statistics ms. sellanes

10/31/2015

1

STATISTICS Algebra 2Ms. Sellanes

GET YOUR FACTS RIGHT

INFORMATION ON PROJECT

-Will be given throughout the week.

-Start thinking about a statistical questions that intrigues you.

-Due first week in November. Check Project details.

10/31/2015

2

DEFINITIONS

Statistics is the science of collecting, organizing,

analyzing and interpreting data in order to make decisions.

Data consist of information coming from

observations, counts measurements or responses. The singular for data is datum.

DATA SETS

Two types of data sets you will use when studying statisticsPopulation is the collection of ALL outcomes, responses, measurement or counts that are of interest.

Sample: the subset of the population.

EX. 1: IDENTIFYING DATA SETS

In a recent survey, 3002 American adults were asked if they read the news on the Internet at least once a week. Six hundred of the adults said yes. Identify the population and the sample. Describe the data set.

A. Identify the population

B. Identify the sample.

C. What does the data set consist of ?

10/31/2015

3

TRY IT YOURSELF

The US Dept of Energy conducts weekly surveys of 800 gasoline stations to determine the average price per gallon of regular gasoline. On July 10, 1998, the average price was $1.066 per gallon. Identify the population and the sample.

A. Identify the population

B. Identify the sample.

C. What does the data set consist of ?

DEFINITIONS

PARAMETER: a numerical description of a population characteristic.

STATISTIC: a numerical description of a samplecharacteristic.

EXAMPLE

Decide whether the numerical value describes a population parameter or sample statistic. Explain your reasoning.

1. A recent survey of a sample of MBA’s reported that the average starting salary for an MBA is less than $65,000l

2. Starting salaries for the 667 MBA graduates from the University of Chicago School of Business increased 8.5% from the previous year.

10/31/2015

4

EX. 2: DISTINGUISHING BETWEEN A PARAMETER AND A STATISTIC.


1. A recent survey of a sample of MBA’s reported that the average starting salary for an MBA is less than $65,000

Because of the numerical measure of $65,000 is based on a subset of the population (MBA’s), it is a sample statistic.

EX. 2: DISTINGUISHING BETWEEN A PARAMETER AND A STATISTIC.


2. Starting salaries for the 667 MBA graduates from the University of Chicago School of Business increased 8.5% from the previous year.

Because the numerical measure of 8.5% is based on all 667 graduates’ starting salaries, it is a population parameter.

WHAT IS BIAS?

How can we avoid it?

~Brainstorm~

10/31/2015

5

BIASED QUESTIONS

Questions may be biased in the following ways:

The wording of the question may encourage or pressure the respondent to answer in a particular way.

The question may be perceived as too sensitive to answer truthfully.

The questions may not provide the respondent with enough information to give an accurate opinion.

The order the questions are asked.

Respondents giving answers they believe will please the questioner.

RANDOMIZATION

Importance of Random Selection

“Randomly selecting the members of a sample is important because it helps prevent bias in your results. Random selection allows impersonal choice to choose the

sample, rather than the individual performing the poll (the sampler) to select their own participants or self-selection of respondents as in the voluntary response poll mentioned above.”

(http://www.nedarc.org/statisticalhelp/selectionAndSampling/randomSelection.html)

DIFFERENT TYPES OF STUDIES

Sample Survey

Experiment

Observational Study

Depending on what we want to study, one type might be better than another.

10/31/2015

6

SAMPLE SURVEY

Statistical surveys are used to collect quantitative information from a specific population. A survey may focus on opinions or factual information depending upon the purpose of the study. Surveys may involve answering a questionnaire or being interviewed by a researcher. The census is a type of survey.

Advantages: can be administered in a variety of forms (telephone, mail, on-line, mall interview, etc.), and efficient for collecting data from a large population

Disadvantages: are dependent upon the respondent's honesty and motivation when answering, can be flawed by non-response, and can possess questions or answer choices that may be interpreted differently by different respondents (such as the choice "agree slightly") .

EXPERIMENT

Experimental Study: Investigators apply treatments to experimental units (people, animals, plots of land, etc.) and then proceed to observe the effect of the treatments on the experimental units.

In a randomized experiment investigators control the assignment of treatments to experimental units using a chance mechanism (like the flip of a coin or a computer's random number generator)

OBSERVATIONAL STUDY DEFINITION

Observational Study: Researchers observe or measure characteristics of the subjects (people or things), but do not attempt to influence or change these characteristics.

How do me Randomize?

Example: For example, suppose we want to study the effect of smoking on lung capacity in women

10/31/2015

7

EXPERIMENT EXAMPLE

Find 100 women age 20 who do not currently smoke.

• Randomly assign 50 of the 100 women to the smoking treatment and the other 50 to the no smoking treatment.

• Those in the smoking group smoke a pack a day for 10 years while those in the control group remain smoke free for 10 years.

• Measure lung capacity for each of the 100 women.

• Analyze, interpret, and draw conclusions from data.

OBSERVATIONAL EXAMPLE

• Find 100 women age 30 of which 50 have been smoking a pack a day for 10 years while the other 50 have been smoke free for 10 years.

• Measure lung capacity for each of the 100 women.

• Analyze, interpret, and draw conclusions from data

PRACTICE IDENTIFYING THE TYPES

In groups, work to create an experiment using one of the three types of types of studies (I will go around and assign it).

Please explain:

1) The study

2) The type of study

3) Whether there is any bias or randomization

-Your group will present example to the class.

10/31/2015

8

HOMEWORK

Make a Venn Diagram comparing and contrasting the three different types of studies (survey, observation, experiment). It should look like this and every spot should have at least two bullets of valuable information.

DIFFERENT TYPES OF GRAPHS

FREQUENCY DISTRIBUTION

The organization of raw data in table form, using classes and frequencies

Class – a subset of the data

Frequency – number of times in a class

Class limit – beginning and end of a class

10/31/2015

9

Suppose a researcher wants to do a study on the number of

miles that employees of Wal-Mart travel each day. She

collects the following data: (see data set 1)

1 2 6 7 12 13 2 6 9 5

18 7 3 15 15 4 17 1 14 5

5 16 4 5 8 6 5 18 5 2

9 11 12 1 9 2 10 11 4 10

9 18 8 8 4 14 7 3 2 6

1 2 6 7 12 13 2 6 9 5

18 7 3 15 15 4 17 1 14 5

5 16 4 5 8 6 5 18 5 2

9 11 12 1 9 2 10 11 4 10

9 18 8 8 4 14 7 3 2 6

Class limits Tally Frequency

1-3

4-6

7-9

10-12

13-15

16-18

1 2 6 7 12 13 2 6 9 5

18 7 3 15 15 4 17 1 14 5

5 16 4 5 8 6 5 18 5 2

9 11 12 1 9 2 10 11 4 10

9 18 8 8 4 14 7 3 2 6

Class limits Tally Frequency

1-3 ////’////’ 10

4-6 ////’////’//// 14

7-9 ////’////’ 10

10-12 ////’/ 6

13-15 ////’ 5

16-18 ////’ 5

10/31/2015

10

CATEGORICAL FREQUENCY DIST.

Used when data can be placed in specific categories, such as nominal or ordinal data. For example, political affiliation, religion, class…

SEE DATA SET 2

Twenty five army inductees were given a blood test to determine their blood type. Data set 2 is the result.

Create a frequency distribution for the data.

A B B AB O

O O B AB B

B B O A O

A O O O AB

AB A O B A

Class Tally Frequency Percent

10/31/2015

11

A B B AB O

O O B AB B

B B O A O

A O O O AB

AB A O B A


A

B

O

AB

A B B AB O

O O B AB B

B B O A O

A O O O AB

AB A O B A


A ////’ 5 20

B ////’// 7 28

O ////’//// 9 36

AB //// 4 16

MEAN, MEDIAN, MODE, AND RANGE

Algebra 2

Ms. Sellanes

10/31/2015

12

OBJECTIVE

Calculate and interpret the mean, median and mode of a set of data.

Central Tendency:

MEAN

Mean is the average of a set of data. To calculate the mean, find the sum of the data and then divide by the number of data.

Example: 12, 15, 11, 11, 7, 13

1. Add: 12 + 15 +11 + 11 + 7 + 13 = 69

2. How many datums? 6

3. Divide: Sum/(number of data)= 69/6

Conclusion: The mean or average is 11.5

An electronics store sells

CD players at the

following prices: $350,

$275, $500, $325, $100,

$375, and $300. What is

the mean price?

10/31/2015

13

An electronics store sells

CD players at the

following prices: $350,

$275, $500, $325, $100,

$375, and $300. What is

the mean price?

The mean or average price of a

CD player is $317.86.

MEDIAN

Median is the middle number in a set of data when the data is arranged in numerical order.

The median of a set of data separates the data into two equal parts.

Example: 12, 15, 11, 11, 7, 13

1. Arrange in Order: 7, 11, 11, 12, 13, 15

2. Find the middle number:

7, 11, 11, 12, 13, 15

3. If there are two numbers in the middle, take the average of the two number:

(11+12)/2=23/2=11.5

Conclusion: The mean or average is 11.5

YOUR TURN

An electronics store sells CD players at the following prices: $350, $275, $500, $325, $100, $375, and $300. What is the median price?

The median price is $325

10/31/2015

14

MODE

The mode is the number that occurs themost. There are 3 possibilities:

One mode Only one number is the most frequent

More than one mode A few numbers appear the same about of time

No more Every number appears the same amount of times

Example: 12, 15, 11, 11, 7, 13

ON YOUR OWN

Make sets of data with the three different possibilities.

The range of a set of data is the difference between the

largest and the smallest number in the set.

For example, consider the following set:

40, 30, 43, 48, 26, 50, 55, 40, 34, 42, 47, and 50

To find the range you would take the largest number, 55,

and subtract the smallest number, 26.

55 – 26 = 29

The range is 29!

RANGE

10/31/2015

15

Mean The average

Median The number or

average of the

numbers in the

middle

Mode The number that

occurs most

BOX AND WHISKER PLOT & STEM AND LEAF PLOT

Algebra 2

Ms. Sellanes

BOX AND WHISKER PLOT: THE RECIPE

The Ingredients:

Median

First (lower) quartile: Q1The median of the first half

Third (upper) quartile: Q3The median of the second half

Max

Min

To “decorate”:

Outliers

The Final Product

10/31/2015 45

10/31/2015

16

GATHERING THE INGREDIENTS:

10/31/2015 46

Write the data in numerical

order and find the first quartile,

the median, the third quartile,

the smallest value and the

largest value.

median = 80

first quartile = 70

third quartile = 90

smallest value = 65

largest value = 100

DIRECTIONS

10/31/2015 47

Place a circle

beneath each

of these

values on a

number line.

DIRECTIONS

10/31/2015 48

Draw a box with ends

through the points for

the first and third

quartiles. Then draw

a vertical line through

the box at the median

point. Now, draw the

whiskers (or lines)

from each end of the

box to the smallest

and largest values.

10/31/2015

17

SPECIAL CASE:

10/31/2015 49

You may see a box-and-whisker plot, like the one

below, which contains an asterisk.

Sometimes there is ONE piece of data that falls well outside the

range of the other values. This single piece of data is called an

outlier. If the outlier is included in the whisker, readers may think

that there are grades dispersed throughout the whole range from

the first quartile to the outlier.

EXAMPLE

82 77 49 84 44 98 93

71 76 65 89 95 78 69

89 64 88 54 87 91 80

44 85 93 89 55 62 79

90 86 75 74 99 62 9610/31/2015 50

In pairs, you will make a box plot of the following data set.

That’s a lot of numbers so FIRST, let’s order the numbers with a stem-and-leaf plot

STEM-AND-LEAF PLOT

Stem Leaf

4 4 4 9

5 4 5

6 2 2 4 5 9

7 1 4 5 6 7 8 9

8 0 2 4 5 6 7 8 9 9 9

9 0 2 3 3 5 6 8 9

10/31/2015 51

8|2 represents a score of 82

10/31/2015

18

STEM-AND-LEAF PLOT

Stem Leaf

4 4 4 9

5 4 5

6 2 2 4 5 9

7 1 4 5 6 7 8 9

8 0 2 4 5 6 7 8 9 9

9

9 0 2 3 3 5 6 8 9

For the median:

There are 35 values, so the median number will be 35/2 or 17.5 or the 18th number.

That value is 80.

10/31/2015 52


STEM-AND-LEAF PLOT

Stem Leaf

4 4 4 9

5 4 5

6 2 2 4 5 9

7 1 4 5 6 7 8 9

8 0 2 4 5 6 7 8 9 9

9

9 0 2 3 3 5 6 8 9

For the median:

There are 35 values, so the median number will be 35/2 or 17.5 or the 18th number.

That value is 80.

10/31/2015 53


odd

Round up

STEM-AND-LEAF PLOT

Stem Leaf

4 4 4 9

5 4 5

6 2 2 4 5 9

7 1 4 5 6 7 8 9

8 0 2 4 5 6 7 8 9 9 9

9 0 2 3 3 5 6 8 9

Note: Half of the first half is

17/2 is 8.5 or 9

So 9 from the bottom, will determine your lower quartile

Count 9 from the top and that will determine your upper quartile

10/31/2015 54


10/31/2015

19

STEM-AND-LEAF PLOT

Stem Leaf

4 4 4 9

5 4 5

6 2 2 4 5 9

7 1 4 5 6 7 8 9

8 0 2 4 5 6 7 8 9 9 9

9 0 2 3 3 5 6 8 9

Now please with a partner “buy” the ingredients:

Median

First quartile: Q1The median of the first half

Third quartile: Q3The median of the second half

Max

Min

And make the box plot.

10/31/2015 55


FINAL PRODUCT

10/31/2015 56

40 50 60 70 80 90

100

Even though the whiskers are different lengths, each whisker

contains at least one fourth of the data while the box contains at

least one half of the data.

OUTLIERS

The interquartile range (IQ) is the

Upper quartile (Q3)- lower quartile (Q1)

Q3 – Q1 = 89 – 65 = 24

To find the outliers, you multiply the IQ by 1.5:

IQ ● 1.5 = 24 ● 1.5 = 36

10/31/2015 57

10/31/2015

20

OUTLIERS DIRECTIONS Now you need to take this value and subtract it from the LQ and add it to the UQ.

Remember:

Q1 = 65 and Q3 = 89

65 – 36 = 29 and 89 + 36 = 125

10/31/2015 58

29 125

Any scores less than 29 or greater than 125 will be

considered outliers. We don’t have any outliers in this case

EXAMPLE 2

Member #

Sold

Member #

Sold

George 51 Kendra 55

Vinette 69 Carmen 39

Tomas 46 Danny 81

Anthony 27 Eddie 54

Marlene 60 Jason 46

Kohana 53 Nashoba 2310/31/2015 59

Twelve members of the Beck High School Pep Club are selling programs of the football game. The number of programs sold by each person is listed below:

In pairs, make the box

and whisker plot for the

data shown.

10/31/2015 60

30

20

40

50

60

70

80

Median (M) = 52

Extreme Values

LV = 23 and GV = 80

Lower Quartile (LQ) = 42.5

Upper Quartile (UQ) = 57.5

Outlier = 81

A point is plotted for 69 since it is the

last value THAT IS NOT AN

OUTLIER. The whisker is drawn to

this point as shown. Outliers are

plotted as isolated points.

10/31/2015

21

B. ANALYZE THE BOX-AND-WHISKER PLOT TO DETERMINE IF ANY OF THE MEMBERS DID AN EXCEPTIONAL JOB SELLING PROGRAMS.

10/31/2015 61

30

40

50

60

70

80

STANDARD DEVIATION

OBJECTIVES

Student should be able to compute standard deviation given data.

Student will be able to interpret what the standard deviation means in a given problem.

10/31/2015

22

INTRODUCTION

Consider two students, each of whom has taken five exams.

Student A has scores 84, 86, 83, 85, and 87.

Student B has scores 90, 75, 94, 68, and 98.

Compute the mean for both Student A and Student B

COMPUTING THE MEAN

85

5

425

5

9868947590

85

5

425

5

8785838684

x

x

x

x

x

xThe mean for Student A is 85

The mean for Student B is 85

AND . . .

For each of these students, the mean (average) of 5 tests is 85. However, Student A has a more consistent record of scores than Student B. One way to measure the consistency or “clustering” of data near the mean is the standard deviation.

10/31/2015

23

TO CALCULATE THE STANDARD DEVIATION

1. Sum the squares of the differences between each value of data and the mean.

2. Divide the result in Step 1 by the number of items in the set of data.

3. Take the square root of the result in Step 2.

HERE IS THE CALCULATION FOR STUDENT A. THE SYMBOL FOR STANDARD DEVIATION IS THE GREEK LETTER SIGMA, DENOTED BY -- THIS IS STEP 1

x (x – x) (x – x)2

84 (84 – 85) (-1)2 = 1

86 (86 – 85) (1)2 = 1

83 (83 – 85) (-2)2 = 4

85 (85 – 85) (0)2 = 0

87 (87 – 85) (2)2 = 4

Total 10

STEP 2

10/5 = 2

10/31/2015

24

STEP 3:

414.12 The standard deviation for Student A’s score is

approximately 1.414.

Following a similar procedure for Student B, the standard

deviation for Student B’s score is approximately 11.524.

Since the standard deviation of Student B’s scores is

greater than that of Student A’s (11.524 > 1.414), Student

B’s scores are not as consistent as those of Student A.

YOUR TURN

To calculate standard deviation:

Find the mean of the weights

Use the procedure for calculating standard deviation.

The weights in pounds of the five-man front line of a college

football team are 210, 245, 220, 230, and 225. find the

standard deviation of the weights.

THE WEIGHTS IN POUNDS OF THE FIVE-MAN FRONT LINE OF A COLLEGE FOOTBALL TEAM ARE 210, 245, 220, 230, AND 225. FIND THE STANDARD DEVIATION OF THE WEIGHTS.



Use the procedure for calculating standard deviation.

226

5

1130

5

225230220245210

x

x

x

10/31/2015

25

THE WEIGHTS IN POUNDS OF THE FIVE-MAN FRONT LINE OF A COLLEGE FOOTBALL TEAM ARE 210, 245, 220, 230, AND 225. FIND THE STANDARD DEVIATION OF THE WEIGHTS.



226

5

1130

5

225230220245210

x

x

x

HERE IS THE CALCULATION FOR STUDENT A. THE SYMBOL FOR STANDARD DEVIATION IS THE GREEK LETTER SIGMA, DENOTED BY -- THIS IS STEP 1

x (x – x) (x – x)2

210 (210 – 226) (-16)2 = 256

245 (245 – 226) (19)2 = 361

220 (220 – 226) (-6)2 = 36

230 (230 – 226) (4)2 = 16

225 (225 – 226) (-1)2 = 1

Total 670

STEP 2

670/5 = 134

10/31/2015

26

576.11134 The standard deviation of the weights is approximately

11.576 lb.

NORMAL DISTRIBUTION

DIFFERENT SHAPES OF DISTRIBUTIONS

Distributions can be described as:

Roughly symmetric

Skewed right

Skewed left

10/31/2015

27

DISTRIBUTIONS OF DATAA distribution of data shows the observed or theoretical frequency of each possible data value.

Shape Definitions:

Symmetric: if the right and left sides of the graph are approximately mirror

images of each other.

Skewed to the right (right-skewed) if the right side of the graph is much longer

than the left side.

Skewed to the left (left-skewed) if the left side of the graph is much longer

than the right side.

Symmetric Skewed-left Skewed-right

10/31/2015

28

SKEW IN BOX PLOTS

SYMMETRIC AND SKEWED DISTRIBUTIONS

If the distribution is relatively symmetric, the mean and the standard deviation can be used.

If the distribution is skewed or has outliers, use the five-number summary to describe the center and spread of the data.

DESCRIBING DATA

Ms. Shroyer’s students each gave a presentation as part of their class project. The length of each presentation is shown in the table and histogram.

Describe the distribution of data.

10/31/2015

29

DESCRIBING DATADescribe the center of spread using either the mean and standard deviation or the five-number summary. Justify your choice.

COMPARING DATAThee points scored per game by a professional football team for the 2008 and 2009 football seasons are shown.

Create a box-and-whisker plot for each set. Describe the shape of each distribution. Then compare the distributions using either the means and standard deviations or the five-number summaries. Justify your choice.

COMPARING DATA

10/31/2015

30

COMPARING DATAThe lower quartile for the 2008 season and the upper quartile for the 2009 season are both 20.5. This means that 75% of the scores from 2008 were greater than 20.5 and 75% of the score from the 2009 season were less than 20.5

The minimum score of the 2008 season is approximately equal to the lower quartile for the 2009 season. This means that 25% of the scores from the 2009 season are lower than any score achieved in the 2008 season.

11-5 THE NORMAL DISTRIBUTION

Normal DistributionsAll Normal curves are symmetric, single-peaked, and bell-

shaped

A Specific Normal curve is described by giving its mean µand standard deviation σ.

Two Normal curves, showing the mean µ and

standard deviation σ.

10/31/2015

31

Normal Distributions

• We abbreviate the Normal distribution with mean µ and standard deviation σ as N(µ,σ).

• Any particular Normal distribution is completely specified by two numbers: its mean µ and standard deviation σ.

• The mean of a Normal distribution is the center of the symmetric Normal curve.

• The standard deviation is the distance from the center to the change-of-curvature points on either side.

Normal Distributions are Useful…

Normal distributions are good descriptions for some distributions of real data.

Normal distributions are good approximations of the results of many kinds of chance outcomes.

Many statistical inference procedures are based on Normal distributions.

Although there are many Normal curves, they all have properties in common.

The 68-95-99.7 Rule

The 68-95-99.7 Rule (“The Empirical Rule”)

In the Normal distribution with mean µ and standard deviation σ:

•Approximately 68% of the observations fall within σ of µ.

•Approximately 95% of the observations fall within 2σ of µ.

•Approximately 99.7% of the observations fall within 3σ of µ.

10/31/2015

32

The distribution of Iowa Test of Basic Skills (ITBS) vocabulary scores for 7th grade

students in Gary, Indiana, is close to Normal. Suppose the distribution is N(6.84, 1.55)

and the range is between 0 and 12.

a)Sketch the Normal density curve for this distribution.


students in Gary, Indiana, is close to Normal. Suppose the distribution is N(6.84, 1.55).

b) What percent of ITBS vocabulary scores are less than 3.74?

10/31/2015

33


students in Gary, Indiana, is close to Normal. Suppose the distribution is N(6.84,

1.55).?

c) What percent of the scores are between 5.29 and 9.94?

Importance of Standardizing There are infinitely many different Normal distributions; all with unique standard deviations and means.

In order to more effectively compare different Normal distributions we “standardize”.

Standardizing allows us to compare apples to apples.

The Standardized Normal DistributionAll Normal distributions are the same if we measure in units of size σ from the mean µ as center.

The standardized Normal distribution is the Normal distribution with mean 0

and standard deviation 1.

10/31/2015

34

How to Standardize a Variable :

1. Draw and label an Normal curve with the mean and standard deviation.

2. Calculate the z- score

x= variable

µ= mean

σ= standard deviation

3. Determine the p-value by looking up the z-score in the Standard Normal table.

4. Conclude in context.

The Standard Normal Table

Because all Normal distributions are the same when we standardize, we can find areas under any Normal curve from a single table.

10/31/2015

35

Using the Standard Normal Table

Using the Standard Normal Table, what is the p-value for a z-score of -2.33?

Using the Standard Normal Table

Using the Standard Normal Table, find the following:

Z-Score P-value

-2.23

1.65

.52

.79

.23

Let’s PracticeIn the 2008 Wimbledon tennis tournament, Rafael Nadal averaged 115 miles per hour (mph) on his first serves. Assume that the distribution of his first serve speeds is Normal with a mean of 115 mph and a standard deviation of 6.2 mph.

About what proportion of his first serves would you expect to be less than 120 mph? Greater than?

10/31/2015

36

1. Draw and label an Normal curve with the mean and standard deviation.

2. Calculate the z- score

x= variable

µ= mean

σ= standard deviation

3. Determine the p-value by looking up the z-score in the

Standard Normal table.

Z .00 .01 .02

0.7 .7580 .7611 .7642

0.8 .7881 .7910 .7939

0.9 .8159 .8186 .8212

P(z < 0.81) = .7910

4. Conclude in context.

We expect that 79.1% of Nadal’s first serves will be less than 120 mph.

We expect that 20.9% of Nadal’s first serves will be greater than 120 mps.

10/31/2015

37

Let’s Practice

When Tiger Woods hits his driver, the distance the ball travels can be described by N(304, 8). What percent of Tiger’s drives travel between 305 and 325 yards?

Normal Distribution Calculations


Step 1: Draw Distribution



Step 2: Z- Scores

10/31/2015

38


Step 3: P-values

Using Table A, we can find the area to the left of z=2.63 and the area to the left of z=0.13.


Step 4: Conclude In Context

Finding Areas Under the Standard Normal Curve

Find the proportion of observations from the standard Normal distribution

that are between -1.25 and 0.81.

Step 1: Look up area to the

left of 0.81 using table A.

10/31/2015

39




Step 2: Find the area to the left of -1.25




Step 3: Subtract.


Can you find the same proportion using a different approach?

1 - (0.1056 + 0.2090) = 1 – 0.3146= 0.6854

10/31/2015

40

11-6 CONFIDENCE INTERVALS AND HYPOTHESIS TESTING

CONFIDENCE INTERVALS

Inferential statistics are used to draw conclusions or statistical inferences about a population using a sample.

Example: In a recent poll, 1514 teens who owned a portable media player had an average of 1033 songs. The poll had the following disclaimer: “For results based on the total sample of national teens, one can say with 95% confidence that the margin of sampling error is ±31 songs.

CONFIDENCE INTERVALS

Is an estimate of a population parameter stated as a range with a specific degree of certainty.

A 95% confidence interval for a population means that we are 95% sure that the mean will fall within the range of z-values.

Example: We were 95% confident that the population mean is within 31 songs of

1033.

10/31/2015

41

MAXIMUM ERROR OF ESTIMATE

MAXIMUM ERROR ESTIMATE EXAMPLEORAL HYGIENE A poll of 422 randomly selected adults showed that they brushed their teeth an average of 11.4 times per week with a standard deviation of 1.6. Use a 99% confidence interval to find the maximum error of estimate for the number of times per week adults brush their teeth.

MAXIMUM ERROR OF ESTIMATE CONT. Maximum Error of Estimate

10/31/2015

42

EXAMPLE

A. 0.49

B. 0.80

C. 0.96

D. 1.26

READING A poll of 385 randomly selected adults showed that they read for recreation an average of 39.3 minutes per week with a standard deviation of 9.6 minutes. Use a 95% confidence interval to find the maximum error of estimate for the number of minutes per week adults read for recreation.

CONFIDENCE INTERVAL FOR THE POPULATION MEAN

CONFIDENCE INTERVAL PACKAGING A poll of 156 randomly selected members of a golf course showed that they play an average of 4.6 times every summer with a standard deviation of 1.1. Determine a 95% confidence interval for the population mean.

Confidence Interval for

Population Mean

statistics ms. sellanes

Documents