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Stats > 250A Midterm > Flashcards

250A Midterm Flashcards

1
Q

advantages and disadvantages of the mode

A

+ actually occurs in data
+ only thing that makes sense for nominal data
+ not affected by outliers
- not able to manipulated mathematically bc no formula

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2
Q

advantages and disadvantages of the median

A 
\+ not affected by outliers
\+ does not require interval assumptions
\+ makes absolute error as small as possible
- does not enter into equations nicely
- cannot be decomposed
- poor estimator of the population value
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3
Q

advantages and disadvantages of the mean

A

+ unbiased (best estimate of pop mean)
+ makes average squared errors as small as possible
+ has a mathematical formula so can be manipulated
- affected by outliers
- need interval scale

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4
Q

+ and - of IQR

A

+ good for boxplots
+ does not assume normality of data
- throws away too much data

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5
Q

what does variance mean?

A

variance tells us the average squared deviation from the mean – how much, on average, each observation differs from the mean in squared units.
proportional to average squared difference between all pairs of observations so it summarizes both how different scores are from each other and how different they are from the mean!

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6
Q

what is standard deviation?

A

SD is average deviation from the mean: how much on average each observation differs from the mean

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7
Q

why do we use variance and sd instead of mean deviation and mean absolute deviation?

A

sum of mean deviations is always 0 so this doesn’t tell us anything
var and sd are useful mathematically because you can partition them.
mean absolute deviation is biased and inconsistent.

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8
Q

difference between trimmed and windsorized means/variances

A

trimmed = chop off top x% and bottom x% of data and recompute mean and var
windsorized: same as trimmed but replace missing data with new lowest and highest values
(if these procedures don’t change your statistics, your stats are robust. cool)

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9
Q

why would we want trimmed/windsorized stuff?

A

mean and var are especially sensitive to outliers in small samples, which can ruin statistical tests. so we may want indices that are more robust (varies little from one sample to another)
decrease influence of extreme values

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10
Q

why divide by n-1 when computing sample variance?

A

because dividing by n leads to a biased estimate of variance – the long run average will be too small
also, because we lose one degree of freedom in estimating xbar from the sample (now that we have estimated xbar, not all data points are free to vary - one is fixed)

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11
Q

chief problem in interpreting variance?

A

it’s in a squared metric and that don’t make no sense, bruh

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12
Q

how is standard deviation interpreted?

A

it’s in the units of your variable – average difference between an observation and the mean

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13
Q

what is expected value?

A

long run average of a statistic – if you resample infinity times and compute the statistic, the value that the statistic converges to is its expected value

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14
Q

what is an unbiased estimator?

A

when the expected value of the sample statistic is the population parameter

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15
Q

true/false: if a statistic is an unbiased estimator of a parameter, the statistic must have a symmetric sampling distribution.

A

False

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16
Q

how would you go about empirically (without equations) determining the bias and efficiency of sample mean and median in terms of estimating the population mean?

A

take a bunch of samples from your population and take the mean, median, and sd of each one. then construct a sampling distribution from this iterative procedure. if the mean of the sampling distribution = the population mean, it’s unbiased. if the standard error of the sampling distribution is small, it’s efficient.

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17
Q

define degrees of freedom

A

degrees of freedom is how many values in your sample are free to vary. e.g., if you calculate the mean, you lose a degree of freedom because now all your observations are not free to vary. one is fixed.

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18
Q

define linear transformations

A

t = a*x + b

adding or subtracting a constant or multiplying or dividing by a constant

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19
Q

effect of linear transformation on mean, sd, variance, relative ordering, and statistical tests

A

adding/subtracting: add/subtract same amount from mean (just shifts distribution left or right)
multiplying/dividing: multiplies/divides mean by the constant, variance by the square of the constant, sd by the constant
DO NOT AFFECT relative ordering or results of statistical tests

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20
Q

describe standardization transformation

A

z = (x - xbar)/s
standardization gives you a distribution with mean = 0 and sd = 1
so tells you how many deviations each value is from mean
to calculate probabilities from this distribution, need normality of data

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21
Q

goal of box-cox (power transformations)

A

optimize normality of predictors
box-cox considers all possible power transformations and computes likelihod of data under normal distribution…then finds the exponent that makes the data most likely

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22
Q

are all transformations monotonic (order preserving)?

A

yes, even nonlinear

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23
Q

what do nonlinear transformations preserve/not preserve

A 
preserve order (bc monotonic) but not shape -- changes relative standing of data points
so results of statistical tests may not be preserved
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24
Q

nominal scale properties

A

scale that classifies people
no numeric/cardinal ordering
classifications mutually exclusive

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25
Q

ordinal scale properties

A

scale that conveys order but no equal distance

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26
Q

interval scale properties

A

dif between scores has same meaning throughout the scale
no true 0
minimum requirement for most statistics

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27
Q

ratio scale properties

A

dif between scores has same meaning but now we have a true 0 so we can talk about ratios of scores

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28
Q

when linear transformations are performed on interval scales, do we maintain same ratios?

A

well it’s no good to talk about ratios in interval scales but if it’s a ratio scale, then yeah you should keep the same ratio after a linear transformation

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29
Q

Why is the normal distribution so important in psychological research?

A

Because it allows us to compute probabilities of observing a score or test statistic

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30
Q

How can we explore the degree to which sample data are normally distributed?

A

Graph your data and look at it
Impose a normal density over your data and look at it
Examine mean, sd, skewness, and kurtosis
normal has skew = 0 and kurtosis = 3

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31
Q

How are areas under normal curve linked to probabilities?

A

p(selecting a case in some range) corresponds to area under the curve between those values

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32
Q

True/false: if a sampling distribution is unbiased, then it must be symmetric and normal as well.

A

False

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33
Q

Why do we need to know theoretical probability distributions like the normal, chi-square, t, etc.?

A

because important things like test statistics follow these distributions and we want to know the probability of obtaining a particular test statistic under different assumptions

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34
Q

How does one find the area under a curve?

A

area under curve from x to y: integrate density function from x to y or use software

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35
Q

Why can’t we compute the exact probability of a sample result instead of a probability for a range of values?

A

In continuous distributions, the probability of obtaining any one value is 0 so you must look at ranges

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36
Q

What do tabled values of the standard normal distribution actually tell you?

A

Probability (area under the curve) to the left of whatever the given value is

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37
Q

Under what conditions can we use a z-score table to compute probabilities of a sample result?

A

If your data are normal or sample size is greater than 30

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38
Q

Distinguish between measures of absolute standing and relative standing.

A

Uhh I don’t know

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39
Q

What is sampling error?

A

Sampling error is random variation from sample to sample due to chance
the numerical value of a statistic will probably deviate from the parameter it’s estimating as a result of the particular observations that happened to be included in the sample

40
Q

What do sampling distributions tell us?

A

Degree of sample-to-sample variability we can expect by chance through sampling error

41
Q

Why do we need sampling distributions?

A

Sampling distribution tells us the expected sample results under some assumptions, so we can calculate the probability of our sample results!

42
Q

Why is random sampling so important? What are some consequences of nonrandom sampling?

A

Random sampling leads to external validity (the generalizability of your results). A representative sample can generalize results. If your sample is biased, your results depend on your sample and don’t generalize.

43
Q

If the mean of a sampling distribution for a statistic equals the population parameter, then we say it is ________.

A

unbiased

44
Q

What is standard error of the mean and how is it different from a standard deviation of X?

A

Standard error = sd of sampling distribution = how much on average a sample value is from its corresponding population parameter
Different from SD(X) because it is the SD of xbar calculated from sampling distribution

45
Q

How does standard error change as a function of N?

A

Standard error decreases as N increases

46
Q

Can the shape (not just variance) of a sampling distribution change as a function of N?

A

Yes – variance

47
Q

Describe completely what the central limit theorem is.

A

the central limit theorem states that if X~Normal in the population or n > 30, your sampling distribution of xbar will be ~Normal(mu, sigma^2/n)

48
Q

Under what conditions can you compute a one sample z test?

A

Need to know population variance

X~Normal or n > 30

49
Q

Differentiate sample statistics from test statistics. Do each have their own sampling distributions?

A

A sample statistic describes characteristics of samples. A test statistic is associated with a specific statistical procedure and has its own sampling distribution

50
Q

True/false: the lower the p value, the less likely the sample result is a fluke of chance sampling.

A

True bruh

51
Q

Contrast research vs null hypothesis.

A

Null hypothesis: all samples drawn from the same underlying population and thus have the same means
Alternative hypothesis: your research hypothesis that you are testing

52
Q

Directional vs non directional alternative hypothesis

A

Directional: prior research suggests the DIRECTION of a difference (easier to reject null bc critical value smaller)
Non-directional: mu1 =/= mu2 without specifying direction

53
Q

Two most common significance levels. How does choice of directional/non

A

.05 and .10

it’s easier to reject the null in directional hypotheses because the critical value is closer to 0

54
Q

What is the relation between Type I and Type II error rates?

A

If you reduce alpha to make Type I errors less likely, you are more likely to make a Type II error (failing to reject false null)
as alpha goes down, type ii error rate increases and decreases power

55
Q

Type I error

A

False positive: p(reject null | null is true)

56
Q

Type II error

A

Miss: p(fail to reject null | null is false)

57
Q

Usually want to avoid Type I errors over Type II. When would you want to avoid Type II errors?

A

You may want to avoid Type II errors in clinical settings. If you tell a patient they do not have a disease (null hyp) when in fact they do (alternative hyp), that could be very dangerous.

58
Q

What does p value in a hypothesis test tell you?

A

p value tells you the probability of obtaining a sample result as extreme or more extreme than your sample result under the null hypothesis

59
Q

What is actually tested in hypothesis testing?

A

Whether the difference in group means we obtained could reasonably have arisen if we drew our samples from the same underlying population (null distribution)

60
Q

Distinguish between P(null | data) and P(data | null).

A

In hypothesis testing, we test the probability of obtaining our data GIVEN the null is true.
In Bayesian statistics, we test the probability of a null hypothesis being true GIVEN our data.

61
Q

Steps of a hypothesis test.

A
  1. Research/alternative hypothesis
  2. Set up appropriate null hypothesis
  3. Construct sampling distribution of particular statistic on assumption that null is true
  4. Collect data
  5. Compare sample statistic to that distribution
  6. Reject or fail to reject null
62
Q

How is hypothesis testing similar to innocent until proven guilty?

A

We assume the null is true and do not reject it until we have sufficient evidence to do so.

63
Q

What are the merits of one vs. two tailed tests in terms of type ii errors?

A

A two tailed test at alpha = 0.05 is more liberal than a one-tailed test at alpha = 0.01

64
Q

What are some criticisms of NHST?

A
  1. Theories never get confirmed or disconfirmed, people just lose interest
  2. The better researchers make their design (more power, better control of confounding variables) the weaker the test of theory
  3. Null is always quasi-false – two groups won’t be exactly equal on everything
  4. Just gives you a binary decision
  5. With large enough sample size, SE goes down enough that all effects eventually become statistically significant
65
Q

The better researchers make their design (more power, better control of confounding variables) the weaker the test of theory – why?

A

Because these things make it easier to reject the null!

66
Q

What are some solutions to the criticisms of NHST?

A
  • Include confints: tells you all the nulls that would have been rejected and all the nulls that would fail to reject
  • report effect sizes to give an indication of how practically big or small a statistically significant difference is
67
Q

What is theory testing by scorecard, and why is it intellectually flabby?

A

theories used to be (and sometimes still are) tested by seeing how many experiments give significant results. The analogy is you basically have a scorecard and the theory with more points (significant results) “wins”, but this is a very weak, inconsistent, and error-prone way to evaluate scientific theories. Steve also calls this “research by tabular asterisks.”

68
Q

What is the difference between hypothesis testing in agronomy and in psychology and what are the consequences?

A

In agronomy it’s like either your crops grow or they don’t. In psych, you have to make a bunch of other auxiliary hypotheses (like the construct exists, and is measured perfectly by your dependent variable).

69
Q

Why do people say null hypothesis testing is worthless once you know the confidence interval?

A

Confidence intervals give you all the information contained in a NHST result and ~MOAR~

70
Q

What is effect size and wat does it mean?

A

It is a standardized difference between means.

  1. used as a complement or counter-point to significance tests where it gives and indication of how practically big or small a statistically significant difference is
  2. To provide a common metric on which to compare effects when dependent variables may be measured on different scales
71
Q

How does one interpret confints?

A

If we were to repeat the experiment over and over based on n cases, each time computing a sample mean, and then computing a confidence interval around that mean, 95% of those confidence intervals would contain the true population parameter

72
Q

What is the relation between a 95% confidence interval and a two-tailed null hypothesis test, alpha = .05?

A

Any null hypothesis mean within the range of the 95% confidence interval would have led to a fail to reject the null hypothesis
Any hypothesized mean that falls outside the range of the confidence interval would have led to the null hypothesis being rejected.
Thus you really don’t need NHST, you just need the confidence interval. That interval tells you all null hypotheses that would have been accepted or rejected, alpha = .05, two-tailed

73
Q

In a t distribution, what happens as df increases?

A

The distribution looks more and more normal

74
Q

What does it really mean to have 95% confidence that an interval contains a parameter?

A

This means that if we repeat our experiment hella times, 95% of the confidence intervals we compute will contain the parameter

75
Q

Compare one sample t-test with one sample z-test and discuss sampling distribution and its form.

A

In t-tests, you don’t know the population variance so you have to estimate it. The t distribution has thicker tails (more extreme results by chance, more extreme critical values)

76
Q

Is t or z test more powerful?

A

Z test because critical values closer to 0

77
Q

What happens if we use a z test in situations where a t test is more appropriate?

A

You will falsely reject more nulls than you should

78
Q

Variance sum law and consequences

A

var(x1) + var(x2) - 2 * cor(x1, x2) * s1 * s2
if x1 and x2 are independent, just sum the variances
if x1 and x2 are dependent, smaller sd because subtract correlation

79
Q

Why is a dependent samples test of a mean difference just a simple one sample t test?

A

Because you can just take difference scores and do all the computations on the one sample of difference scores

80
Q

In a dependent (correlated, repeated) samples t-test, what is the effect of better matching (i.e., a higher correlation between repeated trials)?

A

Better matching increases correlation between groups –> decreases SE –> more POWER

81
Q

What are the two main purposes of matching (experimental groups)?

A
  1. Control confounding variables, which gives you more internal validity
  2. Smaller SE for more powa
82
Q

What is the role of the normality assumption and homogeneity of variance assumptions in the independent samples t-test?

A

In terms of the process of the test, the normality assumption is required for accurate p-values, and the homogeneity of variance assumption is required to pool the variances. In terms of which type of t-test to use, Student’s (Steve called this Gosset’s) t-test is relatively robust to violations of either, Welch’s t-test is more so, and bootstrapping is best.

83
Q

What is the logic of computing the “pooled” estimate of the population variance?

A

With a larger sample size (combining n1 and n2) we get a more precise estimate of variance

84
Q

Compare and contrast the dependent samples t-test with the independent samples t-test. Include discussion of power, degrees of freedom, critical values, interpretability, and costs.

A 
In dependent samples, the scores are dependent on one another in some way.
POWER: dep has higher power
DF: indep has higher df
CRITICAL VALUES: dep has lower crit vals
INTERPRETABILITY: uhh
COSTS: if you do indep you lose power
85
Q

Cohen’s conventions for mean difference size of an effect

A 
small = .20
medium = .50
large = .80
86
Q

Sampling distribution of variance and implications conducting statistical tests of means when pop var is unknown.

A

Positively skewed, especially for small n

Resulting value of t is likely to be larger than z that we would have obtained if sigma was known and used

87
Q

Can the shape (not just variance) of a sampling distribution change as a function of N?

A

Yaaa – variance becomes less positively skewed as N increases

88
Q

If a sampling distribution is unbiased, then it must be symmetric and normal as well?

A

Heyell naw

89
Q

How does Levene’s test evaluate homogeneity of variance assumption in an independent groups t test?

A

It’s a 2 sample t test on (x1 - xbar1) vs (x2 - xbar2)

90
Q

Describe the robustness of the independent groups t-test to violations of its assumptions under N1 = N2 conditions and unequal sample size conditions. What can be done if the t-test is simply not appropriate for the specific research situation?

A

If n1 = n2, t test is pretty p robust to small deviations

if t-test is not appropriate… use Welch-Satterthwaite test

91
Q

How does Chi-square distribution change as a function of degrees of freedom?

A

more symmetric as df increases, and mean and var increase as df increases
mean = df, var = 2*df

92
Q

What are problems with small expected frequencies?

A

For a given sample size, there are limited numbers of contingency tables that you could construct and thus limited number of chi sq values. So these discrete possibilities of chi sq cannot approximate the continuous chi sq distribution if the frequency of 1+ of the cells is small

93
Q

Prospective vs Retrospective study

A

prospective: treatments applied THEN future outcome determined/measured/collected
retrospective: select people who had or had not experienced something and then look BACK to measure outcome

94
Q

Describe in words how you would go about empirically (without equations) determining the bias and efficiency of the sample mean and median in terms of estimating a population mean.

A

You would run a bunch of simulations – draw hella samples and compute hella statistics, then make a sampling distribution for your statistics and examine it!

95
Q

General form of a test statistic

A

signal: how different than what is expected / noise: expected difference by chance

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