# VCU STAT 210 - Lecture10 (46 pages)

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## Lecture10

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- Pages:
- 46
- School:
- Virginia Commonwealth University
- Course:
- Stat 210 - Basic Practice of Statistics

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STAT 210 Lecture 10 Measures of Center September 18 2017 Test 2 Monday September 25 Sections III IV pages 47 95 Combination of multiple choice questions and short answer questions and problems Bring a calculator and writing instrument Practice Problems Pages 94 through 97 Relevant problems IV 1 IV 2 IV 3 IV 5 IV 6 a and b and IV 11 c Recommended problems IV 1 IV 2 and IV 6 a and b Additional Reading and Examples Pages 90 through 93 Top Hat Motivating Example A statistics course at a large university provides free to students statistics review sessions that students can use to answer questions with help solving problems and with help studying for tests The course instructor is interested in the number of students who attend each hour of review session and selects a sample of 15 review session hours spread out over a month s time As described what is the population of interest for this example Motivating Example A statistics course at a large university provides free to students statistics review sessions that students can use to answer questions with help solving problems and with help studying for tests The course instructor is interested in the number of students who attend each hour of review session and selects a sample of 15 review session hours spread out over a month s time As described what is the population of interest for this example Answer one could say all students currently taking statistics but the data in the sample is collected on the review sessions so the population of interest would be all review sessions held for this statistics course Motivating Example A statistics course at a large university provides free to students statistics review sessions that students can use to answer questions with help solving problems and with help studying for tests The course instructor is interested in the number of students who attend each hour of review session and selects a sample of 15 review session hours spread out over a month s time As described what is the population of interest for this example What type of characteristic is the number of students who attend each hour of review session Motivating Example A statistics course at a large university provides free to students statistics review sessions that students can use to answer questions with help solving problems and with help studying for tests The course instructor is interested in the number of students who attend each hour of review session and selects a sample of 15 review session hours spread out over a month s time As described what is the population of interest for this example What type of characteristic is the number of students who attend each hour of review session Answer number of students attending a review session is countable and hence is a discrete quantitative variable Motivating Example A statistics course at a large university provides free to students statistics review sessions that students can use to answer questions with help solving problems and with help studying for tests The course instructor is interested in the number of students who attend each hour of review session and selects a sample of 15 review session hours spread out over a month s time The number of students who attended these 15 review session hours is as follows This data will be used throughout the rest of this chapter 6 1 8 3 1 5 11 7 4 28 12 9 2 10 13 Goal Determine the value of a population parameter designated using Greek letters such as m s and p 1 Central location parameter locate the center of the distribution 2 Dispersion parameter measure the spread or variability around the center Statistics All subjects of the population are rarely known Hence the population parameter of interest can rarely be determined and must be estimated using a sample statistic The statistics are denoted using regular letters such as X s and p Such an estimation is a type of statistical inference Notation n number of observations in the sample x1 value of first observation x2 value of second observation xn value of nth last observation Measures of Central Location 1 Mean Average The mean is the most often used measure of central location including being used in many of the inference procedures we will discuss later in the course Measures of Central Location 1 Mean Average The population mean is denoted by the Greek letter m read mu and is the sum of all observations divided by how many individuals that there are in the population This is usually an unknown parameter Measures of Central Location The population mean is estimated by the sample mean denoted by X read X bar X S x x1 x2 x3 xn n n The sample mean X is a statistic The symbol S implies to sum or add what follows Example 13 In this example are we calculating the population mean m or the sample mean X Example 13 X S x 128 150 183 222 113 154 201 150 n 8 1301 8 162 625 pounds Measures of Central Location The mean is highly influenced by outliers extreme values Example 14 Example 14 X S x 128 150 183 222 113 154 201 150 391 n 9 1692 9 188 pounds Example 14 X S x 128 150 183 222 113 154 201 150 391 n 9 1692 9 188 pounds Hence the outlier value of 391 pounds has increased the sample mean from 162 625 to 188 pounds a 25 375 pound increase Top Hat Measures of Central Location 2 Median The median is more resistant to outliers than the mean and is the central value with half of the observations less than it and half of the observations greater than it Measures of Central Location 2 Median The population median is usually denoted by the Greek letter h read eta and is estimated by the sample median denoted by M Median 1 Order the data from smallest to largest Median 1 Order the data from smallest to largest 2 Calculate the median location n 1 2 Median 1 Order the data from smallest to largest 2 Calculate the median location n 1 2 3 Calculate the median Median If n is odd then n 1 2 is a whole number and the median is the n 1 2 th ordered observation Example n 19 n 1 2 19 1 2 20 2 10 Median is the 10th ordered observation Median If n is even then n 1 2 is a fraction of a whole number and the median is the average of the n 1 2 5 and the n 1 2 5 ordered observations Example n 22 n 1 2 22 1 2 23 2 11 5 Median is the average of the 11th and 12th ordered observations Example 15 1 Ordered observations 113 128 150 150 154 183 201 222 Example 15 1 Ordered observations 113 128 150 150 154 183 201 222 2 n 8 Median location n 1 2 8 1 2 9 2 4 5 Example 15 1 Ordered observations 113 128 150 150 …

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