1. Math 145
September 5, 2007

2. Outline
Recap
Sampling Designs
Graphical methods

3. Statistics
is the science of collecting, analyzing, interpreting, and presenting data.
Two kinds of Statistics:
Descriptive Statistics.
Inferential Statistics.
Population
Sample
 representative sample

4. Methods of Acquiring Information
Census
Sampling
Experimentation
Observational Study – researchers observe characteristics and take measurements, as in sample survey. (Association)
Designed Experiment – researchers impose treatments and controls and then observe characteristics and take measurements. (Cause and Effect)
Consider: #1.27 (p.22), #1.29

5. Sampling Designs Simple Random Sampling. Systematic Random Sampling.
Cluster Sampling.
Stratified Random Sampling with Proportional Allocation.

6. Simple Random Sampling
A sampling procedure for which each possible sample of a given size has the same chance of being selected.
Population of 5 objects: {A, B, C, D, E}
Take a sample of size 2.
Possible samples: {(A,B), (A,C), (A,D), (A,E), (B,C), (B,D), (B,E), (C,D), (C,E), (D,E)}
Random number generators

7. Systematic Random Sampling
Step 1. Divide the population size by the sample size and round the result down to the nearest number, m.
Step 2. Use a random-number generator to obtain a number k, between 1 and m.
Step 3. Select for the sample those numbers of the population that are numbered k, k+m, k+2m, …
Expected number of customers = 1000
Sample size of 30  m = 1000/30 =  33
Suppose k = 5. Then select {5, 5+33, 5+66, …}

8. Cluster Sampling Step 1. Divide the population into groups (clusters).
Step 2. Obtain a simple random sample of clusters.
Step 3. Use all the members of the clusters in step 2 as the sample.

9. Stratified Random Sampling with Proportional Allocation
Step 1. Divide the population into subpopulations (strata).
Step 2. From each stratum, obtain a simple random sample of size proportional to the size of the stratum.
Step 3. Use all the members obtained in Step 2 as the sample.
Population of 9,000 with 60% females and 40% males
Sample of size 80.
 48 females (from 5,400) and 32 males (from 3,600).

10. Descriptive Statistics
Individuals – are the objects described by a set of data. Individuals may be people, but they may also be animals or things.
Variable – a characteristic of an individual. A variable can take different values for different individuals.
Categorical (Qualitative) variable – places an individual into one of several groups or categories. {Gender, Blood Type}
Quantitative variable – takes numerical values for which arithmetic operations such as adding and averaging make sense. {Height, Income, Time, etc.}
Consider: #1.18 (p. 20), #1.21 (p.21)

11. Quantitative Variables
Discrete Variables – There is a gap between possible values.
Counts (no. of days, no. of people, etc.)
Age in years
Continuous Variables – Variables that can take on values in an interval.
Survival time, amount of rain in a month, distance, etc.

12. Graphical Procedures Categorical (Qualitative) Data Quantitative Data
Bar Chart
Pie Chart
Quantitative Data
Histogram
Stem-and-leaf plot (Stemplot)
Dotplot
These plots describe the Distribution of a variable.

13. Length of Stay 5 1 15 9 3 7 2 12 4 18 13 28 24 6 10

14. Fifth-grade IQ Scores
145
101
123
106
117
102
139
142 94
124 90
108
126
134
100
115
103
110
122
136
133
114
128
125
112
109
116
130
131 96
127
113
105
118 81 97 82
137 89

15. Distribution
- The distribution of a variable tells us what values it takes and how often it takes these values
Categorical Data
Table or Bar Chart
Quantitative Data
Frequency Table
Histogram
Stem-and-leaf plot

16. Describing a distribution
Skewness
Symmetric
Skewed to the right (positively skewed)
Skewed to the left (negatively skewed)
Center/Spread
No of peaks (modes)
Unimodal, Bimodal, Multimodal.
Outliers
Extreme values.

17. Homework Exercises: Chapter 1 :
(pp ) #1, 2, 5, 7, 9, 10, 11, 12, 13, , 24, 28.
Chapter 2 :
(pp ) #5, 6, 10.
(pp ) #25, 30.

18. Thank you!