Statistics Final
Complement
(A) only
What is a p-value? How would you describe it?
-A p-value indicates how likely the observed or more extreme data would arise by chance under the statistical model of data describing the null hypothesis. -Changes as a function of data
What is E(X)?
-Expected value or expectation variance → weighted avg (population mean) -Depending on the context, E(X) is also called the population mean of X and denoted by µ
How do you make a decision of NHST based on either p-value or critical-value method?
-If p-value < α (significance level) → reject the null -If p-value > α (significance level) → do not reject the null
What does a (low/high) p-value tell you about the likelihood of your observed data under the null hypothesis?
-If the p-value is low (lower than the significance level which is usually 5%) we say that it would be very unlikely to observe the data if the null hypothesis were true, and hence reject H0 (Null Hypothesis) -If the p-value is high (higher than the significance level) we say that it is likely to observe the data even if the null hypothesis were true, hence we do not reject the H0
How can we use a CI for NHST? What is the decision rule?
-If your 100(1-a)% CI does not contain the null value, then reject the null hypothesis at the a significance level in support of the two-sided alternative hypothesis that the parameter of interest is not the null value; -If your 100(1-a)% CI contains the null value, then fail to reject the null hypothesis at the a significance level
Independent/explanatory and dependent/response variables: What are they and how are they different?
-Independent/explanatory: variables directly manipulated by researcher or hypothesized to have an effect on some other variable. -Dependent/response: variables not manipulated by researcher or hypothetically affected by IV.
Properties of a normal distribution. What are the two quantities that describe a normal distribution? What do you call them and denote them by?
-Most prevalent and most important -Bell curve -Symmetric about μ -Continuous random variable -Described by μ (population mean) and θ2 (population variance)
Difference between one-sided (or one-tailed) and two-sided (or two-tailed) hypothesis tests
-One-tailed: allots all of alpha (α) two-tailed: allots half of alpha (α) -One tailed: testing possibility of relationship in one direction; two tailed: testing possibility of relationship in both directions -One tailed: > or < two tailed: = or ≠
Gambler's fallacy: What is it? What kind of a misconception does it exemplify?
-Other way around from Sally Clark Case. All gambler trials are independent Vs the misconception that they are dependent -the belief that the chances of something happening with a fixed probability become higher or lower as the process is repeated. People who commit the gambler's fallacy believe that past events affect the probability of something happening in the future.
What are the common misinterpretations of a CI? Why are they incorrect?
-People usually interpret C level with 95% probability, the interval covers the true avg -When you mention 95% probability, you don't associate it with previous findings/particular interval
What are the two hypotheses that are considered in statistical significance testing?
null and alternative
What are the two computational methods for performing null hypothesis significance testing (NHST)?
p-value and critical-value (or rejection-region) methods
What is the general construction of a test statistic for NHST?
point estimate - null value ------------------------------ standard error
Outcome
possible result of an experiment
Joint probability
probability of A and B occurring (P(A∩B)); combined distribution of RV (intersection of probabilities)
Conditional Probability
probability of A occurring, given that B occurs (P(B|A))
Marginal probablity
probability of an event occurring (just P(A)) → not conditioned on another event/variable
IQR
range of the middle 50% of scores; upper quartile - lower quartile
What are the measures of variability of a data distribution that we learned in class?
range, IQR, variance, and standard deviation
concept of skewness in a data distribution
refers to where the tail of the data lies --> tail is opposite of the peak
sample space
set of all possible outcomes of an experiment (denoted by S)
Thing(s) set by the user
significance level and its corresponding critical value(s).
standard deviation
sq root of the variance
Event
subset of outcome in the sample space (denoted by A or B)
In any NHST in statistics, a certain quantity is computed from a given sample of data and then compared against the population distribution of that quantity under theoretical assumptions. What is the quantity?
test statistic
Thing(s) varying as a function of data:
test statistic
Thing(s) varying as a function of data
test statistic and its corresponding p-value
sample variance
the avg of the squared differences from the mean
population
the entire group in which we are interested in
Union
AUB (A or B)
What are the three different types of statistical inference about the parameter of interest?
-Point estimation: single value that estimates the parameter calculated from the sample -CI estimation: gives a range of values for the parameter -Hypothesis testing: tests for specific value(s) of the parameter
Definition of the power of a statistical test (either verbally or in relation to Type II error)
-Probability of failing to reject a false null hypothesis (type II error) → inversely related to beta = 1- β -Higher the power, lower the probability of committing a type II error; lower the power, the higher the probability of committing a type II error
Definition of a binomial probability distribution. Under what situation would a binomial random variable arise? What elements do you need to define a binomial probability distribution?
-Probability of having exactly X successes in N trials with probability of success, P → two trials = Bernoulli trial -Trials are independent
In the critical-value method of NHST, what varies as a function of data and what is set by the user?
-Significance level is decided by the user -Test statistic and critical value determined by the data
Influence of a greater/smaller sample size on power
-The greater the sample size, the greater the power. -The smaller the sample size, the smaller the power.
Influence of a higher/lower significance level on power
-The lower the significance level, the lower the power (direct relationship). -The higher the significance level, the higher the power.
Definition of a relative frequency
-The number of times an event occurs during experimental trials, divided by the number of trials conducted -we are looking at the number of times a specific event occurs compared to the total number of events.
What are Type I and Type II errors, the two different kinds of a decision error, in NHST?
-Type I error is rejecting the null when it is in fact true. -Type II error is failing to reject the null when it is false.
What is a rejection region? How is it related to critical values?
-Values where null hypothesis is rejected. -The region within the critical values, if the test is two tailed. -The region between the critical value and the tail of your study.
Rules of summation notations
1. A summation sign can be distributed across the terms of a sum. 2. A constant, k, can be factored out of a summation 3. The sum of a constant, k = the product of the number of terms in the summation, n, and the value of the constant: Distributive property for Summ.(X-Y) = Summ.X-Summ.Y
What is the accurate meaning of the confidence level of a confidence interval (CI)?
95% of the time or with 95% probability, the interval will cover the true average # of applications . This is the meaning of a 95% confidence level (technically, aka the coverage probability.
random variable
A variable all values of which have associated probabilities (or a probability distribution) → variable whose value varies due to change/randomness
Intersection
AnB (A and B)
How is a confidence interval (CI) formed in general?
CI = point estimate +/- margin of error It contains both the point estimate and its precision of inference (i.e., margin of error)
What is central limit theorem (CLT)? How does it apply to the distribution of sample means or sample proportions?
CLT tells distribution of sample means regardless of the shape of the random variable. It applies to the mean. states that the sampling distribution of the mean of any independent, random variable will be normal or nearly normal, if the sample size is large enough
Conditional probability when events are mutually exclusive
Cannot occur together P(A|B) = 0 → P(B|A) = 0 → P(Ā|B) = 1 → P(not)(B|A)
Definition of a probability experiment (or a random process)
Chance effects the result of an experiment → if there are only 2 outcomes, its called a Bernoulli trial (i.e coin flip)
Mutually Exclusive (disjoint)
Events are mutually exclusive if there is no overlap; P(A∩B) = 0 → events can't occur at the same time
Differences between histograms and bar plots
Histograms shows distribution of variables (plot quantitative/numerical data) and Bar plots are used to compare variables (plot categorical data)
What are the two different ways to decrease the margin of error (or width) of a CI? Hint: Adjust your confidence level or your sample size. But in which direction do you want to adjust them?
Increase sample size Lower CI
Difference between measures of central tendency with respect to robustness, i.e., sensitivity to skewness and extreme values
Mean is susceptible to change (sensitive to skewness and extreme values); mode/median are robust (not sensitive to skewness and extreme values)
When performing hypothesis significance testing in statistics, which (either null or alternative) hypothesis is assumed to be true?
Null Hypothesis
Conditional probability when events are independent.
P(A) remains the same regardless of the occurrence of B P(A|B) = P(A) → P(A|(not)B)
Properties of a standard normal distribution. How do you transform a normal distribution to a standard normal distribution?
P(X-M/σ > x-M/σ) → P(Z> x-M/σ) → zₓ → substitute z value in for x into second equation Normal curve is symmetrical about the mean (μ) Mean is in the middle and divides area into 2 halves Total area under the curve is 1 Completely determined by its mean and SD σ or σ2
What is the meaning of "statistical significance" with respect to a p-value? Recall what learned in our last class on Nov 22.
P-value helps determine significance of results -Small \-value: strong evidence against null hypothesis → reject null -Large p-value: weak evidence against null hypothesis → fail to reject null -P-value close to significance level: marginal (could go either way)
parameter
Parameter is an avg. or percentage that describes the population (greek letters β), and is opposite of statistic
Concepts and symbols of population/sample mean and population/sample variance (or SD) Which is a parameter? Which is a statistic?
Parameter is an avg. or percentage that describes the population (greek letters β), and is opposite of statistic, which is an avg or percentage that describes the sample (latin letters x)
Conditional probability
Probability of an outcome based on the occurrence of a previous outcome
Independent Events
Probability of event A doesn't affect probability of event B → P(A∩B) = P(A) x P(B)
Conventional format of a data matrix: What do columns and rows represent?
Row = observations; columns = variables
Sally Clark case: What is it? What kind of a misconception does it exemplify?
Sally Clark case is both babies of the same mother died from SIDS. Doctor testified that the probability of two babies dying from SIDS is low. -The misconception is two events are assumed to be independent when in fact they are dependent. -Sally Clark was eventually release from prison but died later from grief
When is the statistical significance (or statistically significant difference) obtained?
Statistical significance (or evidence for a statistically significant difference between the observed value and the null value under H0) is obtained when the p-value is lower than the significance level
In the p-value method of NHST, what varies as a function of data and what is set by the user?
Test statistic
What is the key difference between z-test and t-test? What is the additional source of uncertainty or randomness in a t statistic as opposed to a z statistic?
The difference is t is sample population standard error (thicker tail bc has additional source of variability) vs z test is population standard error.
How would you interpret the case of a test statistic exceeding critical values?
The null hypothesis is being rejected
Definition of a cumulative distribution function of a RV. Note the direction of the inequality in the equation. X represents a RV and x represents a value of the RV.
The probability that X will take a value less than or equal to x → F(x) = P(X ≤ x)
What is your interpretation of a significance level?
The significance level determines the critical value(s) and sets the chance level with which one is willing to reject the null even if it leads to false rejection when in fact the null is true.
Rationale behind the use of the Type I error rate as the significance level in NHST
The significance level is the probability that a Type I error occurs
What is your interpretation of a significance level?
The significance level sets the chance level, serving as the upper limit for a p-value, with which one is willing to reject the null even if it leads to false rejection when in fact the null is true.
What is the meaning of "statistical significance" with respect to critical values? Recall what learned in our last class on Nov 22. More specifically,
The test statistic exceeding the critical value(s) indicates unusually extreme data that would arise by chance if the null hypothesis were true.
A discrete RV can be represented by a table, but a continuous RV cannot. Any continuous RV can be described by a probability density function (pdf). For example, a normal random variable with its associated normal distribution is described by a pdf that has a bell-like shape.
The total area under the pdf curve of any continuous RV, by definition, is 1. This implies that any (total or partial) area under the pdf is less than or equal to 1. With a continuous RV, the probability of observing a range (or ranges) of values is given by the corresponding area(s) under the pdf curve, which is always greater than or equal to zero. With a continuous RV, the probability of observing any single value of it is always zero; that is, P(X=a)=0 regardless of a if X is a continuous RV. An implication of this is: P(a≤X≤b) = P(a<X≤b) = P(a≤X<b) = P(a<X<b)
Concept of the normal approximation to the binomial distribution. What is it? Do not need to memorize any formula.
Used for large samples → (n)(p) > 5 & n(1-p) > 5 Correction for continuity: adding/subtracting .5 to attain more accurate probability
How do we interpret a CI by convention?
We are 95% confident that the avg # (true population mean) is between the interval we have computed
percentile
a score and not necessarily limited from 0 to 100, can be less or more. Example: A student scoring at the 35th percentile scored as well as, or better than, 35 percent of students in the same grade in the norm group
Mean of z-scores
always 0
SD of z-scores
always 1
percentage
always ranges from 0 to 100; a proportion out of 100
Variance
average squared difference of the scores from the mean (σ2 = population & s2 = sample)
statistic
avg or percentage that describes the sample
Descriptive statistics
describes the data collected. (avg. standard deviation)
Z-score
how many SD away from the mean the observation is.
range
largest score - smallest score; relies only on extreme values which is misleading
inferential statistics
makes inferences about a population based on a sample of data from the population. (inferring about DCs population height avg using the avg height of your sample)
Combination
nCx (order doesnt matter) - Repetition v. no repetition
Permutation
nPx (order matters) - Repetition v. no repetition
Properties of each central tendency measure; for example, for (unordered) categorical data, mean or median is not defined, only the mode is defined
nominal data - mode is best used. ordinal data - median/IQR is best used. interval(or ratio) data - mean is best used.