Math 145 September 5, 2007

Outline Recap Sampling Designs Graphical methods

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

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

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

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

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, …}

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.

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).

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)

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.

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.

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

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

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

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.

Homework Exercises: Chapter 1 : (pp. 19-23) #1, 2, 5, 7, 9, 10, 11, 12, 13, 16, 24, 28. Chapter 2 : (pp. 36-40) #5, 6, 10. (pp. 50-53) #25, 30.

Thank you!