what distribution?

N (0,1)

a normal distribution has a mean of 25 and a SD of 5. What is the z score that corresponds to a score of 47?

4.4 (47-25)/5 = 4.4

Which of the following variables would be distributed according to the binomial distribution? Select one: a. The number of pets owned by a family in the U.S. b. The age of a student in a class of 30 students if the mean age is 19 years, 6 months. c. The number of grades of A in a class of 30 students if the probability that any student gets an A is .25. d. The time it takes a student to finish an exam if the mean time to finish is 45 minutes.

The number of grades of A in a class of 30 students if the probability that any student gets an A is .25. (The binomial distribution is the distribution of the number of times an outcome occurs in n trials, given that the outcome has fixed probability p. This is the case for the grading example, where the number of trials is 30 and the probability is .25)

normal distribution

a continuous probability distribution can be called the 'bell curve' has a specific shape one hump in the middle, or symmetrical in which there is a certain probability of a value within one standard deviation of the mean, within two standard deviations of the mean, within 3 standard deviations and so on

z - score

a way of expressing each value in terms of how many standard deviations it is from the mean

what best describes the law of large numbers?

as the number of trials increases, the proportion of times that an outcome occurs converges on the true probability of that outcome (ex. the proportion of heads when flipping a fair coin 5000 times is likely to be closer to 50% than when flipping a coin only 5 times)

probability density function

density curve to show the probability distribution when the outcomes are continuous

Two intro to biology classes are offered this semester. Suppose that the grades in class A are normally distributed with a mean of 79 and a standard deviation of 10, while the grades in class B are also normally distributed, but with a mean of 70 and a standard deviation of 5. Ingrid is in class A and got a grade of 95; Frank is in class B and got a grade of 86. Who has a higher z-score, Ingrid or Frank?

frank (frank z score (86-70)/5 = 3.2 ingrids z score (95-79)/10 = 1.6)

the multiplication rule

if the two events are independent, the probability of A and B is equal to the probability of A times the probability of B p(A and B) = p(A) x p (B)

the addition rule

if two events are independent, the probability of A or B is equal to the probability of A plus the probability of B, minus the probability of both p(A and B) = p(A) + p(B) - p(A and B)

probability distribution

instead of keeping track of how many times each value actually occurs, we represent the probability of each value (each outcome) occuring

how does a probability behave

it is restricted to a range between 0 and 1, there are no negative probabilities, and are no probabilities greater than 1 the sum of the probabilities of the various possible events that can happen must be equal to 1 the probability of an event happening, plus the probability of that event not happening is always 1

Since the probabilities of all the outcomes sum to one, the area under the curve always sums to what?

one

a value that is one standard deviation above the mean will have a z score of what?

one

any value above the mean will have a _________ z score and any value below the mean will have a ___________ z score

positive, negative

the law of large numbers says what?

that as n increases, p hat n converges to (gets closer to) the true probability of the outcome p

discrete probability distribution is also called what

the binomial distribution

A success

the event we are interested in is often called this

N (0,1) distribution

the mean of the z scores will be 0 and the SD of the z scores will be one

continuous probability distribution is also called what

the normal distribution

p hat

the number of times the event occurs, over the total number of times it could have occurred

binomial distribution

the probability distribution of the number of times that an event happens in n trials of some process, if the event has a fixed probability the probability distribution of a set of discrete outcomes

68-95-99.7 rule

the probability of an outcome between one SD above and one SD below the mean is 0.68 the probability of an outcome between two SD above and two SD below the mean is 0.95. the probability of an outcome between 3 SD above and 3 SD below the mean is 0.977

p hat n

the proportion of times that the event occurs out of n occasions on which it could have occurred

the probability of an outcome is what?

the proportion of times the outcome would occur if we were to run the process we are interested in an infinite number of times

what is the meaning of the y axis values on a graph of a probability density function (pdf)?

the y axis values have no specific meaning, they are scaled so the the area under the curve is equal to one

The four binomial distributions correspond to p = .1, .25, .5, or .75, with n = 18. In which case is the probability of 4 or fewer successes equal to about .5?

to determine the probability of 4 or fewer successes, you sum the heights of the bars corresponding to 0, 1, 2, 4, and 4 successes, in this case only in the graph where p=0.25 is the sum near 0.5

true or false : the binomial distribution doesn't have to be symmetrical

true

true or false : when you sum up the values of a large number of independent variables, this sum will tend to be normally distributed

true

a value that is two standard deviations above the mean will have a z score of what?

two

the normal distribution is what shape?

unimodal