DECK 12 INFERENCE MIXED Flashcards
what is a biased estimator?
When the sampling distribution (pile of sample stats, x bars or p hats) is NOT centered on the true population parameter. If you were weighing people and there was a 1 pound weight on the scale, the pile would be centered 1 pound higher. Baised.
What are the three chi-squared models?
goodness of fit, test for homogeneity, test for independence
What is a critical value?
It is the amount of standard errors you’ll reach out, depending on your confidence (a t or z). Example.. 68% crit z = 1 .. For 95% crit z = 2 (well, 1.96).. For means.. Use t crits
you reject when _____________ evidence
you reject when YOU HAVE EVIDENCE
In order to reject a null hypothesis, you need ___________
evidence
What is the missed opportunity error? (the “I didn’t notice” error)
Type 2
What is alpha?
It is the rejection area. Generally, we use .05. The significance level.
What is a sampLING distribution?
a pile of statistics. A pile of p-hats or x-bars.
What is the null model (the sampling distribution) in a 2 sample mean t-test?
a pile of differences of TWO MEANS samples, taken from a bunch of PAIRS of samples. Take two samples, calculate two means, subtract to get a difference, PUT THE DIFFERENCE IN THE PILE.
What is the NULL HYPOTHESIS?
The DULL HYPOTHESIS, the nothing changed hypothesis, the no-difference hypothesis, the “he’s telling the truth” hypothesis, the “No trickery” hypothesis
How is a margin of error different from a standard error?
A margin of error is a NUMBER OF STANDARD ERRORS. It is how far up or down you go in a confidence interval. A standard error tells you about the spread of a pile of statistics (a sampling distribution).
What is the differnce between standard error and standard deviation?
Standard error is the typical distance a STATISTIC is from the mean in a sampling distribution (pile of a bunch of sample’s statistics) and Standard Error is the typical distance a DATUM is from the mean in a pile of raw data.
For both a chi squared test for independence and a regression t test you are looking for an association, how do the null hypotheses differ?
The chi squared will be in words.. Ho: the variables are not associated. The regression t test will be with symbols (and words). Ho: Beta = 0 (the slope is zero) . Saying “beta=0” is the same as saying “there is no association”
when do you need crits?
in confidence intervals (and old fashioned hyp tests.. We look at Z to see if greater than crit.)
you fail to reject when ____________ evidence
you fail to reject when you DON’T HAVE EVIDENCE
notation: what is x-bar
mean of your sample
What is the null for a chi squared test for homogeneity?
The [samples of —] are similarly distributed.
How is a paired T test different from a 2 sample mean T test?
A paired test talks about an AVERAGE OF DIFFERENCES from one list, whereas a 2 sample mean t-test talks about a DIFFERENCE OF AVERAGES between two samples.
when is data “paired”
when you have 2 measurements of the same variable on the same subject (or matched subjects)
What is the null for a 2 sample mean T?
mu1=mu2 OR mu1-mu2=0 there is no diff
When you are doing PAIRED or MATCHED or BLOCKED tests.. What are you finding?
The average difference.. You are doing 1 sample procedures on a NEW THIRD LIST OF DIFFERENCES
What is a confidence interval?
it is a parameter catcher.. Like a fishing net. We stand at our statistic, and reach up and down a margin of error, and hope to CATCH the parameter? sometimes we do, sometimes we don’t? but we never know.. Mooo hooo hooo haaaa haaa haaa (evil laugh)
What does CLT say about the distribution of the population?
Not much… just that it doesn’t matter what it is.. With large samples.. The SAMPLING dist will be approx normal (dist of stats.. NOT DATA)
What is difference between population of interest and parameter of interest?
Population is the WHO (subjects you measure, beads people) Parameter is the actual number you want (like % of or AVG)
what is a parameter?
some numerical summary of a population. Often called “the parameter of interest.” It is what we are often trying to find.. It doesn’t vary. It is out there and STUCK at some value, it is the truth, and you’ll probably not ever know it! We try to catch them in our confidence intervals, but sometimes we don’t (and we don’t know it!). It Could be the mean of a population, the standard deviation of a population, the proportion of successes in a population, the slope calculated from a population, a difference of 2 means from 2 population, a difference of 2 proportions from population
How can you tell if it is a T or a Z procedure?
YES-NO-PROP-Z. Remember t for means, z for proportions. Think of the subjects. Could you get the info in a yes/no fashion? if so, then z-props. Do you need to get a number from each subject? if so, then t-means.
Can you make a 100% confidence interval?
Sure, I’m 100% confident that it will snow between 0 and 500 feet tomorrow.
How do you find Margin of Error from an inteval?
It is half the width.. (HI-LO divided by 2) Remember you stand at statistic (point estimate) and reach up and down a Margin of Error. So an inteval is always exactly 2 margins of error wide)
what does “95% confidence” in a 95% confidence interval mean? (explain the confidence level)
It means if we took a ton of samples, and made confidence intervals from each of them,ABOUT 95% of the intervals would contain the parameter, 5% would not.
sample size calcs FOR PROP AND MEANS
n= (z^2 * p * q )/ (ME ^2) and n = ( t*s / ME) ^ 2 (start with Z then do T)
What does the CLT say about the distribution of actual sample data?
Nothing? The sample will be distributed similar to the population. Bimodal populations have bimodal samples. The CLT only talks about distributions (histograms) of sample statistics, of summaries, which are groups of means.., NOT OF INDIVIDUALS!!!! NOT DATA
Which hypothesis shows what you are trying to prove?
The alternative.
What happens to a pile of statistics if you take larger samples?
All of the x-bars or all of the p-hats will get closer to eachother, and closer to the parameter ( mu or p)
If you are doing a 2 tailed test with alpha=.05.. What confidence interval goes with that?
95% confidence interval (there is .025 in each tail)
who invented the t model?
Bill Gosset, guiness brewing company.
how do you find deg freedom?
n-1 for one sample, for 2 samples you must use calculator. For PAIRED use n-1, REGRESSION IS n-2
What is beta?
It is probability that you’ll make a Type II error.. P(Type II error)
What is “statistically significant?”
When our sample statistic is so far away from what we were expecting that we don’t think that it was due to random sampling error. Then is statistically significant. When p-value is below the alpha, we say “statistically significant”.. Low p-values are statistically significant.
how are t models like Normal models?
both are unimodal and symmetric. T models aren’t as high and have more area in tails, that’s why you have to reach out a little further than z for same confidence.
What is a way to think about the three conditions?
- Sample is random2. Sample is small enough (<10%)3. Sample is large enough (np&nq>10 for props, n>30 for means or the histogram is normalish)EXTRAS: chi squared exp at least 5 in each cell, regression- random resid
What is the null model(the sampling distribution) in a 1-sample mean T test?
A pile of means from a bunch of samples.
notation: what is mu
true population mean (average)
notation: what is z*
critical z, how many SE you are reaching up and down in a confidence interval for proportions
What is the null for a chi squared GOF test?
The distribution fits [the expected distribution]
How do you find point estimate from an interval?
It is in the dead center of interval, so take the average of the upper and lower bounds.
How do you find df in 2 samples?
USE CALCULATOR.(or smaller sample-1). you have to run an interval or a test on your TI and read the output (unless you want to use the equation.)
What if you want more cofidence with same size interval?
increase your sample size
Can you draw the alpha/beta/power diagram?
BE ABLE TO SKETCH THE ALPHA BETA POWER DIAGRAM from the original pregnancy worksheet. Know where everything is. This helps you understand how alpha, beta and power interact.
How else can you explain power?
The likelihood you correctly reject a false null.. The likelihood you correctly detect what you were trying to detect
What is a null model?
It is a sampling distribution. It tells us how sample statistics would vary if the null were true. It is centered at the null. A pile of p-hats or x-bars.
notation: what is t*
critical t, how many SE you are reaching up and down in a confidence interval for means
Describe the distribution of a sample
It will look like the population. The distribution of a sample is a histogram made from the sample, which will look kind of like the population. If the population is bimodal, then the distribution of the sample is bimodal. The SAMPLING distribution of a bunch of means, however, will look normalish.
What is an unbiased estimator?
When the sampling distribution (pile of sample stats) is centered on the true population parameter.
What is the “you think it worked but it didn’t” error?
Type 1
What is a point estimate?
Your p-hat or your x-bar. Your best guess. What you got in your sample. It is in the middle of the interval.
how can you decide the right test? What are the 3 questions?
1 or 2 samples? Proportions (z) or Means (t)? Test or Interval?(YES/NO/PROP/Z)
If the p-value is low, (below alpha), how do you write conclusion?
With p-value this low (show p value < alpha) I reject the null hypothesis. There is strong evidence that the proportion of students who eat rice has changed.
What is a t-crit?
It is the same as z crit. It is the number of sd you reach out in your CI. To find it, do INVT(area in one tail, degrees of freedom)
How are power and alpha related?
they go up and down together
What are we confident in?
our confidence lies in our interval. if we took another sample.. We’d have a different interval..
What are the xbar and mu in t-test? (on calc)
xbar is your sample mean, mu is your hypothesized population mean
What if you want more confidence?
get a bigger net.. (wider conficence interval) (or increase sample size)
Will 95% of other statistics be within my interval?
NO!!! You have no idea where your statistic is (or your interval) in regards to true parameter
What are the mean and standard deviation of a sampling distribution for a mean?
mean is mu and standard deviation is sigma/root n (look at formula sheet) N(mu, sigma/rootn)
What are the names of four two-sample procedures we do?
2 proportion Z interval2 proportion Z test2 sample mean T interval2 sample mean T test
If there is a screening test for mathphobia, describe a type 1 and a type 2 error
Type 1: You think the person has mathphobia, but they don’tType 2: They have mathphobia, but you didn’t notice
What are conficence intervals for?
PARAMETER CATCHERS. They are an attempt to say what the true population parameter is.. It is our best guess. “We think that there will be between 8 and 12 inches of snow”
What is the difference between the distribution of a sample and a sampling distribution?
A distribution of a sample is just a histogram of the DATA in a sample. A sampling distribution is made from an bunch of sample STATISTICS. It is the distribution of the statistic that was calculated from those many many samples.
N ( ?1 , ?2 ) what does this mean?
it means NORMAL models centered at ?1 With a standard deviation of ?2
What are the 3 steps in hypothesis testing AFTER YOU CHECK CONDITIONS?
- Make your Ho and Ha 2. Make a Null Model (centered at null, use your Ho as center and in calculations, use your sample size).. This is a sampling distribution for the statistics if the null were true. 3. THINK then CHECK. use your statistic (p-hat, x-bar, phat1-phat2, xbar1-xbar2) to calculate your test statistic and then p value
notation: what is p-hat
sample proportion (percent in our sample)
how are beta and power related
as one increases, the other decreases, and vice versa. They have to because they BOTH ADD TO ONE!!! Power + Beta = 1
what is another name for alpha level?
a significance level
Can you decrease alpha while increasing power (even though they move together?)..
Yes.. increase samle size. They move together with constant sample size.
What does Central Limit Theorem Say?
It basically says.. NO MATTER WHAT SHAPE THE POPULATION IS (normal, bimodal, uniform, skewed, crazy.. ) If you make a histogram of a bunch of means taken from a bunch of samples, that histogram will be unimodal and symmetric WITH LARGE ENOUGH SAMPLES.. Close to normal. So.. A nerdy way to say it is: The sampling distribution of means is approximately normal no matter what the population is shaped like. The larger the sample size, the closer to normal. (the normal curve is just a model.. the sampling distribution is close to it, but not it! we use the model anyway!)
If you take two bits of information from each subject there are two possible tests you could do. If the data is quantitative, you can do___ and if it is categorical, you can do a ______
regression t test, chi-squared test for independence.
notation: What is Ha
The alternative. This is what you are trying to prove.
What is the null model (the sampling distribution) in a 2-proporiton Z test?
a pile of differences of TWO PROPORTIONS, taken from a bunch of PAIRS of samples. Take two samples, calculate proportions, subtract to get a difference, PUT THE DIFFERENCE IN THE PILE.
With regression computer output, how is the t-ratio and the p-value calculated?
T ratio is just SLOPE/ST ERROR and the p value is just TCDF(T ratio, 9999, n-2)
What is a Chi squared model?
A sampling distritubion. If you took a bunch of samples and calculated a bunch of Chi-square statistics, the pile of chi squareds would look like that.
what are the names of four one-sample procedures we do?
1 proportion Z interval1 proportion Z test1 sample mean T interval1 sample mean T test
What is the mean and mode of a chi squared model? What is the 5% cutoff chi-squared?
The mean is the degrees of freedom and the mode is df-2. The cutoff is at 1.5df+3.
What is alpha?
It is the rejection threshold. You reject p-values below it.. It is how willing you are to make a Type 1 error? alpa=P(Type I error)
notation: what is a p-value
At the end of a hypothesis test, it is the likelihood of getting your results if the null was true.
what is a statistic
some numerical summary of a sample.. Could be the mean of a sample, the standard deviation of a sample, the proportion of successes in a sample, the slope calculated from a sample, a difference of 2 means from 2 samples, a difference of 2 proportions from 2 samples, a difference of 2 slopes from 2 samples.. you can make sampling distributions for any of these, and they will all be centered around the parameter…
what does “ 90% confidence” mean in a 90% interval for avg weight of mice (2.3, 3.5) in ounces
90% of intervals made this way would catch the true mean weight. If you took 1000 samples and made 1000 intervals, about 900 intervals would catch the true weight, 100 would not.
Do parameters vary?
NO!!! Statistics do because they are calculated from samples, different samples have different statistics. they vary from sample to sample. The parameter doesn’t vary because there is only one.
If you have one group and ask them two questions, what type of chi squared test is it?
Test for independence.
Do you use p-hat or p-null when you calculate your standard error in a null model?
use p-null..
What is diff between homogeneity and test for independence?
homogeneity is more than one sample and asking about one variable, independence is just one sample with two variables.
What is sampling error?
same as sampling variability.. The natural variability between STATISTICS.. NOT DATA!!! . We call it error EVEN THOUGH YOU MADE NO MISTAKES!!!
what is df for goodness of fit?
cells - 1
DEGREES OF FREEEDOM: one sample t, two sample T, regression, chi square gof, chi square hom/indep
one sample t: n-1two sample t: calc or smaller n-1regression: n-2Chi square GOF: cells-1CHi square hom/indep: ROW-1 x COL-1
notation: what is mu - mu
true difference between two populatinon means
What is the null model (the sampling distribution) in a regression t test for slope?
A pile of slopes taken from a bunch of samples
How do you make confidence interval for slope?
STAT +/- CRIT SE of slope
What are the mean and standard deviation of a sampling distribution for a proportion?
mean is p and sdandard deviation is root pq/n (look at formula sheet) N(p, root (pq/n) )
How can you increase power?
Increase alpha or increase sample size..
What is a standard error?
The typical, or expected, error. It is how far off you are expecting your statistic to be from the parameter. It is calculated like the standard deviation, but we are using sample statistics.. We don’t know the true parameters, so we estimate with statistics adding error to our calculation
If you are testing to see if a math program works in your town (that will cost 4 million to implement) describe a type 1 and type 2 error.
Type 1: you think it worked, but it didn’t so you spend 4 million on a program that isn’t good.Type 2: It worked, but you didn’t notice, so you miss the opportunity to adopt a good math program.
If you take two bits of information from each subject there are two possible tests you could do. If the a chi squared test would be for _____ data and a regression t-test would be for ______ data
categorical, quantitative.
notation: what is p - p
true difference between two population proportions (percents).
How do you write conclusion if you reject?
With such a low p-value, (p-value < alpha) I reject the null hypothesis. There is strong evidence that the proportion of students who eat rice has changed.
how do you find z and t crit?
for z crit.. INVNORM(area in 1 tail) for t crit. INVT(area in 1 tail, deg freedom). area in 1 tail is just ( 1-CL) / 2
WHat is the null for a chi squared test for independence”
The [two variables in context] are independent.
What is the “nearly normal” condition for means? (aka large enough sample)
If n>30, good to go. If n<30, then you have to make sure the histogram of the sample looks normalish.
What is difference between 2 Samp T test and a PAIRED T Test
In a two sample T test you are comparing TWO SAMPLE AVERAGES to eachother. In a PAIRED T test you are looking just at JUST ONE average of the THIRD LIST… They are paired.. So you find each individual BEFORE-AFTER and take the average of all of those differences. You do ONE SAMPLE T TEST on it because you really have one mean. You just the average or the difference list.
What is a p-value
It is the probability of getting your sample randomly if the null were true. Basically, how likely is it that your sample statistic came from the Null Model.
What is the null model (the sampling distribution) in a chi squared test?
a pile of chi-squared statistics calculated from a bunch of samples
How do you find Expected Count?
for GOF: Exp %(total).. For indep and homog: ROW*COL/TOTAL
Can you prove a null hypothesis true?
NO.. We just fail to reject it.
What is the other conditions for regression inference?
Random residuals (equal random scatter, no pattern).
How can you decrease alpha and beta at the same time?
increase sample size. this will also increase power.
interpret this 90% confidence interval for avg weight of mice (2.3, 3.5) in ounces
I am 90% confident that the mean weight of mice is between 2.3 and 3.5 ounces
What is the “nearly normal” condition for proportions? (aka large enough sample)
np>10 and nq>10. actually Show this calculation
IF you are testing to see if a marketing program increases sales, describe a type 1 and type 2 error.
Type 1: you think it increased sales but it didn’gType 2: It actually increased sales but you didn’t notice
What do you have to look for with two sample procedures:
- You have to check the conditions for each sample AND2. The samples have to be independent from eachother
Is a confidence interval a PROBABLILITY?
NO
When do you know it is GOF test?
When you have ONE ROW or ONE COLUMN… then it gives you a ratio , like 1:2:5 or it gives you expected percents.
What is the general formula for ALL CONFIDENCE INTERVALS?
STAT +/- CRIT SE
notation: what is xbar- xbar
difference between two sample means
what is df for chi squared homogeneity or independence?
(rows-1)(columns - 1) (remove a column and row.. Count boxes)
What is the null model (the sampling distribution) in a 1-proportion Z test?
A pile of proportions (%) from a bunch of samples.
How do you write conclusion if you fail to reject?
With a p-value this high (p-value > alpha) I fail to reject the null. There is not enough evidence to say that more students like eggs now.
what is a test statistic?
a t or z score (or chi squared) that you use to find a p value
WHat is the null model (the sampling distribution) in a paired T-Test?
A pile of average differences. Remember that in a paired test, you are getting an individual differenc from each pair of data, then finding the average of differences.
What is the “ large enough sample” condition for Chi Square?
Make sure that there are at least 5 in each expected cell.
One tail or 2 tailed? How do you tell?
if it just says “changed” or “different”.. Then it is 2 sided.. DOUBLE THE P VALUE!If it says “more” “less than” “greater” etc.. Then it is just one sided..
notation: what is p
true population proportion (percent in the population)
Can you accept a null hypothesis? Can you say “keep the null?”
Never accept a Ho, don’t keep the Null. simply “FAIL TO REJECT THE NULL”
How wide is a confidence interval? (how many ME?)
It is 2 margins of error wide ALWAYS (DON’T CONFUSE WITH NUMBER OF SE)
what happens to t models as n gets larger?
The models look more like the normal model. An infinite sample size would give a t model identical to the normal model.
If you have one group and ask them one question, what type of chi-squared test is it?
Goodness of fit test.
What does a “significance level of .02” mean?
set alpha= .02 and reject only p-values below that
What do we compare our p value to in a hypothesis test?
compare it to alpha. if p value
If you are doing a one tailed test with alpha=.05.. What confidence interval goes with that?
90% confidence interval tests a one tailed test. There is 5% in the tail.
notation: what is phat - phat
difference between two sample proportions
If you have multiple groups and ask them one question, what Chi-squared test is it?
Test for homogeneity.
What is the Fundemental Theorem of Statistics?
The CLT!! The Central Limit Theorem!
Are models what really happen?
No. A model train is not a real train. We use models to say what kind of happens.
What is the “nearly normal” condition for regression? (aka large enough sample)
Make sure the histogram of the residuals is normalish.
How do statistics from big samples compare to small? (notice this doesn’t ask about DATA)
Larger sample statistics have less variablility, so statistics from larger samples are closer to eachother and to the parameter. Statistics from smaller samples are more spread out, further away from true parameter.
What is sampling variability?
same as sampling error. The natural variation of sample statistics.. NOT DATA.. Samples vary. so do their statistics.. Parameters do not vary!
How is a confidence interval made?
statistic +- margin of error. Statistic +- (crit * s.d ). Stand at the statistic, reach out up and down a margin of error, and hope that you catch the parameter.
What is a margin of error?
critical * s.d. It is how far you reach out in a confidence interval.. You reach up and down one of these, so the interval is actually 2 margins of error wide.
What is the null for a paired T test?
xbar diff=0 (the average diff is zero)
What is statistical inference?
Using a statistic to infer something about a parameter.. Basically, using a sample to say something about a population.
CONDITIONS: What are the three conditions that you have to check with pretty much every inference procedure (every test and every interval)
- Randomly chosen sample (or assigned treatment). Circle the word random or explain why you think it is. 2. Sample size is less than 10% of the population. Show that 10n is less than N. Example, for 50 students, write 10(50)=500 is less than all students.3. Nearly normal (or large enough sample)- this differs based on the type of data and test.
What is the null for a 2 prop Z?
p1=p2 OR… p1-p2=0, there is no diff
notation: what is Ho
The NULL, the dull, the “things haven’t changed” hypothesis
If the p-value is high, (above alpha), how do you write conclusion?
With a p-value this high(show p value < alpha) I fail to reject the null. There is not enough evidence to say that more students like eggs now.
