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AP Statistics > DECK 7 Confidence Intervals and Hypothesis Tests part A > Flashcards

DECK 7 Confidence Intervals and Hypothesis Tests part A Flashcards

1
Q

notation: what is mu

A

true population mean (average)

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2
Q

notation: what is p

A

true population proportion (percent in the population)

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3
Q

notation: what is x-bar

A

mean of your sample

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4
Q

notation: what is p-hat

A

sample proportion (percent in our sample)

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5
Q

notation: what is a p-value

A

At the end of a hypothesis test, it is the likelihood of getting your results if the null was true.

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6
Q

notation: what is z*

A

critical z, how many SE you are reaching up and down in a confidence interval for proportions

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7
Q

notation: what is t*

A

critical t, how many SE you are reaching up and down in a confidence interval for means

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8
Q

notation: what is mu - mu

A

true difference between two populatinon means

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9
Q

notation: what is p - p

A

true difference between two population proportions (percents).

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10
Q

notation: what is xbar- xbar

A

difference between two sample means

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11
Q

notation: what is phat - phat

A

difference between two sample proportions

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12
Q

notation: what is Ho

A

The NULL, the dull, the “things haven’t changed” hypothesis

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13
Q

notation: What is Ha

A

The alternative. This is what you are trying to prove.

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14
Q

What is the difference between the distribution of a sample and a sampling distribution?

A

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.

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15
Q

What is a sampLING distribution?

A

a pile of statistics. A pile of p-hats or x-bars.

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16
Q

Are models what really happen?

A

No. A model train is not a real train. We use models to say what kind of happens.

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17
Q

What is “statistically significant?”

A

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.

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18
Q

What is the differnce between standard error and standard deviation?

A

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.

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19
Q

What does CLT say about the distribution of the population?

A

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)

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20
Q

What are the mean and standard deviation of a sampling distribution for a proportion?

A

mean is p and sdandard deviation is root pq/n (look at formula sheet) N(p, root (pq/n) )

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21
Q

What does Central Limit Theorem Say?

A

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!)

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22
Q

What is difference between population of interest and parameter of interest?

A

Population is the WHO (subjects you measure, beads people) Parameter is the actual number you want (like % of or AVG)

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23
Q

What happens to a pile of statistics if you take larger samples?

A

All of the x-bars or all of the p-hats will get closer to eachother, and closer to the parameter ( mu or p)

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24
Q

What does the CLT say about the distribution of actual sample data?

A

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

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25
Q

N ( ?1 , ?2 ) what does this mean?

A

it means NORMAL models centered at ?1 With a standard deviation of ?2

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26
Q

Describe the distribution of a sample

A

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.

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27
Q

What is a standard error?

A

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

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28
Q

How do statistics from big samples compare to small? (notice this doesn’t ask about DATA)

A

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.

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29
Q

What is statistical inference?

A

Using a statistic to infer something about a parameter.. Basically, using a sample to say something about a population.

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30
Q

what is a statistic

A

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…

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31
Q

what is a parameter?

A

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

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32
Q

What is the Fundemental Theorem of Statistics?

A

The CLT!! The Central Limit Theorem!

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33
Q

What is sampling variability?

A

same as sampling error. The natural variation of sample statistics.. NOT DATA.. Samples vary. so do their statistics.. Parameters do not vary!

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34
Q

What is sampling error?

A

same as sampling variability.. The natural variability between STATISTICS.. NOT DATA!!! . We call it error EVEN THOUGH YOU MADE NO MISTAKES!!!

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35
Q

what happens to t models as n gets larger?

A

The models look more like the normal model. An infinite sample size would give a t model identical to the normal model.

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36
Q

What is an unbiased estimator?

A

When the sampling distribution (pile of sample stats) is centered on the true population parameter.

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37
Q

how are t models like Normal models?

A

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.

38
Q

what is a biased estimator?

A

When the sampling distribution (pile of sample stats) 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.

39
Q

What are the mean and standard deviation of a sampling distribution for a mean?

A

mean is mu and standard deviation is sigma/root n (look at formula sheet) N(mu, sigma/rootn)

40
Q

What if you want more confidence?

A

get a bigger net.. (wider conficence interval) (or increase sample size)

41
Q

when do you need crits?

A

in confidence intervals (and old fashioned hyp tests.. We look at Z to see if greater than crit.)

42
Q

how do you find z and t crit?

A

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

43
Q

What is a margin of error?

A

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.

44
Q

How is a confidence interval made?

A

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.

45
Q

What is a critical value?

A

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

46
Q

what does “95% confidence” in a 95% confidence interval mean? (explain the confidence level)

A

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.

47
Q

Do parameters vary?

A

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.

48
Q

How wide is a confidence interval? (how many ME?)

A

It is 2 margins of error wide ALWAYS (DON’T CONFUSE WITH NUMBER OF SE)

49
Q

What is a confidence interval?

A

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)

50
Q

Can you make a 100% confidence interval?

A

Sure, I’m 100% confident that it will snow between 0 and 500 feet tomorrow.

51
Q

Is a confidence interval a PROBABLILITY?

A

NO

52
Q

What are we confident in?

A

our confidence lies in our interval. if we took another sample.. We’d have a different interval..

53
Q

Will 95% of other statistics be within my interval?

A

NO!!! You have no idea where your statistic is (or your interval) in regards to true parameter

54
Q

What if you want more cofidence with same size interval?

A

increase your sample size

55
Q

What are conficence intervals for?

A

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”

56
Q

How is a margin of error different from a standard error?

A

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).

57
Q

What is a point estimate?

A

Your p-hat or your x-bar. Your best guess. What you got in your sample. It is in the middle of the interval.

58
Q

What is a t-crit?

A

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)

59
Q

How do you find Margin of Error from an inteval?

A

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)

60
Q

How can you tell if it is a T or a Z procedure?

A

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.

61
Q

how do you find deg freedom?

A

n-1 for one sample, for 2 samples you must use calculator. For PAIRED use n-1, REGRESSION IS n-2

62
Q

How do you find point estimate from an interval?

A

It is in the dead center of interval, so take the average of the upper and lower bounds.

63
Q

who invented the t model?

A

Bill Gosset, guiness brewing company.

64
Q

interpret this 90% confidence interval for avg weight of mice (2.3, 3.5) in ounces

A

I am 90% confident that the mean weight of mice is between 2.3 and 3.5 ounces

65
Q

what does “ 90% confidence” mean in a 90% interval for avg weight of mice (2.3, 3.5) in ounces

A

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.

66
Q

Can you accept a null hypothesis? Can you say “keep the null?”

A

Never accept a Ho, don’t keep the Null. simply “FAIL TO REJECT THE NULL”

67
Q

What does a “significance level of .02” mean?

A

set alpha= .02 and reject only p-values below that

68
Q

what is another name for alpha level?

A

a significance level

69
Q

How do you write conclusion if you reject?

A

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.

70
Q

How do you write conclusion if you fail to reject?

A

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.

71
Q

What is a p-value

A

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.

72
Q

What is the NULL HYPOTHESIS?

A

The DULL HYPOTHESIS, the nothing changed hypothesis, the no-difference hypothesis, the “he’s telling the truth” hypothesis, the “No trickery” hypothesis

73
Q

Which hypothesis shows what you are trying to prove?

A

The alternative.

74
Q

One tail or 2 tailed? How do you tell?

A

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..

75
Q

If the p-value is low, (below alpha), how do you write conclusion?

A

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.

76
Q

If the p-value is high, (above alpha), how do you write conclusion?

A

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.

77
Q

you reject when _____________ evidence

A

you reject when YOU HAVE EVIDENCE

78
Q

you fail to reject when ____________ evidence

A

you fail to reject when you DON’T HAVE EVIDENCE

79
Q

In order to reject a null hypothesis, you need ___________

A

evidence

80
Q

Can you prove a null hypothesis true?

A

NO.. We just fail to reject it.

81
Q

Do you use p-hat or p-null when you calculate your standard error in a null model?

A

use p-null..

82
Q

What are the 3 steps in hypothesis testing AFTER YOU CHECK CONDITIONS?

A
  1. 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
83
Q

what is a test statistic?

A

a t or z score (or chi squared) that you use to find a p value

84
Q

What is a null model?

A

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.

85
Q

What are the xbar and mu in t-test? (on calc)

A

xbar is your sample mean, mu is your hypothesized population mean

86
Q

What is alpha?

A

It is the rejection area. Generally, we use .05. The significance level.

87
Q

What do we compare our p value to in a hypothesis test?

A

compare it to alpha. if p value

88
Q

how can you decide the right test? What are the 3 questions?

A

1 or 2 samples? Proportions (z) or Means (t)? Test or Interval?

(YES/NO/PROP/Z)

89
Q

what are the names of four one-sample procedures we do?

A

1 proportion Z interval
1 proportion Z test
1 sample mean T interval
1 sample mean T test

90
Q

What are the names of four two-sample procedures we do?

A

2 proportion Z interval
2 proportion Z test
2 sample mean T interval
2 sample mean T test

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