Math 121 exam 3 Questions fully solved & verified for accuracy(A+graded)

p

fixed constant (qualitative)

^p (p hat)

sample proportion (quantitative), random variable

descriptive statistics

organizing and summarizing data

inferential statistics

formal methods for drawing conclusions and making generalizations from good data

case

individual or object for which we obtain information

variable

characteristic recorded for each case

stacked data format

rows represent cases and columns represent variables

unstacked data format

columns represent variables but rows don't represent cases

ordinal categorical

categories can be ordered or ranked

nominal categorical

categories can't be ordered or ranked

continuous quantitative

value to be measured

discrete quantitative

value to be counted

frequency table

frequency or count of a categorical variable

relative frequency

percentage or proportion for each category of variable

two-way table

used to show relationship between 2 categorical variables

levels

possible values of an explanatory variable

associated

if values of 1 variable tend to be related to values of other variable

causally associated

if changing the value of explanatory variable influences response variable value

lurking variable

variable with important effect on relationship among study variables, but isn't 1 of the explanatory variables studied

observational studies

study in which the researcher doesn't actively control value of any variable, only observes as exist (can't control for lurking variables or be used to establish causation)

experimental study

study in which the researcher actively controls value of one or more explanatory variables

randomized experimental study

study in which explanatory variable for each unit is determined randomly (can be used to establish causation)

single-blind

subjects don't know which group they're in

experimenter bias

distortion that can arise on the part of the experimenter due to how the subjects are assigned to groups

double-blind

neither subjects nor researchers know which group participant has been assigned to

distribution

pattern of variability, provides possible values variable can take on and how often these possible values occur

histogram

visualization of distribution of quantitative variable in which data are binned into discrete groups

skewed right

data piled on left and tail extends to the right

skewed left

data piled on right and tail extends to the left

bar chart

1 bar for each categorical variable (bars don't touch)

categorical variability

diversity of data values (many categories = high variability, few categories = low variability)

measures of center

mean, median, and mode

order statistics

ranked order

resistance measure

measure relatively unaffected by outliers (mean and mode)

measures of variability

numerical values that describe how spread out data are (variance, standard of deviation, range, and IQR (sum of deviations always 0))

mean absolute deviation

average of absolute value of deviations from values to mean

z-score

measures how many standard deviation above/below mean

percentile

measures indicating the value below which a given percentage of observations in a dataset falls

5 number summary

minimum, Q1, Q2/median, Q3, maximum

range

difference between maximum and minimum

IQR

difference between Q1 and Q3

residual

difference between observed response (y) and predicted response (y hat)

least squares regression line/line of best fit

y hat = a + bx

extrapolation

using a regression line to predict y for a value of x that is outside the range of data used to determine the regression line

theoretical probability

relative frequency of the event is the process was repeated infinitely many times

empirical probability

relative frequency of the event based on an experiment or real-life process (or simulation in some cases)

sample space

collection/set of all possible outcomes of an experiment

event

specific collection/set of outcomes, subset of sample space

mutually exclusive/disjoint

A ∩ B = 0

random variable

numerical quantity that changes trial to trial in a random process

parameter

numerical value that describes a population

statistic

numerical value that describes a sample

standard error

standard deviation of sampling distribution, measures how much statistic varies between samples

CLT

good approximation of sampling distribution without needing other simulations

confidence interval (parameter)

captures parameter for specified proportion of all samples

null hypothesis

claim that there's no effect/difference

alternative hypothesis

claim for which we seek evidence

reject H0

p-value < α

fail to reject H0

p-value >= α

What are the steps of a hypothesis test?

State hypotheses and define parameters, state significance level, check assumptions, calculate sample statistic, calculate standardized test statistic, find p-value, make formal decision, and write conclusion.

Population (N)

The entire group of individuals or instances about whom we hope to learn.

Sample (n)

A subset of a population, examined in hope of learning about the population.

Sample mean (x̄)

The arithmetic average of the values in a sample.

Population mean (μ)

The arithmetic average of all values in a population.

Sample variance (s²)

A measure of how spread out the data in a sample is from the sample mean.

Population variance (σ²)

A measure of how spread out the data in a population is from the population mean.

Sample standard deviation (s)

The square root of the sample variance, representing the average distance of sample data points from the mean.

Population standard deviation (σ)

The square root of the population variance, representing the average distance of population data points from the mean.

What is the regression equation ŷ = a + bx?

A linear model where ŷ is the predicted response, a is the y-intercept, b is the slope, and x is the observed explanatory variable.

Additive Rule of Probability

P(A ∪ B) = P(A) + P(B) - P(A ∩ B), used to find the probability of the union of two events.

Complement Rule

P(Ac) = 1 - P(A), used to find the probability of an event not occurring.

Null Hypothesis (H0)

The statement of no effect or no difference, which is assumed to be true until evidence suggests otherwise.

Alternative Hypothesis (HA)

The statement that there is an effect, a difference, or a relationship in the population.

What is the Interquartile Range (IQR)?

The difference between the third quartile (Q3) and the first quartile (Q1), representing the middle 50% of the data.

What is a p-value?

The probability of observing a test statistic at least as extreme as the one calculated, assuming the null hypothesis is true.

What is a confidence interval?

A range of values calculated from sample data that is likely to contain the true population parameter with a specified level of confidence.

What is the Margin of Error?

The range of values above and below the sample statistic that accounts for sampling variability.

What is the Binomial Coefficient?

A formula, denoted as nCr or (n choose k), used to calculate the number of ways to choose k successes in n trials.

What is the Central Limit Theorem (CLT) for proportions?

A theorem stating that the sampling distribution of a sample proportion will be approximately normal if the sample size is large enough.