• On average, how many hours of sleep do BYU-Idaho students get?

• How can we accurately predict who will win apresidential election before the election actuallytakes place?

• How can we tell if taking Vitamin C Supplementsreduces the risk of colds?

These types of questions can be answered withstatistics.

TWO DEFINITIONS OF STATISTICS:

• Statistics is the science of collecting, organizingand interpreting data.

• Statistics are the data that describe orsummarize something.

5 BASIC STEPS IN A STATISTICAL STUDY

1. State the goal of your study precisely; that is,determine the population you want to study andexactly what you’d like to learn about it.

2. Choose a representative sample from thepopulation.

3. Collect raw data from the sample andsummarize these data by finding sampleststistics of interest.

4. Use the sample statistics to infer the populationparameters.

5. Draw conclusions; determine what you learnedand whether you achieved your goal.

STATISTICS AT WORK

1. Our class counts the m&ms in a single bag todetermine what percentage of all bagged m&msare brown. Identify the population, sample,population parameters and sample statistics inthis study.

2. The Gallop Organization conducted a poll of1200 adults to determine how Americansrespond to the question “Do you think collegecoaches use physical force with their athletes?”

Identify the population, sample, populationparameters and sample statistics in this study.

3. How could we apply the five basic steps of astatistical study to determine what percentageof high school students in the United States aremembers of the church?

WHAT KIND OF STUDY SHOULD BE USED?

EXAMPLE 1: Can listening to classical music whilestudying improve a students grades?

EXAMPLE 2: Are high levels of artificial flavoringharmful to humans when eaten infoods?

EXAMPLE 3: Does talking to plants help them growbetter?

EXAMPLE 4: What is the average number of cars perhousehold in America?

IDENTIFY THE CASES AND CONTROLS OR

CONTROL GROUP AND TREATMENT GROUP

WHERE APPLICABLE IN THE ABOVE STUDIES

SHOULD YOU BELIEVE A

STATISTICAL STUDY?

GUIDELINES:

1. IDENTIFY THE GOAL, POPULATION AND TYPE OFSTUDY

2. CONSIDER THE SOURCE

3. LOOK FOR BIAS IN THE SAMPLE

4. LOOK FOR PROBLEMS IN DEFINING ORMEASURING VARIABLES OF INTEREST

5. WATCH OUT FOR CONFOUNDING VARIABLES

6. CONSIDER THE SETTING AND WORDING INSURVEYS

7. CHECK THAT THE RESULTS ARE PRESENTEDFAIRLY

8. STAND BACK AND CONSIDER THE CONCLUSIONS

WORDING IN SURVEYS

Consider the following two survey questions:

Question 1: What is your favorite cola?

Question 2: Is Pepsi your favorite cola?

Do you think that the wording will affect theparticipants’ responses?

IS BIAS AN ISSUE IN THESE STUDIES?

EXA. 1: Market researchers conduct a survey at asupermarket on a weekday between 10:00a.m. and noon to decide which of twobrands of beer customers prefer.

EXA. 2: A start-up pharmaceutical companyconducts it’s own trials on 1000 subjects todetermine whether it’s new allergy drug isbetter than it’s competitor’s drugs.

FINAL PAPER GRADES

A F B B C B B C C D C C d A F A

B B D C C D F F B C B A d A B F

QUALITATIVE VS. QUANTITATIVE

Determine whether the variable described ineach example is qualitative or quantitative.

1. The number of credits taken by BYU-Istudents.

2. The responses of people to the question“Do you think Bush is doing a good job?”

3. Favorite colors of individuals.

4. Favorite numbers of individuals.

5. The number of hours of sleep that BYU-Istudents get.

6. Political parties that individuals belong to.

7. Grade point averages of collegestudents.

HOURS OF SLEEP DATA

5 5 5 5 5 5 6 6 6 6

6 6.5 6.5 7 7 7 7 7 7 7

7 7 7 8 8 8 8 8 8

CHANGE DATA

0 0 0 0 0 0 0 0 0 .35

.75 .86 .87 .9 1 2 2 5 5 7

8 10.13 11 20.01 24 39 53 62.5 144

HEIGHT DATA

64 64 64 64 66 66 66 66.75 67 68

68 68 68 68 69 69 70 70 70 72

72 72 72 72 73 74 75 75 75

EXAMPLE:

The homework scores on a given assignment for 13 peopleare as follows:

1, 6, 6, 7, 7, 8, 8, 8, 8, 9, 9, 9, 10

Find the “average” of the scores in three different ways byfinding the mean, median, and mode of the data.

EXAMPLE:

A track coach wants to determine an appropriate heart ratefor her athletes during their workouts. She chooses 5 of herbest runners and asks them to wear heart rate monitorsduring a workout. In the middle of the workout, she readsthe following heart rates for the five athletes:130, 135, 140, 145, 325.

Which is a better measure of the “average” in this case, themean or the median? Why?

CHARACTERIZING A DISTRIBUTION BY ITS SHAPE

HOW MANY PEAKS?

Would you expect the following distributions to be unimodal,bimodal, or uniform?

1. The weights of adult American men.

2. The weights of all adult Americans.

3. IQ scores for all adult Americans.

4. The last digit of telephone numbers of Idaho residents.

5. The times it takes to run a mile for BYU-I students.

6. The times it takes to run a mile male BYU-I students.

SKEWED OR SYMMETRIC?

Would you expect the following distributions to be right-skewed, left-skewed, or symmetric?

1. The exam scores (out of 100) on a very easy exam.

2. The weights of all adult Americans.

3. The family income for American families.

4. The number of times that people change jobs duringtheir careers.

5. IQ scores for all adult Americans.

RANGE CAN BE MISLEADING

Suppose that I ask two different classes (of 10 peopleeach) how much money they have with them. Supposefurther that I obtain the following results:

Class A

$1, $2, $2, $3,

$5, $5, $6, $8,

$9, $10

1 2 3 4 5 6 7 8 9 10 11

Class B

$0, $0, $0, $0,

$0, $0, $0, $0,

$0, $95

0 95

For which class would you say that the amount of moneyper student varies more?

Which class has the bigger range?

QUARTILES AND THE FIVE NUMBER SUMMARY

The values in the two data sets below are waiting times, inminutes, at checkout lines in two supermarkets: Big Martand Super-Duper Mart. Find the 5-number summary foreach supermarket.

Big Mart

4 5 5 6 7 7 8 9 10

Super-Duper Mart

1 6 6 7 7 7 8 8

Draw boxplots for both supermarkets on the same axis.

Find the standard deviation for both supermarkets.

CALCULATING THE STANDARD DEVIATION

STEP 1: Compute the mean of the data set. The findthe deviation from the mean for every datavalue by subtracting the mean from the datavalue.

DEVIATION = DATA VALUE – MEAN

STEP 2: Find the squares of all the deviations.

STEP 3: Add all the squares together.

STEP 4: Divide this sum by one less than the totalnumber of data values.

STEP 5: The standard deviation is the square root ofthis quotient.

RANGE RULE OF THUMB

1. To estimate the standard deviation (given the

range), we can use the following rule of thumb:

S.D. . range / 4

2. To estimate the range (given the mean and

the standard deviation), we can use the following rule of thumb:

low value . mean - 2(S.D.)

high value . mean + 2(S.D.)

EXAMPLES:

1. Use the range rule of thumb to estimate the standarddeviation for Big Market and Super-Duper Market.

Do we get a good estimate?

2. Given that the mean IQ score in a group of people is

105 with a standard deviation of 15, use the rangerule of thumb to estimate the range of IQ scores forthe group.

WAITING TIMES AT BUS TERMINALS

Atlanta:

5.5 6.0 8.0 5.0 7.0 6.5 5.0 7.5 5.5 4.0

Boston:

5.5 8.0 2.0 5.0 8.5 12.0 1.5 6.5 9.5 10.0 6.0

Find the mean, median, and range for the Atlanta data and forthe Boston data.

Find the five number summary for each data set.

Draw a boxplot for each data set.

DATA SETS WITH ANORMAL DISTRIBUTION

Which of the following sets of data would you expectto be normally distributed?

1. The ACT scores of all students who take it.

2. The heights of all BYU-I students.

3. The weights of cans of A&W root beer.

4. The scores on a very easy test.

5. IQ scores for all adult Americans.

6. The heights of all adult men.

7. Shoe sizes of adult women.

8. The family income for American families.

68-95-99.7 RULE

EXA. 1: One way that a vending machine detects counterfeitcoins is by weighing them. The weight of U.S.quarters is normally distributed with a mean of 5.67grams and a standard deviation of .07 grams.

If a certain machine is adjusted to reject quarters thatweigh more than 5.88 grams and less than 5.46grams, approximately what percent of actual U.S.quarters will be rejected.

EXA. 2: Suppose that 1000 students take an exam and thescores are normally distributed with a mean of 75%and a standard deviation of 7%.

• Approximately what percent of the students scoredabove an 89%?

• Approximately how many students scored abovean 89%?

• Approximately what percent of the students scoredbelow 54%?

• Suppose a 68% was required in order to pass. Approximately how many of the students passed?

• Approximately what percent of the students scoredabove a 70%?

STANDARD SCORES AND

PERCENTILES

Find the standard score and percentile for each of thefollowing data values:

• A data value that is 1.2 standard deviations belowthe mean.

• A data value that is 3 standard deviations abovethe mean.

• A data value that is 0.7 standard deviations abovethe mean.

Use the 68-95-99.7 rule to approximate the percentilesthat correspond to the following data values: (check yourapproximation by looking up the actual percentiles in thetable.)

• A data value with a standard score of z=1.

• A data value with a standard score of z=0.

• A data value with a standard score of z=-2.

PERCENTILE EXAMPLE

Suppose that Mark, a college student at BYU-Idaho, hasdecided that as a rule he will not date girls that are tallerthan him. (Mark is 5' 7".) If Mark sticks by this decision,approximately what percentage of girls has Mark cut outwith his height rule?

To answer this question, use the fact that heights of adultgirls are normally distributed with a mean height of 63.5inches and a standard deviation of 2.5 inches.

ANOTHER PERCENTILE EXAMPLE

Suppose that IQ test scores are normally distributed with amean of 100 and a standard deviation of 15.

• About what percent of the population has an IQ between100 and 120?

• About what percent of the population has an IQ above125?

• If Marilyn (from the newspaper column“Ask Marilyn”) has an IQ of 180, aboutwhat percent of the population has an IQhigher than her?

• Suppose that in order to be a member of a certain eliteclub, you have to have an IQ that is in at least the 95th

percentile. How high does your IQ score have to be?