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Probability and Statistics – Mrs. Leahy

Chapter 1 – Introduction to Statistics

1.1 An Overview of Statistics STATISTICS is the science of collecting, organizing, analyzing, and interpreting data in order to make decisions. Reliable statistical conclusions require reliable data.

____________________________ statistics: methods of organizing, picturing, and summarizing information from samples or populations.

____________________________ statistics: methods of using information from a sample to draw conclusions regarding the population.

Example 1: A survey of 2000 third- to twelfth-grade students found that they devoted an average of 7 hours and 38 minutes each day to using electronic media.

Descriptive: Make an inference: ___________ : information coming from observations, counts, measurement, or responses

_____________________ : the collection of ALL outcomes, ALL measurements, ALL responses. Sometimes this is called a census.

A numerical description of a characteristic of a population is called a _______________________.

_____________________ : a subset, or part, of a population.

A numerical description of a characteristic of a sample is called a _______________________.

Example: BEADS – Variability in Samples

Why Study Statistics?

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Example 2: Identify the population and the sample in the following study. A study of the dietary habits of 20,000 men was conducted to find a link between high intakes of dairy products and prostate cancer. Example 3: Would you collect data using a population (a census) or a sample (a sampling) to determine the following? Why?

a) The most popular sports team among people in New York.

b) The average salary of the 30 employees of a company.

c) Whether wedding guests are allergic to the planned main course.

d) Amount of time high school students spend on their cell phones each week. Example 4: Determine if each of the following is a population parameter or a sample statistic.

a) A survey of 1000 U.S. adults found that 40% think that the Internet is the best way to get news and information.

b) At a college, 90% of the members of the Board of Trustees approved the contract of the new president.

c) A survey of 733 small business owners found that 17% have a current job opening.

d) The 2182 students who accepted admission offers to Northwestern University in 2009 have an average SAT score of 1442. (Source: Northwestern University)

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Example 5: In a recent survey, 1500 adults in the United States were asked if they thought there was solid evidence for global warming. Eight hundred fifty-five of the adults said yes. (Adapted from: Pew Research Center) Identify the population and the sample. Describe the data set. Example 6: A large sample of men, aged 48, was studied for 18 years. For unmarried men, approximately 70% were alive at age 65. For married men, 90% were alive at age 65. (Source: The Journal of Family Issues) HIGHLIGHT or UNDERLINE any descriptive statistics. What is a possible inference that could be drawn from the data?

1.2 Data Classification OK. You’ve got data. Now what?

TYPES OF DATA: Variables can be quantitative: _________________ measurements or counts for which operations such as addition or averaging make sense. Variables can be qualitative: _________________ descriptions or labels that place an individual into a category or group such as male or female. Example 1: You conduct a survey and collect data. UNDERLINE the QUANTITATIVE variables. CIRCLE the QUALITATIVE variables.

Name Height Age Gender

Marital Status GPA Student ID# Pulse Rate

Why Study Statistics? Pt 2

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Levels of Measurement: What statistical calculations make sense with this particular data?

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*Great Charts on Page 12 of Textbook* A great read on Levels of measurement:

Example 2: State if the variable is quantitative or qualitative. Then categorize the measurement as nominal, ordinal, interval, or ratio. a) Length of time to complete an exam b) Course evaluation scale: poor, acceptable, good c) Major field of study d) Score on last exam out of 100 points e) Postal ZIP Code f) Rating of tech support call: Rate from 1 to 5, 1 = unsatisfied, 5 = extremely satisfied g) Classroom temperature in °F h) Annual income of all certified employees

** i) Time of first class (Example: 8:00AM)

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1.3 (Part 1) Experiment Design

GOAL: Collect data and use the data to make a decision.

Textbook: pg 17 GUIDELINES FOR DESIGNING A STATISTICAL STUDY

What kind of study should we use?

Observational Study Researcher observes and measures characteristics of interest, but does not change existing conditions.

Experiment A treatment if applied to part of a population (called the treatment group) and the results are measured. Often a placebo is given to a second portion of the population (called the control group) to make sure the results are not just by chance.

Simulation Use a mathematical or physical model to reproduce the condition of a situation or process. Useful when the actual condition would be dangerous or impractical.

Survey Data is collected by people asking questions. Extra care must be taken in survey design to prevent biased questioning.

Example 3: Which should be used? Observational Study, Experiment, Simulation, or Survey?

a) Study of the effect of stopping the cooling process of a nuclear reactor.

b) Study of the amount of time college students taking a full course load spend watching television.

c) Study of the effect on bone mass of a calcium supplement given to young girls.

d) Study of the reaction of individuals who find a lost wallet.

e) Whether wedding guests are allergic to the planned main course.

f) Effect of a new medicine for heart disease.

e) Effect of a large scale earthquake.

g) Study of how often preschoolers choose gender-neutral toys over gender-specific toys.

Three Questions For Statistics: 1. How big is it? 2. What difference does it make? 3. Are you sure that’s not just dumb luck? (source: US Census Bureau)

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HOW TO SET UP AN EXPERIMENT 1. CONTROL Potential Problems:

Confounding Variables – An experimenter cannot tell the difference between the effects of different factors on the variable.

This can ruin an experiment!

The Placebo Effect – A subject in the control group responds favorably to a placebo when in fact they have received no treatment. Help control this by “blinding” (subject doesn’t know which group they are in) or “double blinding” (subject AND experimenter don’t know which group they are in).

2. RANDOMIZATION Subjects are randomly selected into groups to prevent bias. Randomized Block is very common:

3. REPLICATION

An experiment should be able to be replicated with a large number of subjects in order for the results to be considered valid. Often experiments with controversial or cutting-edge results are replicated in the scientific community to validate and improve the original findings.

BIASED QUESTIONS These questions are bad. Can you improve them?

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Sampling Techniques

We must be able to create a RANDOM SAMPLE for our experiments, surveys, and samplings. There are several techniques for creating a random sample.

How to we get a random sample?

1. Number all members of the population sequentially 2. Use a table, calculator, or computer to select random numbers from the numbers assigned to the population members. 3. Create a sample by using population members with numbers corresponding to those randomly selected. Using a random number table Example 1: Use a random number table to pick a random sample of 10 cars from a population of 500 cars. (Arbitrarily choose a place to start in the table, say, row 4, column 2) Example 2: Use a random number table to generate a list of 10 numbers from 1 to 60 (inclusive). (Arbitrarily choose where to start, say row 6, column 4)

Sampling Technique: Simple Random Sampling (SRS) Features: 1. Each element in the population has an equal probability of selection 2. Each possible sample of a given size (n) has an equal probability of being the sample actually selected. 3. Each element is selected independently of every other element.

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Examples:

People are lined up to buy tickets to a rock concert. Every 5th person is chosen for the sample. The first person is randomly selected from the first 5 people.

A fabric mill worker is required to systematically check the product produced by a certain loom. She checks every 20th yard of fabric for flaws. (Problem: the machine could be making an error on every 7th yard. The error would not be discovered until 140 yards of fabric had been produced!)

Examples:

A population of all undergraduate college students is divided into four strata: freshman, sophomores, juniors, and seniors. Ten students from each strata are randomly selected and their GPAs are recorded.

Every cardholder at the public library is divided into three strata: men, women, and children under 12. Four cardholders from each strata are selected at random and asked how many times they visit the library each month.

Sampling Technique: Stratified Sampling Features: 1. Divide population into (at least 2) groups or classes (strata) that share common characteristics 2. A simple random sample (SRS) of a certain size is drawn from each stratum 3. Information is carefully adjusted/weighted based on size of stratum

Sampling Technique: Systematic Sampling Features: 1. Assign each element in the population a sequential number, or arrange the elements in a natural sequential order. 2. Select a random starting point and then select every kth element for the sample.

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Example:

We want to take a survey of all the school children in a large city. Randomly select 4 schools and then include all the children from each selected school in the sample.

Stratified Sampling vs. Cluster Sampling Divide into groups with similar characteristics Divide into groups (often geographically) Randomly select a few from each group. Randomly choose a few groups. Your sample contains a few from each group. Your sample contains all members of those groups. All groups are represented. Some groups will not be represented.

Example:

Newspaper reported interviews the first 5 people leaving a local music score about their opinion on a tax to support the local symphony.

Sampling Technique: Cluster Sampling Features: 1. Used extensively by government agencies and private research organizations 2. Divide the demographic area into sections (or clusters). 3. Randomly select sections (clusters) and include all the members of the clusters in the sample.

Sampling Technique: Convenience Sampling Features: 1. Create a sample by using data from population members that are readily available. 2. Sample runs the risk of being extremely biased

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Example 3: Identify the sampling technique used: Simple Random Sample (SRS), Convenience, Systematic, Stratified, or Cluster.

a) A Los Angeles Times reporter gets a reaction to a breaking story by polling people as they pass the front of his office building.

b) Dr. Smith randomly selects 5 students in his class by assigning each a number and drawing five numbers from a hat.

c) The Orange County Commissioner of Jurors obtains a list of 55,014 car owners and constructs a group of possible jurors by selecting every 50th name on the list.

d) In a Harris poll of 1,011 adults, the interview subjects were selected by using a computer to randomly generate telephone numbers that were then called.

e) A Ford Motor Company researcher has partitioned all registered cars into categories of compact, mid-sized, and family-size. He is surveying 75 car owners from each category.

f) Motivated by a student who died from binge drinking, Indiana State conducts a study of student drinking by randomly selecting 10 different classes and interviewing all of the students in each of those classes.

g) A statistics student obtains height/weight data by interviewing the members of his fraternity.

h) A UCLA researcher surveys all cardiac patients in each of 30 randomly selected hospitals.