Math 145

1. Math 145 June 19, 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 = 33.33  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

14. Fifth-grade IQ Scores

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!