1. Statistics Test # 2 Review
Chapter 1
Statistics Test # 2 Review
Algebra II
Larson/Farber 4th ed.
Larson/Farber 4th ed

2. What is Data?
Data Consist of information coming from observations, counts, measurements, or responses.
“People who eat three daily servings of whole grains have been shown to reduce their risk of…stroke by 37%.” (Source: Whole Grains Council)
“Seventy percent of the 1500 U.S. spinal cord injuries to minors result from vehicle accidents, and 68 percent were not wearing a seatbelt.” (Source: UPI)
Larson/Farber 4th ed.

3. What is Statistics?
Statistics The science of collecting, organizing, analyzing, and interpreting data in order to make decisions.
Larson/Farber 4th ed.

4. Data Sets
Population
The collection of all outcomes, responses, measurements, or counts that are of interest.
Sample
A subset of the population.
Larson/Farber 4th ed.

5. Parameter and Statistic
A number that describes a population characteristic.
Average age of all people in the United States
Statistic
A number that describes a sample characteristic.
Average age of people from a sample of three states
Larson/Farber 4th ed.

6. Branches of Statistics
Descriptive Statistics Involves organizing, summarizing, and displaying data.
e.g. Tables, charts, averages
Inferential Statistics Involves using sample data to draw conclusions about a population.
Larson/Farber 4th ed.

7. Types of Data
Qualitative Data Consists of attributes, labels, or nonnumerical entries.
Major
Place of birth
Eye color
Larson/Farber 4th ed.

8. Types of Data Quantitative data Numerical measurements or counts. Age
Weight of a letter
Temperature
Larson/Farber 4th ed.

9. Data Collection Observational study
A researcher observes and measures characteristics of interest of part of a population.
Researchers observed and recorded the mouthing behavior on nonfood objects of children up to three years old. (Source: Pediatric Magazine)
Larson/Farber 4th ed.

10. Data Collection Experiment
A treatment is applied to part of a population and responses are observed.
An experiment was performed in which diabetics took cinnamon extract daily while a control group took none. After 40 days, the diabetics who had the cinnamon reduced their risk of heart disease while the control group experienced no change. (Source: Diabetes Care)
Larson/Farber 4th ed.

11. Data Collection Simulation
Uses a mathematical or physical model to reproduce the conditions of a situation or process.
Often involves the use of computers.
Automobile manufacturers use simulations with dummies to study the effects of crashes on humans.
Larson/Farber 4th ed.

12. Data Collection Survey
An investigation of one or more characteristics of a population.
Commonly done by interview, mail, or telephone.
A survey is conducted on a sample of female physicians to determine whether the primary reason for their career choice is financial stability.
Larson/Farber 4th ed.

13. Key Elements of Experimental Design: Control
Placebo effect
A subject reacts favorably to a placebo when in fact he or she has been given no medical treatment at all.
Blinding is a technique where the subject does not know whether he or she is receiving a treatment or a placebo.
Double-blind experiment neither the subject nor the experimenter knows if the subject is receiving a treatment or a placebo.
Larson/Farber 4th ed.

14. Sampling Techniques Simple Random Sample
Every possible sample of the same size has the same chance of being selected. x
Larson/Farber 4th ed.

15. Simple Random Sample
Random numbers can be generated by a random number table, a software program or a calculator.
Assign a number to each member of the population.
Members of the population that correspond to these numbers become members of the sample.
Larson/Farber 4th ed.

16. Other Sampling Techniques
Stratified Sample
Divide a population into groups (strata) and select a random sample from each group.
To collect a stratified sample of the number of people who live in West Ridge County households, you could divide the households into socioeconomic levels and then randomly select households from each level.
Larson/Farber 4th ed.

17. Other Sampling Techniques
Cluster Sample
Divide the population into groups (clusters) and select all of the members in one or more, but not all, of the clusters.
In the West Ridge County example you could divide the households into clusters according to zip codes, then select all the households in one or more, but not all, zip codes.
Larson/Farber 4th ed.

18. Other Sampling Techniques
Systematic Sample
Choose a starting value at random. Then choose every kth member of the population.
In the West Ridge County example you could assign a different number to each household, randomly choose a starting number, then select every 100th household.
Larson/Farber 4th ed.

19. Example: Identifying Sampling Techniques
You are doing a study to determine the opinion of students at your school regarding stem cell research. Identify the sampling technique used.
You divide the student population with respect to majors and randomly select and question some students in each major.
Solution:
Stratified sampling (the students are divided into strata (majors) and a sample is selected from each major)
Larson/Farber 4th ed.

20. Example: Identifying Sampling Techniques
You assign each student a number and generate random numbers. You then question each student whose number is randomly selected.
Solution:
Simple random sample (each sample of the same size has an equal chance of being selected and each student has an equal chance of being selected.)
Larson/Farber 4th ed.

21. Graphing Quantitative Data Sets
Stem-and-leaf plot
Each number is separated into a stem and a leaf.
Similar to a histogram.
Still contains original data values. 26 2 3 4 5
Data: 21, 25, 25, 26, 27, 28, , 36, 36, 45
Larson/Farber 4th ed.

22. Solution: Constructing a Stem-and-Leaf Plot
Include a key to identify the values of the data.
From the display, you can conclude that more than 50% of the cellular phone users sent between 110 and 130 text messages.
Larson/Farber 4th ed.

23. Measures of Central Tendency
Measure of central tendency
A value that represents a typical, or central, entry of a data set.
Most common measures of central tendency:
Mean
Median
Mode
Larson/Farber 4th ed.

24. Measure of Central Tendency: Mean
Mean (average)
The sum of all the data entries divided by the number of entries.
Sigma notation: Σx = add all of the data entries (x) in the data set.
Population mean:
Sample mean:
Larson/Farber 4th ed.

25. Example: Finding a Sample Mean
The prices (in dollars) for a sample of roundtrip flights from Chicago, Illinois to Cancun, Mexico are listed. What is the mean price of the flights?
Larson/Farber 4th ed.

26. Solution: Finding a Sample Mean
The sum of the flight prices is
Σx = = 3695
To find the mean price, divide the sum of the prices by the number of prices in the sample
The mean price of the flights is about $
Larson/Farber 4th ed.

27. Measure of Central Tendency: Median
The value that lies in the middle of the data when the data set is ordered.
Measures the center of an ordered data set by dividing it into two equal parts.
If the data set has an
odd number of entries: median is the middle data entry.
even number of entries: median is the mean of the two middle data entries.
Larson/Farber 4th ed.

28. Example: Finding the Median
The prices (in dollars) for a sample of roundtrip flights from Chicago, Illinois to Cancun, Mexico are listed. Find the median of the flight prices
Larson/Farber 4th ed.

29. Solution: Finding the Median
First order the data.
There are seven entries (an odd number), the median is the middle, or fourth, data entry.
The median price of the flights is $427.
Larson/Farber 4th ed.

30. Example: Finding the Median
The flight priced at $432 is no longer available. What is the median price of the remaining flights?
Larson/Farber 4th ed.

31. Solution: Finding the Median
First order the data.
There are six entries (an even number), the median is the mean of the two middle entries.
The median price of the flights is $412.
Larson/Farber 4th ed.

32. Measure of Central Tendency: Mode
The data entry that occurs with the greatest frequency.
If no entry is repeated the data set has no mode.
If two entries occur with the same greatest frequency, each entry is a mode (bimodal).
Larson/Farber 4th ed.

33. Example: Finding the Mode
The prices (in dollars) for a sample of roundtrip flights from Chicago, Illinois to Cancun, Mexico are listed. Find the mode of the flight prices
Larson/Farber 4th ed.

34. Solution: Finding the Mode
Ordering the data helps to find the mode.
The entry of 397 occurs twice, whereas the other data entries occur only once.
The mode of the flight prices is $397.
Larson/Farber 4th ed.

35. Solution: Comparing the Mean, Median, and Mode
Sometimes a graphical comparison can help you decide which measure of central tendency best represents a data set.
In this case, it appears that the median best describes the data set.
Larson/Farber 4th ed.

36. The Shape of Distributions
Symmetric Distribution
A vertical line can be drawn through the middle of a graph of the distribution and the resulting halves are approximately mirror images.
Larson/Farber 4th ed.

37. The Shape of Distributions
Uniform Distribution (rectangular)
All entries or classes in the distribution have equal or approximately equal frequencies.
Symmetric.
Larson/Farber 4th ed.

38. The Shape of Distributions
Skewed Left Distribution (negatively skewed)
The “tail” of the graph elongates more to the left.
The mean is to the left of the median.
Larson/Farber 4th ed.

39. The Shape of Distributions
Skewed Right Distribution (positively skewed)
The “tail” of the graph elongates more to the right.
The mean is to the right of the median.
Larson/Farber 4th ed.

40. Range
Range
The difference between the maximum and minimum data entries in the set.
The data must be quantitative.
Range = (Max. data entry) – (Min. data entry)
Larson/Farber 4th ed.

41. Example: Finding the Range
A corporation hired 10 graduates. The starting salaries for each graduate are shown. Find the range of the starting salaries. Starting salaries (1000s of dollars)
Larson/Farber 4th ed.

42. Solution: Finding the Range
Ordering the data helps to find the least and greatest salaries.
Range = (Max. salary) – (Min. salary)
= 47 – 37 = 10
The range of starting salaries is 10 or $10,000.
minimum
maximum
Larson/Farber 4th ed.

43. Interpreting Standard Deviation
Standard deviation is a measure of the typical amount an entry deviates from the mean.
The more the entries are spread out, the greater the standard deviation.
Larson/Farber 4th ed.

44. Interpreting Standard Deviation: Empirical Rule (68 – 95 – 99.7 Rule)
For data with a (symmetric) bell-shaped distribution, the standard deviation has the following characteristics:
About 68% of the data lie within one standard deviation of the mean.
About 95% of the data lie within two standard deviations of the mean.
About 99.7% of the data lie within three standard deviations of the mean.
Larson/Farber 4th ed.

45. Interpreting Standard Deviation: Empirical Rule (68 – 95 – 99.7 Rule)
99.7% within 3 standard deviations
2.35%
95% within 2 standard deviations
13.5%
68% within 1 standard deviation
34%
Larson/Farber 4th ed.

46. Example: Using the Empirical Rule
In a survey conducted by the National Center for Health Statistics, the sample mean height of women in the United States (ages 20-29) was 64 inches, with a sample standard deviation of 2.71 inches. Estimate the percent of the women whose heights are between 64 inches and inches.
49.85%
Larson/Farber 4th ed.

47. Quartiles
Quartiles approximately divide an ordered data set into four equal parts.
First quartile, Q1: About one quarter of the data fall on or below Q1.
Second quartile, Q2: About one half of the data fall on or below Q2 (median).
Third quartile, Q3: About three quarters of the data fall on or below Q3.
Larson/Farber 4th ed.

48. Interquartile Range Interquartile Range (IQR)
The difference between the third and first quartiles.
IQR = Q3 – Q1
Larson/Farber 4th ed.

49. Example: Finding the Interquartile Range
Find the interquartile range of the test scores. Recall Q1 = 10, Q2 = 15, and Q3 = 18
Solution:
IQR = Q3 – Q1 = 18 – 10 = 8
The test scores in the middle portion of the data set vary by at most 8 points.
Larson/Farber 4th ed.

50. Box-and-Whisker Plot Box-and-whisker plot
Exploratory data analysis tool.
Highlights important features of a data set.
Requires (five-number summary):
Minimum entry
First quartile Q1
Median Q2
Third quartile Q3
Maximum entry
Larson/Farber 4th ed.

51. Drawing a Box-and-Whisker Plot
Find the five-number summary of the data set.
Construct a horizontal scale that spans the range of the data.
Plot the five numbers above the horizontal scale.
Draw a box above the horizontal scale from Q1 to Q3 and draw a vertical line in the box at Q2.
Draw whiskers from the box to the minimum and maximum entries.
Whisker
Maximum entry
Minimum entry
Box
Median, Q2 Q3 Q1
Larson/Farber 4th ed.

52. Example: Drawing a Box-and-Whisker Plot
Draw a box-and-whisker plot that represents the 15 test scores. Recall Min = 5 Q1 = 10 Q2 = 15 Q3 = 18 Max = 37
Solution: 5 10 15 18 37
About half the scores are between 10 and 18. By looking at the length of the right whisker, you can conclude 37 is a possible outlier.
Larson/Farber 4th ed.