Thinking Mathematically by Robert Blitzer

1
Statistics, p. 630-680 (Chapter 12)

• OBJECTIVES
• Determine the measure of central tendency
• Determine the measure of dispersion
• Recognize normal distributions

2
Statistics p. 631

• Statistics is the science of data. This involves
  collecting, classifying, summarizing, organizing,
  analyzing, and interpreting numerical information.

3
Types of Statistics p. 631

• Descriptive Statistics utilizes numerical and
  graphical methods to look for patterns in a data
  set, to summarize the information revealed in a
  data set, and to present that information in a
  convenient form.
• Inferential Statistics utilizes sample data to
  make estimates, decisions, predictions, or other
  generalizations about a larger set of data.

4
Random Samples p. 632

• A random sample is a sample obtained in such a
  way that every element in the population has an
  equal chance of being selected for the sample.
• Select a random sample from a large city to
  determine how the citys citizens feel about
  Wal-Mart.
• Randomly select neighborhoods of the city and
  then randomly survey people within the selected
  neighborhoods.
• Selecting people in the Wal-Mart parking lot does
  not give everyone an equal chance of being
  selected.

5
The MEAN p. 645

• The mean of a set of quantitative data is the sum
  of the measurements divided by the number of
  measurements contained in the data set. (the
  average)
• The mean is the sum of the data items divided by
  the number of items.

6
p. 738 12

• Out of 10 possible points, a class of 16 students
  made the following test scores
• 4, 4, 5, 6, 6, 6, 7, 8, 8, 8, 8, 9, 9, 9, 10, 10
• Mean 445666788889991010
• 16
• 117/16 7.3125

7
The MEDIAN p. 647

• To find the median of a group of data items,
• 1. Arrange the data items in order, from
  smallest to largest.
• 2. If the number of data items is odd, the
  median is the item in the middle of the list.
• 3. If the number of data items is even, the
  median is the mean of the two middle data items.

8
Position of the Median

• If n data items are arranged in order, from
  smallest to largest, the median is the value in
  the
• (n1)
• 2
• position.

9
p. 738 12

• Out of 10 possible points, a class of 16 students
  made the following test scores
• 4, 4, 5, 6, 6, 6, 7, 8, 8, 8, 8, 9, 9, 9, 10, 10
• Mean 445666788889991010
• 16
• 117/16 7.3125
• b) Median (88)/2 16/2 8

10
p. 738 16

• The carry-on luggage weights for a random sample
  of 10 passengers during a domestic flight were

30, 30, 32, 35, 37, 40, 40, 40, 42, 44
Mean 30 30 32 35 37 40 40 40 42
44 10
370/10 37 Median (40 37)/2 77/2
38.5 Mode 40
11
The MODE p. 652

• The mode is the data value that occurs most often
  in a data set.
• For example, the mode for the following set of
  numbers 7, 2, 4, 7, 8, 10 is 7 because the
  number 7 occurs more often than any other.

12
The Midrange p. 653

• The midrange is found by adding the lowest and
  highest data values and dividing the sum by 2.
• Midrange lowest data valuehighest data value
• 2

13
The Range p. 658

• The range, the difference between the highest and
  lowest data values in a data set, indicates the
  total spread of the data.
• Range highest data value - lowest data value
• The ten most expensive markets for new homes in
  U.S. has mean home cost in thousands of dollars
• 332, 256, 251, 235, 223, 215, 215, 213,
  210, 210.
• The range in costs is 332 - 210 122

14
The 68-95-99.7 Rule for the Normal Distribution
p. 666

• Approximately 68 of the measurements will fall
  within 1 standard deviation of the mean.
• Approximately 95 of the measurements will fall
  within 2 standard deviations of the mean.
• Approximately 99.7 (essentially all) the
  measurements will fall within 3 standard
  deviations of the mean.

15
The 68-95-99.7 Rule for the Normal Distribution
99.7
95
68
- 3
- 2
-1
2
3
1
16
The Normal Distribution

• Data
• 4, 4, 5, 6, 6, 6, 7, 8, 8, 8, 8, 9, 9, 9, 10, 10
• 1 3 5 7 9 11 13

8 8 8 8
6 6 6
9 9 9
4 4
10 10
7
17
EXAMPLE

• The carry-on luggage weights for a random sample
  of 10 passengers during a domestic flight were

30, 30, 32, 35, 37, 40, 40, 40, 42, 44
Mean 30 30 32 35 37 40 40 40 42
44 10
370/10 37 Median (40 37)/2 77/2
38.5 Mode 40
18
Standard Deviation
x
30 30 32 35 37 40 40 40 42 44
30 37 -7 30 37 -7 32 37 -5 35 37
-2 37 37 0 40 37 3 40 37 3 40 37
3 42 37 5 44 37 7
49 49 25 4 0 9 9 9 25 49
19

• 49 49 25 4 0 9 9 9 25 49
• 9
• 228/9

Standard deviation is 5 Mean is 37.
20
34
34
13.5
13.5
2.35
2.35

• 37 42 47 52

22 What is the probability that a person selected
at random will have luggage between 37 and 43
pounds?
32
27
21

• z43 43 37
• 5
• z43 6/5 1.20
• Look at the z table on page 672.
• Look at the 1.2 row 88.49.
• Therefore, 88.49 of data lies below 43 and
  38.49 of data is between 37 and 43.
• The probability of selecting a person at random
  whose luggage weighs between 37 and 43 pounds is
  0.3849

z-score data item mean
standard deviation
22
Computing z-Scores p. 669

• A z-score describes how many standard deviations
  a data item in a normal distribution lies above
  or below the mean. The z-score can be obtained
  using
• z-score data item mean
  standard deviation
• Data items above the mean have positive z -
  scores. Data items below the mean have negative
  z-scores. The z-score for the mean is 0.

23
Percentiles p. 671

• If n of the items in a distribution are less
  than a particular data item, we say that the data
  item is in the nth percentile of the
  distribution.
• For example, if a student scored in the 93rd
  percentile on the SAT, the student did better
  than about 93 of all those who took the exam.

24
Finding the Percentage of Data Items between Two
Given Items in a Normal Distribution p. 675

• Convert each given data item to a z-score
• z data item - mean
  standard deviation
• Use the table to find the percentile
  corresponding to each z-score in step 1.
• Subtract the lesser percentile from the greater
  percentile and attach a sign.

25
Margin of Error in a Survey p. 677

• If a statistic is obtained from a random sample
  of size n, there is a 95 probability that it
  lies within 1/?n of the true population
  statistic, where 1/ ?n is called the margin of
  error.

26
HOMEWORK

• Office hours M-F 900-1015
• or by appointment
• Math Tutoring Walker Bldg M-Th 430-630
• Read p. 630-680
• P. 641- 32 1-32 alternate odd
• P. 655-657 1-59 alternate odd
• P. 663-664 1-6
• P. 680-682 1-106 alternate odd