Psych 320 Statistics final
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- 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?
-
Use addition rule:
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
n=total number of grades
~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