AP Stats Unit 1 (Part 2) Notes

Ways to Collect Data to Answer Investigative Question:

1. Census

2. Sample

3. Experiment

4. Observational Study

Statistical Inference

Using statistics gathered in sample to make judgements, predictions, or estimations of the parameter from the population

Survey

observational study in which the data are collected from humans using a standard set of questions

Confounding variable

• provides an alternative explanation for observed relationship between the explanatory and response variables determined in the study

  • basically, undermines the creditability of the cause and effect relationship with in a study

  • this isn’t the explanatory variable we’re measuring

  • Examples: gender of the rater, harsh rater in general, experience with previous teachers

To be a confounding variable…

a variable must be associated with both the explanatory variable and the response variable

Confidence Interval

used when we use a sample statistic to estimate a population parameter

Hypothesis Test

using statistics from a sample to determine if a claim about a population parameter is true

Bivariate Data

analyzing two variables to see what relationship they (might) have

response variable

measures an outcome of a study

explanatory variable

Helps explain or predict changes in the response variable.

Be careful: sometimes…

there might not be an obvious explanatory variable.

Association/Correlation is NOT

Causation

Observational Studies

• Observes units and measures variables of interest, but does not attempt to influence the responses

  • Retrospective: Examine existing data for a sample.

  • Prospective: Tracks individuals into the future.

Experiments

• Deliberately imposes treatments on different groups of units to compare their responses.

  • If done well, can help prove cause & effect.

  • Observational units become Experimental Units

Levels of Treatments

• Factors

• Levels

• Control Group

An investigative question must contain the following components:

• Variables: Specify the explanatory and response variables

• Parameter: Mean or Proportion?

• Direction: Indicate the direction of association or causal effect

• Conclusion/Population: Name the population of interest

If needed:

Comparison Group: Specify what group you are comparing to.

Investigative Question Sentence Starters

For Causation: For [population of interest], does [explanatory variable] cause [direction of outcome] in [response variable] compared to [comparison group]?

For Association: For [population of interest], is [explanatory variable] associated with [direction of outcome] in [response variable] compared to [comparison group]?

Sampling With or Without Replacement

Referring to what to do with a person or object after they have been randomly selected.

• With replacement - we put them back to possibly be picked again

• Without replacement - we do not put them back so they cannot be picked again (most used option when sampling)

Sample NON Random Bias

• Volunteer Response Bias

• Convenience Sample Bias

Sample is Random Bias

• Undercoverage Bias

• Nonresponse Bias

• Response Bias

Voluntary Response Bias

may occur when a sample consists entirely of volunteers

Bias

occurs when your sampling method could result in a severe underestimation or overestimation of the true parameter

If our sample is not a very good estimate of the population…

our statistic could be biased.

Convenience Sample

Selecting individuals from the population that are easy to reach.

Example: Circling the first words you see

Sample Random Sample (SRS)

sampling where every set of individuals has an equal chance to be chosen as the sample

Example: using RNG

Voluntary Response

• When the individuals themselves choose to be in the sample based on a general invitation.

• Most people only volunteer themselves if they have an extreme opinion or measurement.

Undercoverage

When some members of the population are less likely to be chosen or cannot be chosen in a sample.

Nonresponse Bias

When an individual chosen for the sample can’t be contracted or refuses to participate.

Response Bias

When there is a pattern of inaccurate results due to the way the sample is collected.

Precise

• Statistics are precise when they are consistent (different samples should be similar to each other).

• Precision is the opposite of VARIABILITY / SPREAD

Accuracy

• Statistics are accurate when they are close to the truth (the truth is the parameter).

• Accuracy is the opposite of BIAS.

Steps of SRS

1. Label Individuals

   1. Assign a number to each individual or write names on a slip of paper

2. Randomize

   1. Use RNG to find random sample of size n or put written names in hat, shuffle the hat, and choose a random sample of size n

Systematic Random Sampling

1. Pick a random starting point

2. Pick a number (k)

3. Select individuals by choosing every kth individual

Best Used: When you don’t know many details about your population, or if you’re unsure of its size.

Examples: - Surveying voters after they exit the polling station

- Surveying customers at a mall

Stratified Random Sampling

Step 1: Separate your population into separate stratas

Step 2: Take a small SRS from every strata

Best Used: When your population is very heterogeneous (diverse)

in a way that will affect your answer.

Examples: - Sampling college students based on their major

(some English majors, some Math majors, etc)

Strata

• Groups of a population that share similar characteristics

• Example: Concert seats that had similar costs (rows)

Clustered Random Sampling

Step 1: Randomly choose a number of clusters

Step 2: Survey every individual in those clusters

Best Used: When your population is extremely large or dense.

Examples: - Small groups of trees in a forest

- Taking samples of blood from a patient

Cluster

• A group of a population that are located near each other

• Example: Columns of seats at a concert

Strata should be…

homogeneous (similar). “Sample some from all groups”

Clusters should be…

heterogeneous (diverse). “Sample all from some groups”

Treatment

The condition applied to the individuals in an experiment.

Ex: Which resume you receive (John/Jennifer)

Placebo

A treatment that has no active ingredient, but appears like a regular treatment

Example: a pill with no medicine

Differences between Observation Studies and Experiments

Observational Studies:

• No treatment

• Takes multiple samples from a population

Experiments

• imposes a treatment

• takes samples and puts them into experiment units

Subjects

experimental units that are people

Experimental Unit

what/who a treatment is imposed on

Random Sampling

You pick a random sample so that you can apply your results to the general population.

Random Assignment

You randomly assign units to groups so that you can:

• Have groups of similar size

• prevent confounding variables from affecting your results.

This helps to prove a cause-and-effect relationship.

Steps to an Experiment

Step 1: Gather experiment units (ideally through random sampling)

Step 2: Randomly assign units to different groups (ideally at least 30 in each group)

Step 3: Apply different treatments to each group (specifically correspond each value/symbol to the group)

Step 4: Compare responses

Example: Label 40 slips of paper “A” “B” “C” “D” “E” and “F.” Put all 240 (total) slips of paper in a large bowl. Have each patient take a slip of paper out of the bowl without replacement. “A” patients receive a placebo pill, “B” patients receive a placebo liquid, “C” patients receive a 125 mcg pill, “D” patients receive a 125 mcg liquid, “E” patients receive a 250 mcg pill, “F” patients receive a 250 mcg liquid.

Single-Blind

Either the subjects or the experimenters don’t know who is receiving which treatment.

Double-Blind

Neither the subjects nor the experimenters know who is receiving which treatment.

Experimenter

the researcher that’s actually interacting with the subjects

• statistician always knows who receives which treatment

Benefits of Blinding

Placebo Effect: Placebo effect helps prove whether or not the real treatment actually has an effect.

Ethical Concerns: Blinding the experimenters means there is no risk of preferential treatment between groups.

Control

Keeping other variables constant for all experimental units. This keeps variability LOW.

Example (Vitamin D Experiment):

• All patients are the same age

• All patients have similar diets and exercise

Blocking

If you know different units can be grouped together into categories, we call those categories blocks. Blocking is one way to control.

Example (Vitamin D Experiment):

• One block of young people (<30) and one block of older people (>40)

• One block of overweight people, one block of healthy weight people

Replication

Using enough experimental units to have replicable results.

Example (Vitamin D Experiment):

• How many patients were in the sample? (240)

• Larger samples mean more replication which means more reliable results

(replication doesn’t mean repeat but it means sample size of experiment)

If you don’t do random sampling…

your results only apply to those similar to people in the sample

Factors

The different explanatory variables being tested in an experiment.

Levels:

• Specific values/options for the factors in an experiment.

• Different levels of factors are then assigned to each treatment.

Control Group

• The ‘baseline’ group used to compare different factors/levels to.

• Usually either given a placebo or no treatment at all.