1. Math 145 June 19, 2007

2. Outline 1. Recap 2. Sampling Designs 3. Graphical methods

3. Statistics is the science of collecting, analyzing, interpreting, and presenting data. is the science of collecting, analyzing, interpreting, and presenting data. Two kinds of Statistics: 1. Descriptive Statistics. 2. Inferential Statistics. 1. Population 2. Sample  representative sample

4. Methods of Acquiring Information 1. Census 2. Sampling 3. Experimentation 1. Observational Study – researchers observe characteristics and take measurements, as in sample survey. (Association) 2. Designed Experiment – researchers impose treatments and controls and then observe characteristics and take measurements. (Cause and Effect) 3. 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 = 33.33  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  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 51159 37212 418913 282413 1610 569

14. Fifth-grade IQ Scores 145101123106117102 1391429412490108 126134100115103110 122124136133114128 125112109116139114 130109131102101112 96134117127122114 110113110117105102 118811271099782 11811312413789101

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. 19-23) #1, 2, 5, 11, 12, 16, 24, 28 Chapter 2 : (pp. 36-40) #5, 6, 10. (pp. 50-53) #25, 30, 32.

18. Thank you!