Psych 320 Statistics final

Terms

undefined, object
copy deck
Which distribution is used to determine the p value?
sampling distribution
definition of critical rejection region
CRR = the place where the sample mean, x bar, is greater than the cutoff.
How is the alpha level used?
The alpha level is used to determine whether we'll retain or reject the null hypothesis
P value represents...
tail area probability
when p< alpha, what do we with the null?
p< alpha, reject null
alpha levels are usually set at...
p = .05, z=1.96
p=.01, z=2.33
The critical rejection region is a region where..
we are justified in rejecting the null.
Something that is statistically significant...
-is not necessarily scientifically significant. Stats just lets us make a best educated guess.
-occurs whenever p is less than alpha and you reject the null.
if p>alpha, what happens to the null hypoth?
we will retain (fail to reject) Ho (null).
Which type of error is being mimimized in hypothesis testing?
-type I (alpha), in which you reject Ho but it is actually true.
What is type II error?
Type II (Beta) is the prob of retaining Ho when it's false.
What type error if:
the null is false but you retain it?
type II
What type error if:
the null is true but you reject it?
type I
What does the decision rule tell us?
Decision rule- if p value is less than alpha then H1 is true. alpha is most commonly .05
Null is...
the only thing that we can test. Null is the baseline, default, status quo, no effect, no change condition.
Expressing p-value
p value = p (observed data or more extreme given the null)
Alternatively
pval = p(observed or more extreme | Ho)
Frequentist methods
are the standard. Ho and H1 are not on equal footing, bias is in favor of null. Decision is based upon p value.
Bayesian methods
more intuitive, computer-based method. Ho and H1 are put on equal footing.
for H1
P(x bar > x bar observed)
Will this calc produce a p value?
No- this is not a tail area, so it's not a p value
p<.05
reject null and choose alternative
p>.05
retain (fail to reject) null
What terms can you use when referring to Ho?
reject, retain, or fail to reject
CANNOT EVER say accept. The use of accept is confined to the alternative hypothesis.
definition of standard error
standard error = strd dev of the sampling distribution
The p value is the ...
probability of obtaining a sample mean equal to the observed sample mean or one that is more extreme UNDER THE NULL HYPOTHESIS
Standard error represented as:
SE = strd dev/ sq rt of n
When you draw the sampling distributions for 2 different sample hypotheses two overlapping bell shaped curves will appear. Are these sampling distributions of the sample or the population?
sample
If the null hypothesis is the leftmost of two sampling distributions where is the tail area probability located?
The p will be the rightmost tip of the leftmost (null)sampling distribution
If two sampling distributions are drawn such that the one on the left is H1, then where is the tail area probability?
Tail area probability will be the leftmost edge of the rightmost (null) distribution.
Ho is a ....
statistical decision. It's the true state of nature and has not been proven.
Hypothesis testing is also known as ....
null hypothesis significance test
If you're going to reject the null, you'll say that you are rejecting the null and...
accepting the alternative. You can only use the word accept in relation to H1
The p value is the tail area probability under which hypothesis?
Tail area prob is the null.
The large area that remains after you subtract away the tiny sliver of the p is the probability of the alternative probability, but this is NOT considered a p value.
How do you solve for z?
(null-alternative)/Strd Error
Are the null and alternative hypotheses stated in terms of the original population or the sampling distribution?
The null and alternative hypoth are stated in terms of the original population. I think that it's when you calculate z that you'll start considering elements of the sample.
definition of x bar in terms of a sampling distribution
The distribution of all possible sampling means
How do you interconvert between the sm. portion and the p value?
.081 is sm. portion
8.1% is p value
Continuous random variables will have what kind of distribution?
Normal distribution
Does a sampling distribution have anything to do with actual data?
not reallly. The sampling distribution is a hypothetical distribution
Consider:
1.the mean of sample = to mean of poulation
2.variance of sampling distribution = variance of population/n
Are these properties part of the central limit theorem?
No. The central limit theorem deals exclusively with shape.
Definition of central limit theorem
If you use the distribution of all possible means and the distribution is large enough than the shape of the sampling distribution will become normal.
If original distribution is normal, what does the CLT say about the sampling distribution?
original normal, sampling distribution exactly normal
how large does sample size need to get for the central limit theorem to apply?
30
Discuss the importance of CLT in inferential stats.
It simplifies the probability calculation so that you don't have to do numerical integration, you can just use normal distribution.
sigma =
sigma squared =
Mu =
n=
sigma = strd dev
sigma squared = variance
Mu = mean
n= number of observations in a sample
CLT maintains that...
all sampling distributions of any size or kind will be approximately normal regardless of the original
CLT tells us what about the shape of the graph?
The shape will get more normal as n increases
As n goes to infinity, the shape of the ___ distribution will resemble the ___ distribution no matter what original distribution looked like
As n goes to infinity, the shape of the sampling distribution will resemble the population distribution no matter what original distribution looked like
3 pillars of stats
I think that they are:
1.mean sample = mean orig pop
2. variance sample = (variance of pop/n)
3. CLT
What are the two relationships that exist between the sampling distribution and the original population distribution?
1.mean sample = mean orig pop
2. variance sample = (variance of pop/n)
Population measures indicate...
all possible outcomes of an event
sampling distribution of the mean involves...
collecting all possible means that you can from the original population??
The sampling distribution is derived from....
the original population distribution
sampling distribution of the mean is defined as the...
-distribution of sample mean x bar.
-it is obtained from all possible random samples of a given size
-is hypothetical and not something that is obtained from data.
mean to z column in normal distribution table refers to..
area from 0 to z
larger portion column in normal distribution table refers to..
area from negative infinity to z
smaller portion column in the normal distribution table refers to...
area from z to pos infinity
standardized z values use what kind of distribution?
unimodal normal distribution
steps in taking a binomial distribution to a standard normal distribution (with z scores)
1. W/ binomial distribution, get tail-area variables
2.W/ tail area probability (p) scale to a standardized z score
3. Z scores are used in a unimodal, normal distribution
scaling
rewriting scores in terms of z
___ refers to the normal distribution probability.
____ of ____ is not a probability.
An Area refers to the normal distribution probability.
height of curve is not a probability.
Probability is determined as an ___ not a ___
Probability is determined as an area, not a height
You can obtain the probability of something that is ___ zero but not ___ zero.
You can obtain the probability of something that is around zero but not exactly zero.
What is the probability of a single point?
zero
What are the chief differences between normal and binomial distributions concerning height?
In normal distributions, height is not a probability.
In binomial distribution the height represents a relative probability.
If the original x was normally distributed, what kind of distribution will be obtained?
Standard (Unit) normal distribution
*What are always the values of standard deviation and mean for the standard (unit) normal distribution?
strd dev = 1
mean = 0
How is z useful in determining location?
Z gives you a relative location in terms of how far your your number is above or below the mean
A z = -1.2 means...
the original x is below the mean by 1.2 standard deviations.
positive linear transformations
-can be done for interval and ratio scales of x only
-ex. celsius and farenheight
-z= x/strd dev - Mu/strd dev
z = x-Mu/sigma (for populat)--->z=x-xbar/ strd dev (for sample). This is known as a ____ ______
transformation scaling
definition of standardized z score
a transformation of the original measurement score x which gives a relative position of x in a distribution. The z scores that result from this transformation whether the original score (x) was a certain number of strd devs above(+) or below (-) the mean.
What do a series of normal curves with different means but the same strd deviation look like?
different means = each line that vertically bisects each curve at its peak is at a different location.
The same strd dev = same width
What do normal curves with the same means but diff strd dev?
same mean = the curves are superimposed upon each other such that the same vertical line bisects the peak of each of the 3 curves
-diff strd dev - different widths
*What can be said about the symetry of a binomial distribution vs. a normal distribution?
A binomial distribution is asymetric.
Normal distributions are symetric
What are the four characteristics of a normal distribution?
1. continuous
2. unimodal (only one peak in a single curve)
3. always symetric
4. mean = median = mean
For which kind of variable is the normal distribution used?
continuous random variable
definition of p value
-probability of obtaining the null or higher
-probability of obtaining the current data or more extreme under the null hypoth. Graphically you can see that this is the tail area.
For which kind of variable is the binomial distribution used?
discrete random variables
ex. correct or not correct
What kind of distribution would be used for height & weight measurements?
Normal, because height and weight are continuous random variables.
How is the binomial probability distribution obtained?
The binomial probability distribution is a collection of probabilities that were obtained from an independent bernouli process.
Definition of Independent Bernouli process
Bernouli trials that are statistically independent from each other.
P = 0.2
What does this mean in terms of a coin toss?
P= 0.2 means that each time there's a 20% chance of heads
definition of Bernouli trail aka Binomial trial
a sampling trial in which only one of two things can happen (ex heads/tails)
Bernouli process
consists of many bernouli trials
If cards a drawn with replacement...
Each card has an equal chance of being drawn.
If cards are drawn with replacement, it is equally likely that each will be drawn. Why is this?
Independence and Random Sampling
If cards are drawn w/ replacement , what is the probability that their sum will be 4?
Prob of 1 event + Prob of another event....
How do you calculate P(a|b)?
p(a|b) = p(a)*p(b)/p(b).
If A and B are independent events, what can we say about P(A |B)?
If A and B are indep, then:
P(A|B) = P(A)
How calc the P(x and y)?
P (x and y) = P(x |y) * p(y)
What are the two ways to show that a pair of events are independent?
1. P(A and B) = P(A)
2. P(A and B)= P(A)P(B)
*only work if the events are independent
*The same number should result from each
Three events are independent if.....
P x and y and Z) = P(x)* P(y)* P(z)
General multiplication rule (if you don't know whether/not 2 events are indep)
P(A and B) = P(A|B)P(B) = P(B|A)P(A)
If two players draw from three cards without replacement, then what's the prob of player 2 gets a 1 while player 1 gets a 2?
When player 2 goes to pick up a card, the cards left are 1 and 3. Therefore, 1/2.
If cards are drawn without replacement, the probability that player 1 gets a 1 is....
1/3
If 3 cards are present and are drawn w/out replacement, what are the three mutually exclusive events into which the event player 2 gets a 1 can be decomposed?
1. Player 1 gets 1 and P2 gets 1.
2. Player 1 gets 2 and P2 gets 1
3. Player 1 gets 3 and P2 gets 1
For these mutually exclusive events, which rule is used?
1. Player 1 gets 1 and P2 gets 1.
2. Player 1 gets 2 and P2 gets 1
3. Player 1 gets 3 and P2 gets 1
For exlusive events, the addition rule is used.
in this case:
P(1,1 or 2,1 or 3,1) = P(1,1) + P(2,1) + P(3,1)
A population is given a test upon which they receive a certain mean and strd dev. What raw scores correspond to the upper 25 and the lower 20 % of the population?
Once 20% and 25% are converted to decimals, these need to be matched with the appropriate sm or lg portion and its corresponding z value. Plug these numbers into the z equation and solve for x (the null or the ordinary mean of the distribution)to find the raw score, not x bar (the alternative or intervention)
above =
below =
will have a z that is positive
will have a z that is negative
scoring at least 150 would mean that you'd look in which column to find the proportion?
the sm portion column will give you the proportion, .06. Proportion*100 will give you the percent.
sm portion and larger portion
are a proportion in decimal form. if you want to convert it to percent, mult by 100
Placing students in a pop in the upper 25% (above). Look in ___ portion. Z value will be ___
Place students in upper 25%
look in sm. portion. Z will be positive A sm portion of .25 corresponds to a lg portion of .75
Placing students in pop in lower 20%. Look in ___ portion. Be/z it's lower 20% (below), z will be ____
Students in lower 20:
look in sm portion. Z will be negative.
The grades of 2 students are given. The instructor would like to know if they understand the material equally well. What kind of distribution is this?
Binomial distribution
To calculate binomial probability=6 where n=8 and p=1/2....
8!/(2!6!) * (1/2)^8

n!/((n-P)*P) *(p)^(n)
8 grades of 2 students are given. Nicole gets higher grades six of the times.The instructor would like to know if they understand the material equally well. When calculating probabilites, what 3 P=# should be established?
3 values
P (when n=8 and p=1/2) =6
P (when n=8 and p=1/2) =7
P (when n=8 and p=1/2) =8
where
p=1/2 was given in prob

~I think that you start with P=6 in your calc of binomial probability values be/c its the highest number of superior grades obtained by either of the students. Once you've calc P=6,7,8 in the binomial way, add these 3 probs together and compare the result to .05 to determine retain or reject the null.
How do you calculate the Binomial P=7, when n=8 and p=1/2...
8! /(1!7!)*(1/2)^8

n!/((n-P)*P) *(p)^(n)
How do you calc the binomial p=8 when n=8 and p=1/2
(1/2)^8
For multiple choice probs where 4 choices are present, what is the probability that a student makes all 5 problems by sheer guessing?
n=5
p=1/4

(p)^(n)
(1/4)^(5)
For a set of 10 probs, find the sm. # of items that a student has to do correctly to demonstrate that he is not doing them by sheer guessing? For this prob, what are the null and alt hypoth?
Ho=student is doing the probs by guessing
H1 = student is not doing the prob by guessing
8 grades are given for two different students. The instructor would like to know if the understand the material equally well. What are the null and alt hypoth?
Ho= both understand mat'l equally well
H1= 1 of the students understood the mat'l better than the other
Students A,B, and C are playing a shooting game. What kind of distribution will the results from this shooting activity produce?
This will be have a binomial distribution, since the results, hit or no hit, are discrete.
Students A, B, and C are playing a shooting game and each of them shoots once. Suppose that the probability for a hit is 0.3 for all of them and they shoot independently. What is the probability that exactly two of them make a hit?
n!/((n-P)!*(P)!)*(p)^(P) *(1-p)^(n-P)
in this case,
P=2
n=3
p=0.3
If you have independent events, what is the special rule that you can use?
The multiplication rule.
P(A and B)= P(A)*P(B)
If you don't have independent events, what is the rule that you MUST use?
P(A and B) = P(A|B)*P(B)
or P(A and B) = P(B|A) *P(A)
The multiplication rule is used for...
statistically independent events
When do you use the addition rule?
When you have mutually exclusive events, like (P(1,1 or P(2,1) )=P(1,1)+P(2,1) in drawing cards w/out replacement.
Students A,B and C are playing a shooting game and each one of them shoots once. Suppose the prob FOR A HIT is 0.3 for all of them and they shoot indep. What is the P that only A and B hit?
P(A hit) * P(B hit)* P(miss)
0.3 * 0.3 * 0.7
*****P(A|B)=
P(A|B)= P(A and B)/P(B)
Students a,b,c are shooting. Prob of hit is 0.3 & they shoot indep. What is the prob that student a hits and exactly 2 of them hit?
"Exactly hit" is indicative of the presence of a binomial calculation. P(a) and P(exact 2) can be broken down into two tinier mutually exlusive events, P(A hit B hit) and P (A hit C hit). These mutually exlusive events get added together. Within each one of the mut exclus events, P (A hit) is indep of B hit) so you can use the special form of multiplic rule (PAandB=Pa*Pb).You MUST multiply in the P of a miss, however. In sum,
P(A hit) and exactly 2 hit = P(only A and B) + P(A and C)
=.3*.3*.7 + .3*.3*.7
Which hypothesis is being tested in hypothesis testing?
Ho, the null
When the null hypothesis is retained it does not mean that the null hypothesis is proven, it only means that..
it may be true
Do we have any control over type II (Beta) error?
no
____ error is controlled and minimized by setting the __ level in hypoth testing.
Type 1, alpha
Can hypothesis testing tell which of the hypotheses being tested is true?
No. Hypothesis testing can only provide an educated guess about which one is likely to be true.
In hypothesis testing a decision about the null hypothesis is made upon...
actually observed and unobserved data
If Ho = students do equally well in stats classes at all times of day and H1= students do better in afternoon stats classes and p is .0119, what statistical conclusions can be made?
Reject Ho. The data indicate that students in the afternoon may do better in the stats class than those who take the early morning class.
Between what two scores do the middle 95% of sample means fall on a scale in which the mean is 100, std dev is 15, and sample size is 25? In this question, how do you determine the middle 95%?
100-95 = 5 percent, so there's five percent to be divided up between the two ends of the distribution.
5/2 = 2.5
Convert 2.5 to a propotion
2.5/100 = .025
If this will serve as the left end, make it negative (-.025) and look up corresponding z in the sm portion.
At the right end:
95 + .025 = .975
look in lg portion to find .975, then match this with a z value.
Type I error
reject the null when the null is true
Type I is what we try to minimize
=alpha
type II error
retain the null when the null is false
=Beta
If you're not asked for a p value, you can do hypothesis testing with z alone. How?
1. Your alpha level will determine the magnitude of your z critical, for alpha of .05, z=1.64, for alpha of .01 z is 2.33.
2. The sign of your z crit is determined by a comparison of H1 with Ho. If H1 > Ho, then z crit is pos.
3.If the calculated z is GREATER than z critical, then we reject the null. (This is diff than with p in which if calc p is LESS than .05 by the stat decion rule we reject the null).
How do you know which value to use in the alternative hypoth?
Whichever number corresponds to results following the treatment or intervention.
Are the null and alternative hypotheses stated in terms of population parameters or sample parameters?
null and alt are stated in terms of population
Ho does not equal x bar, rather, it is equal to mu
(Ho=mu). Why is this?
X bar refers to a sample statistic. The hypotheses are done in terms of the population.
A sampling distribution is done under the assumption that...
the null is true

135