Midterm

Question 1

Categorical Variables

Answer 1

These variables represent categories or groups and cannot be used in mathematical operations like addition or subtraction to derive meaningful results. They are qualitative in nature.

Question 2

Ordinal Variables

Answer 2

These have a natural order or ranking, but the differences between categories are not quantifiable or uniform. For example:
Shift Sizes: Small, Medium, Large. While there is an order, the difference between Small and Medium may not be the same as between Medium and Large.
Tax Brackets: Low, Medium, High. The brackets are ordered, but the difference between Low and Medium is not necessarily the same as between Medium and High.
Importance: Ordinal variables are useful in surveys and rankings where relative positioning matters, but exact differences do not.

Question 3

Nominal Variables

Answer 3

These represent categories without any inherent order. They are essentially labels or names.
Yes or No: Binary responses like “Yes” or “No” are nominal.
Colors: Red, Blue, Green, etc. These are different but cannot be ordered.
Types of Animals: Dog, Cat, Bird. These are distinct categories without any natural order.
Importance: Nominal variables are crucial in classification tasks where the goal is to group data into distinct categories without implying any hierarchy.

Question 4

Numerical Variables

Answer 4

These variables represent quantities and can be used in mathematical operations. They are quantitative in nature.

Question 5

Discrete Variables

Answer 5

These take on specific, separate values, typically integers. They do not vary continuously.
Number of Times Done: For example, the number of times a person has visited a doctor. This is a count and can only be a whole number.
Importance: Discrete variables are essential in counting and frequency analysis, where the focus is on the number of occurrences.

Question 6

Continuous Variables

Answer 6

These can take on any value within a continuous interval, including fractions and decimals.
Height: A person’s height can be measured to any degree of precision.
Concentration: The concentration of a chemical in a solution can be measured with high precision.
Importance: Continuous variables are vital in measurements and modeling where precision is required, such as in scientific experiments and engineering.

Question 7

Independent Variable (Predictor)

Answer 7

This is the variable that you believe may influence or cause changes in the response variable. It is the “input” in an experiment or study.
Importance: Identifying the independent variable is crucial for designing experiments and understanding causal relationships.
“INPUT”

Question 8

Dependent Variable (Response)

Answer 8

This is the variable that you are trying to explain or predict. It is the “output” in an experiment or study.
Importance: The dependent variable is the focus of analysis in many studies, as it represents the outcome of interest.
“OUTPUT”

Question 9

Causal Relationship

Answer 9

This is a relationship where changing the independent variable directly affects the dependent variable.

Question 10

Observational Studies

Answer 10

These cannot establish causal relationships because they do not involve controlled manipulation of variables.

Question 11

Experimental Studies

Answer 11

These can establish causal relationships by manipulating the independent variable and observing the effect on the dependent variable.

Question 12

Correlation vs. Causation

Answer 12

It is important to remember that correlation does not imply causation. Just because two variables are correlated does not mean that one causes the other.

Question 13

Sampling

Answer 13

Is the process of selecting a subset of individuals from a larger population to estimate characteristics of the whole population.

Question 14

Population

Answer 14

The entire set of elements you are interested in studying.

Question 15

Sample

Answer 15

A subset of the population that you collect data from. The goal is for the sample to be representative of the population.

Question 16

Parameters

Answer 16

Numerical values that describe some characteristic of a population.

Question 17

Statistics

Answer 17

Numerical values that describe some characteristic of a sample.

**The goal is to use statistics to estimate parameters.

Importance: Sampling is essential because it is often impractical or impossible to study an entire population. A well-chosen sample can provide accurate estimates of population parameters.

Question 18

Random Sampling

Answer 18

Each member of the population has an equal chance of being selected. This helps avoid bias and ensures that the sample is representative.
Importance: Random sampling is crucial for generalizing results from the sample to the population.

Question 19

Avoiding Selection Bias

Answer 19

Ensuring that the sample is not systematically overrepresented or underrepresented. This includes eliminating convenience bias, where samples are chosen based on ease of access.
Importance: Selection bias can lead to incorrect conclusions, so it is vital to ensure that the sample is representative.

Question 20

Sample Size

Answer 20

The sample should be large enough to minimize sampling error but not so large that the marginal gain in accuracy is negligible.
Importance: An appropriately sized sample ensures that the results are reliable and valid.

Question 21

Coverage of Key Demographics or Features

Answer 21

Ensuring that all relevant subgroups are included in the sample proportionally.
Importance: This ensures that the sample accurately reflects the diversity of the population.

Question 22

Randomized Allocation

Answer 22

Randomly assigning subjects to different groups to ensure that each group is comparable. This is crucial for establishing causal relationships.

Importance: Randomized allocation helps eliminate confounding variables and ensures that any observed effects are due to the independent variable.

Question 23

Confronting Variance/Counfounding Variable

Answer 23

** Influences both the Predictor and Response Variables

Variance can arise from differences in subjects’ backgrounds, environments, etc. Methods to control variance include:

Matching: Pairing subjects based on similar characteristics to control for differences.
Replication: Using a larger sample size to increase the accuracy of the results.
Blocking: Grouping subjects into blocks based on certain characteristics and then randomizing within each block.
Importance: Controlling variance is essential for ensuring that the results of an experiment are valid and reliable.

Question 24

Hypothesis Testing

Answer 24

A statistical method used to determine whether there is enough evidence to reject a null hypothesis.

Question 25

Null Hypothesis (H₀)

Answer 25

The hypothesis that there is no effect or no difference. It is assumed to be true until evidence suggests otherwise.

Question 26

Alternative Hypothesis (H₁)

Answer 26

The hypothesis that there is an effect or a difference

Question 27

p-value

Answer 27

Probability of observing data w/ the "weakest evidence" in favor of Ha assuming Ho is true. A low p-value (typically < 0.05) suggests that the null hypothesis should be rejected.

Question 28

Type I Error (False Positive)

Answer 28

Rejecting the null hypothesis in favor of Ha, when Ho is actually true. This is determined by the significance level (α). Importance: Controlling Type I errors is crucial to avoid false conclusions.

Question 29

Type II Error (False Negative)

Answer 29

Failing to reject the null hypothesis when it is actually false. This is influenced by the sample size and the effect size. Importance: Minimizing Type II errors is important to ensure that real effects are not missed.

Question 30

Analytical Hypothesis Testing

Answer 30

Analytical methods are used when the data meets certain conditions, such as being normally distributed or having a sufficiently large sample size. Importance: Analytical methods provide precise calculations of probabilities and are essential when the conditions are met. However, if the conditions are not met, simulation methods may be more appropriate.

Question 31

What is the order for Simulation Hypothesis Testing

Answer 31

1) Assume Ho initially and take the general probability (this is done by shuffling the predictor variable to mimic natural variation *hold one column and shuffle the other* 2) Simulate multiple trials to get different proportion values 3) plot # of trials vs (sample/test statistic) (mean1-mean2) 4) Find the p-value (which is the probability of observing data with the "weakest evidence" in favor of Ha assuming Ho is true) *find from prob of finding a value >= to the test statistic 5) If p-value < preset threshold (alpha), Reject Ho in favor of Ha 6)*alpha is known as the significance level*

Question 32

what does the significance level represent

Answer 32

Represents the rarity of our test statistic to reject Ho in favor of Ha (difference in the proportions or means) that we use to compare to the p-value 0.05

Question 33

What is the Central Limit Theorem

Answer 33

States that the distribution of samples means will approximate a normal distribution (Gaussian) as the sample size increases, regardless of the original population's distribution

Question 34

What is the Gaussian curve

Answer 34

normal distribution/bell curve with symmetry and "common" standard dev zones

Question 35

What is the Poisson distribution

Answer 35

A discreteb(not continuous) probability distribution that models the number of events (independent of one another) occurring in fixed intervals given a constant rate of occurrence

Question 36

How does the Poisson Distribution differ from the normal distribution

Answer 36

Poissons counts the number of events and is discrete whreas the normal distribution is continuous and counts the range of mean values

Question 37

What is a z-test?

Answer 37

* St. Dev is known so accurate *difference in the two proportions A z-test compares a sample mean (or proportion) to a population parameter under the assumption that the population standard deviation (σ) is known (or the sample is large enough to estimate σ very accurately). Narrow Tall curve

Question 38

What is a t-test?

Answer 38

* St. Dev of population is unknown so an estimate * Difference in the two means A t-test compares sample means when the population standard deviation (σ) is unknown, and it uses the sample’s standard deviation (s) as an estimate. Flatter curve which get pointer with an increased sample size

Question 39

When do I use a z-test vs. a t-test?

Answer 39

z-test if: The population standard deviation is known, or The sample is large thus gaussian (n(sample size) ≥ 30) and you trust the normal approximation. t-test if: The population standard deviation is unknown, or The sample is small, but you can assume (approximately) normal data.

Question 40

What are the two types of interval estimates

Answer 40

95% confidence interval and standard error

Question 41

What is bootstrapping

Answer 41

It is a resampling technique that draws samples with random replacements from the original data set to estimate the distribution of a dataset (the same observation can be sampled multiple times) to calculate confidence intervals from simulated proportions from resampled data that don't rely on assumptions about the base population

Question 42

What is the significance of the original proportion in confidence interval and standard error

Answer 42

It acts as the basis for which we use bootstrapped data to produce a range/interval around

Question 43

What are the key characteristics of Standard Error

Answer 43

1 standard deviation ~68% chance of capturing the true population parameter Narrower range Represented by +- More Commonly used

Question 44

What are the key characteristics of 95% Confidence Intervals

Answer 44

95% chance of capturing the true population parameter Wider Range, and thus typically shown as a range Less commonly used

Question 45

What is the propagation of uncertainty via bootstrapping

Answer 45

Bootstrapping is used to estimate the standard error (uncertainty) for each measurement by resampling the data. These uncertainties are then propagated through the function or model that calculates the desired 'output,' resulting in a range of possible values for the output that reflects the combined uncertainty of all inputs.

Question 46

what is matching

Answer 46

********also can act as a control for some reason (Primary Definition) Matching: Pairing subjects based on similar characteristics to control for differences. *DONE IN OBSERVATIONAL STUDIES*

Question 47

what is replication

Answer 47

Replication: Using a larger sample size to increase the accuracy of the results.

Question 48

what is blocking

Answer 48

Blocking: Grouping subjects into blocks based on certain characteristics and then randomizing within each block. *DONE IN EXPERIMENTAL STUDIES*

Question 49

Hypothesis testing can only be used to compare means or proportions. (True or False)

Answer 49

False **** we can also compare the medians because of flexibility **** False Explanation: Hypothesis testing is a versatile tool in statistics and is not limited to comparing means or proportions. It can also be used to: Compare variances (e.g., F-test). Test for independence in contingency tables (e.g., Chi-square test). Assess goodness-of-fit (e.g., Chi-square goodness-of-fit test). Evaluate correlation (e.g., testing if a correlation coefficient is significantly different from zero). Test regression coefficients in linear models. And much more!

Question 50

Hypothesis testing can only be used to compare two groups. (True or False)

Answer 50

*** For this class's examples its true*** False Explanation: Hypothesis testing is not limited to comparing two groups. It can be used to compare: More than two groups: For example, ANOVA (Analysis of Variance) is used to compare means across three or more groups. Single groups: For example, a one-sample t-test compares the mean of a single group to a known value. Relationships between variables: For example, regression analysis tests the relationship between a dependent variable and one or more independent variables.

Question 51

Hypothesis testing can only lead to binary decisions. (True or False)

Answer 51

*** True for this class***** False Explanation: While hypothesis testing often results in a binary decision (reject or fail to reject the null hypothesis), it is not limited to binary outcomes. Hypothesis testing also provides: p-values: A measure of the strength of evidence against the null hypothesis. Confidence intervals: A range of plausible values for the parameter being tested. Effect sizes: A measure of the magnitude of the observed effect. These additional outputs provide more nuanced insights beyond a simple "yes or no" decision. For example, a p-value of 0.06 might not lead to rejecting the null hypothesis at the 0.05 significance level, but it still suggests some evidence against the null hypothesis.

Question 52

What are Type I errors controlled by

Answer 52

The significance level (alpha)

Question 53

To minimize Type I errors do what

Answer 53

lower alpha or increase the sample size

Question 54

how can you reduce type II errors

Answer 54

increase the sample size and increase alpha

Question 55

What happens when you increase d in a t-test

Answer 55

the curve becomes more "pointy" as it becomes more Gaussian

Question 56

What statements must be true in order for an analytical approach to be valid

Answer 56

The samples must be drawn from a pop. whose values are distributed along a Gaussian curve If the above condition is not satisfied then we must have a sufficiently large number of samples to thus fit the Gaussian Profile

Question 57

What is the equation for the Population Density

Answer 57

(Number of trials in bin/Number of total trials in sample)/bin width

Question 58

*confusing slide* how are simulated p-values slightly better than those found analytically

Answer 58

Simulated methods can utilize monte carlo simulations which are insentive to the shape of the population distribution, dont need to be a Gaussian fit and are more flexible to perform hypothesis testing on other kinds of numerical summaries of sample data, like using the median and such

Question 59

What is the range of the 95% confidence interval

Answer 59

2.5-97.5

Question 60

Does randomization include replacement?

Answer 60

No it does not, replacement is for bootstrapping only

Question 61

What do simulations randomize

Answer 61

The predictor variable

Question 62

what is recall bias

Answer 62

when participants have poor ability to recall past events accurately as needed for a study

Question 63

what is convenience bias

Answer 63

participants selected by ease of access instead of random sampling

Question 64

p-value is tied with what words

Answer 64

Rare event

Question 65

rank on wideness

Answer 65

95CI>90CI>SE