MAT142 Chapter 5A

Fundamentals of Statistics

Chapter 5B

Should you believe a statistical study?

5-A

Two Definitions of Statistics

 Statistics is the

science

of collecting, organizing, and interpreting data.

 Statistics are the

data

that describe or summarize something.

 Raw Data: All of the data collected in a study.

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More Definitions

 The

population

in a statistical study is the complete set of people or things being studied.



Population parameters

are specific characteristics of the population that a statistical study is designed to estimate.

 The

sample

is the subset of the population from which the raw data are actually obtained.



Sample statistics

are numbers or observations that summarize the raw data in the sample.

Identify the population, sample, population parameter, and sample statistic .

1.

In a survey of 1100 American adults, 38% said that they kept a dog for protection.

2.

In a survey of 450 Arizona residents over the age of 65, 32% said they use the internet on a daily basis.

3.

An election eve poll found that 52% of the 800 surveyed voters plan to vote for Smith.

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Margin of Error

 The measure of accuracy of a properly conducted statistical study  The

margin of error

is used to describe a

confidence interval

that is likely to contain the true population parameter. The confidence interval is From (sample statistic − margin of error)  To (sample statistic + margin of error)

“The sample percentage differs from the population percentage by more than the margin of error less than 5% of the time.”

Margin of Error

 A survey finds that 84% of Americans watched the Super Bowl, with a margin of error of 5%. Interpret the survey results.

 There is a 95% chance that between 58% and 66% of voters will vote for Senator Sam in the next election. What do you know about the sample statistics and the margin of error of the survey?

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Margin of Error

In a survey conducted in a certain city it was found that 4.8% of men and 6.1% of women were unemployed. The margin of error for each report was 0.8 percentage points. Use each sample statistic to find a confidence interval. Interpret your answer in a complete sentence.

Can we conclude that the unemployment rate is higher amongst women than amongst men?

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Margin of Error

MOE

 1

n

 For a sample of size n, the margin of error (MOE) would be:

MOE

 1

n

Note: Population size is not part of the equation!

 chosen SCC students. Suppose our results show that 19% of them are left-handed What is the margin of error?

Determine the Margin of Error and confidence Interval

1.

In a survey of 1100 American adults, 38% said that they kept a dog for protection.

2.

In a survey of 450 Arizona residents over the age of 65, 32% said they use the internet on a daily basis.

3.

An election eve poll found that 52% of the 800 surveyed voters plan to vote for Smith.

Elements of a Statistical Study

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Copyright © 2008 Pearson Education, Inc.

Slide 5-10

 The criterion for selecting a

random

sample is that each unit in the population is as likely to be chosen as any other unit.  A statistical study suffers from

bias

if its design or conduct tends to favor certain results Copyright © 2008 Pearson Education, Inc.

Slide 5-11

BIAS • A systematic as opposed to a random distortion of a statistic as a result of sampling procedure • A statistic is

biased

if it is calculated in such a way that it is systematically different from the population parameter of interest.

Choosing a Sample

 Choosing a sample is the most important step in any statistical study.

 A representative sample is a sample in which the relevant characteristics of the sample members match those of the population.

 It is vital that the sample be as representative (unbiased) as possible in order to have valid data!

Slide 5-13

Copyright © 2008 Pearson Education, Inc.

Common Sampling Techniques Simple Random Sampling Systematic Sampling Stratified Sampling Convenience (Haphazard) Sampling

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Simple Random Sampling

Simple random sample

Suppose we have a population of 3000 and we want a SRS of 100.

Make a

list

of all the units in a population and number them. Obtain 100 random numbers between 1 and 3000 ***It is very important that you measure the actual 100 individuals selected!!!!!***

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Systematic Sampling

List population. Choose every n th person on the list.

Ex. Suppose you want to select a sample of 10 from a population of 100 . Let n=10. Choose units 3, 13, 23, 33 … NOTE: Be careful these don’t have some similarity inherent in the ordering. (Ex. Picking offices 101, 201, 301,… in an office building.)

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Stratified Sampling

Strata

are the “natural” groups populations fall into.

To obtain a stratified random sample, we first divide the population into strata and then take simple random samples of each stratum.

Advantages – We can obtain results not only for the whole population, but also for each stratum.

Convenience (Haphazard) Sampling

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Note: “Random” and “Haphazard” are not the same!!!

Identify the sampling method.

 A college coach randomly chooses two athletes from each sport to attend a banquet.

 A computer randomly selects 300 students from an ASU computer data base to find out if they voted in this year’s election.

 An ASU professor polls all the students in the south parking lot at 4:00 pm to ask if they prefer morning or afternoon classes.

 Every 200th quart of ice cream off the assembly line is tested for taste and consistency.

Copyright © 2008 Pearson Education, Inc.

Slide 5-19

Sampling Bias:

1.

2.

Selection bias: when some individuals or groups are more likely to be selected for the sample than others. This results in biased samples Participation bias – Self-selected polls and Voluntary Response surveys

May prevent the sample from being representative of the population!

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Types of Statistical Studies

 In an

observational study

, researchers observe or measure characteristics of the sample members but do not attempt to influence or modify these characteristics.

 In an

experiment

, researchers apply a treatment to some or all of the sample members and then look to see whether the treatment has any effects.

5-A Slide 5-21

Copyright © 2008 Pearson Education, Inc.

Experiments Treatment and Control Groups

 The

treatment group

in an experiment is the group of sample members who receive the treatment being tested.

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 The

control group

in an experiment is the group of sample members who do

not

receive the treatment being tested.

It is important for the treatment and control groups to be selected randomly and to be alike in all respects except for treatment.

Copyright © 2008 Pearson Education, Inc.

Slide 5-22

Experiments The Placebo Effect

 A

placebo

lacks the active ingredients of a treatment being tested in a study, but is identical in appearance to the treatment. Thus, study participants cannot distinguish the placebo from the real treatment.

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 The

placebo effect

refers to the situation in which patients improve simply because they believe they are receiving a useful treatment.

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Experiments Blinding

 An experiment is

single-blind

if the participants do not know whether they are members of the treatment group or members of the control group, but the experimenters do know.

 An experiment is control group.

double-blind

if neither the participants nor the experimenters (people administering the treatment) know who belongs to the treatment group and who belongs to the

Slide 5-24

Copyright © 2008 Pearson Education, Inc.

Observational Studies Case-Control Studies

 A

case-control study

is an observational study that resembles an experiment because the sample naturally divides into two or more groups.

 The participants who engage in the behavior under study form the

cases

.

 The participants who do not engage in the behavior are the

controls .



Example:

Does smoking elevate blood pressure?

5-A Slide 5-25

Copyright © 2008 Pearson Education, Inc.

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What type of study?

1.

2.

3.

4.

5.

6.

Does garlic lower cholesterol?

Does caffeine cause birth defects?

The Acme Soda Company wants to know if their soda tastes better than Cooky Cola.

What is the average income of elementary school teachers?

Can a new herbal remedy reduce the severity of migraine headaches?

Do seat belts save lives?

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Is the treatment effective?

1.

2.

3.

70% of those in the treatment group showed improvement, whereas 40 % of those in the placebo group showed improvement.

40% of those in the treatment group showed improvement, whereas 10 % of those in the placebo group showed improvement.

70% of those in the treatment group showed improvement, whereas 65 % of those in the placebo group showed improvement.

Copyright © 2008 Pearson Education, Inc.

Slide 5-27

Validity vs. Reliability

Validity:

A valid measurement is one that actually measures what it claims to measure To determine validity you need to know

exactly

what was measured.

Measuring happiness by the number of times a person smiles a day is not valid because smiling does not completely ( or necessarily ) measure happiness.

Validity vs. Reliability

Reliability

– A measurement is reliable if it will give you or anyone else approximately the same result time after time when taken on the same object.

NOTE: Most reliable measurements are physical measurements with a

precise instrument

Biased Measurement

– A measurement that is incorrect in a Ex. consistent direction A scale that overweighs every user A clock that is 5 minutes fast NOTE: Although scale is biased, it is reliable because it will

consistently

over-weigh people

Should you believe a statistical study?

1. Identify the goal, population and type of study.

2. Consider the source.

3. Look for bias in the sample.

4. Look for problems in defining or measuring the variables of interest.

5. Watch out for confounding variables.

Definition: A variable is an item or quantity that can vary or take on different values.

The variables of interest in a statistical study are the items or quantities that the study seeks to measure.

Categorical vs. Measurement

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Guideline 4: Look for Problems in Defining or Measuring

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the Variables of Interest

Examples: How does exercise affect resting heart rate?

How many days per week do you eat breakfast?

Stress, happiness, etc…

Guideline 4: Look for Problems in Defining or Measuring the Variables of Interest

Example: Which of the following quantities of interest would be the most difficult to measure?

(a)

The average height of a volleyball team

(b)

The team member with the highest salary

(c)

The team member with the longest hair

(d)

The most outgoing team member

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Confounding Variables: Variables that are not intended to be part of the study and can sometimes make it difficult to interpret the results properly. They “confound” (confuse) the study’s results

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Example People who meditate have been found to have lower levels of an age related enzyme. But they were also more likely to have vegetarian diets, and perhaps diet and not meditation has an effect on the enzyme. Vegetarian diet (or not) is a confounding variable.

Confounding Variables: Proper randomization eliminates/minimizes confounding factors!!

Randomization is not used in observational studies, so confounding variables cannot be minimized.

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Should you believe a statistical study?

1. Identify the goal, population and type of study.

2. Consider the source.

3. Look for bias in the sample.

4. Look for problems in defining or measuring the variables of interest.

5. Watch out for confounding variables.

6. Consider the setting and wording in surveys.

7. Check that results are presented fairly.

8. Stand back and consider the conclusions.

Guideline 8: Stand back and Consider the Conclusions

1. Did the study achieve its goals?

2. Do the conclusions make sense?

3. Do the results have any practical significance?

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Guideline 8: Stand back and Consider the Conclusions

Example: An experiment is conducted in which the weight losses of people who try a new “Fast Diet Supplement” are compared to the weight losses of a control group of people who try to lose weight in other ways. After 8 weeks, the results show that the treatment group lost an average of ½ pound more than the control group.

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Example: An experiment was performed on male physicians aged 40-84 without serious health problems.

Can the results be applied to: Non-Physicians Women 20-year-olds Cancer patients

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Example: A study conducted by the oil company Exxon Mobile shows that there was no lasting damage from the large oil spill in Alaska.

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Homework – Chapters 5A and 5B



Green

Pg 334 1, 4, 6, 7, 8, 9,11, 12, 14, 16, 17, 25, 29, 39, 57 Pg 345 4, 5, 7, 8, 9, 10, 11, 12, 13,14 

Orange

Pg 307 QQ: 1, 4, 6, 7, 8, 9 Ex: 1, 2, 4, 6, 7, 15, 19, 29, 47 Pg 317 QQ 4, 5, 7, 8, 9, 10 Ex: 1, 2, 3, 4