Statistical Reasoning

Introduction to Probability and StatisticsMs. Young

What is/are statistics?

When singular, statistics is the science of collecting, organizing, and interpreting data Ex. ~ “Statistics was used to determine the

average salary of recent college graduates” When plural, statistics are the data

(numbers or other pieces of information) that describe or summarize something Ex. ~ The average age most people stop

believing in Santa Claus is 8 The number 8 is considered a statistic

Ex. ~ More than 6 billion minutes are spent on Facebook (worldwide) each day

6 billion minutes is the statistic that describes the time spent on Facebook

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How Statistics Works

Advertisers want to know the number of people that watch the Super Bowl game in order to decide if it is worth it for them to spend the nearly $3 million for a 30-second spot. Nielsen Media Research, a company that conducts statistical studies, states that 93.2 million Americans watched the Indianapolis Colts win Super Bowl XLI. How did they come up with this number? Did they ask everyone in America if they watched the game? Of course not! The company has what they call “people meters” that are set up in over 5000 homes which monitor the television viewing habits of the people in those homes. With that data they were able to come up with the statistic of 93.2 million viewers.

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Determine the population in which you are interested in learning about The complete set of people or things being

studied Ex. ~ The goal of Nielsen Media Research during Super

Bowl XLI (41) was to determine the total number of Americans that watched the game. In this particular case, the population is all Americans

Determine the ultimate goal (or goals) of the study These are known as population parameters -

specific characteristics of the population Ex. ~ The population parameter in the Super Bowl

example would be the number of Americans who actually watched the Super Bowl

How Statistics Works Cont’d…

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How Statistics Works Cont’d… Choose a representative sample – a subset of the

population from which data are actually obtained Ex. ~ The sample in the Super Bowl scenario would be the

5000 (ore more) homes (which equate to about 13,000 people) that are equipped with the “people meters”

Use the sample to obtain raw data – the actual measurements or observations collected from the sample that support the population parameter(s) Ex. ~ The raw data from the Super Bowl study would be the

number of viewers (out of the sample of 5000 homes) that tuned in to the Super Bowl TV

Analyze the raw data and come up with a sample statistic (or statistics) – characteristics of the sample found by consolidating or summarizing the raw data. Ex. ~ The total number of people within the sample of 5000

homes that watched the Super Bowl. For example, if 31% of the 5000 homes watched it, then it would

be a good estimate to say that 31% of the entire population (all Americans) watched it

Use the sample statistic(s) to draw conclusions about the entire population Ex. ~ Based on the sample statistic, it is estimated that 93.2 million

viewers tuned in to Super Bowl XLI

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Mapping a Statistical Study

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start

PopulationPopulation

parameters Sample

Raw dataSample statistic

Example 1: The U.S. Labor Department defines the civilian labor force

as all those people who are either employed or actively seeking employment. Each month, the Labor Department reports the unemployment rate, which is the percentage of people actively seeking employment within the entire civilian labor force. To determine the unemployment rate, the Labor Department surveys 60,000 households. For the unemployment reports, describe the population, population parameter, sample, raw data, and sample statistic. Population:

The civilian labor force Population parameter:

The unemployment rate of the civilian labor force Sample:

The 60,000 households that were surveyed Raw data:

The actual number of unemployed civilians from that sample of 60,000 households

Sample statistic: The unemployment rate from the sample (found by analyzing and

summarizing the raw data)

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95% Confidence Intervals Confidence interval – a range of values that is likely to contain

the true population parameter The most common interval is the 95% confidence interval, but there

are other common levels as well (90% and 99%) Margin of Error – the maximum likely difference between the

sample statistic and the true population parameter In other words, it is a value that takes all the fluctuations in the raw

data into account and gives a “cushion” on either end of the sample statistic

A confidence interval is constructed by taking the sample statistic and both adding and subtracting the margin of error in order to identify the likely range of the true population parameter

Sample statistic – margin of error < true population parameter <Sample statistic + margin or error

Ex. ~ In the case of the Super Bowl, the margin of error (for a 95% level of confidence) was found to be about 1% and the sample statistic was 31%. The 95% confidence interval would be:

Based on this data we can be 95% confident that between 30% and 32% of all Americans watched the super bowl

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31% 1% true population percentage 31% 1%

30% true population percentage 32%

Example 2: The Pew Research Center for the People and the Press

interviewed 1,546 adult Americans about their attitudes toward the future. Asked whether humans would land on Mars within the next 50 years, 76% of these 1,546 people said either definitely yes or probably yes. The margin of error for the poll was 3 percentage points. Describe the population and the sample for this survey

The population is all adult Americans and the sample is the 1,546 adult Americans that were interviewed

Explain the meaning of the sample statistic of 76% The 76% represents the percentage of people from the sample

that believe humans will land on Mars within the next 50 years What can we conclude about the percentage of the population

that thinks humans will land on Mars with in the next 50 years?

With the 3 percentage point margin of error, the confidence interval is between 73% and 79%

Based on this information, we can be 95% confident that the true percentage of the population that believes we will land on Mars within the next 50 years is between 73% and 79%

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The purpose of statistics

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Among many uses, perhaps the most important use for statistics is to help us make good decisions about issues that involve uncertainty.

Essential Questions

Reflect on the 4 essential questions from the beginning & try to put them in your own words without referring to your notes.

What are the two different meanings of statistics?

What is the basic process of a statistical study?

What is a confidence interval?

What is the purpose of statistics?

Sec. 1.1 Homework Problems

P.9 #’s 1-4, 6-20 even, 24 & 28