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Section 9.1 Samples and Central Tendency

Section 9.1 Samples and Central Tendency. Objectives: 1. To distinguish population parameters from sample statistics. 2. To find measures of central tendency.

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Section 9.1 Samples and Central Tendency

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  1. Section 9.1 Samples and Central Tendency

  2. Objectives: 1. To distinguish population parameters from sample statistics. 2. To find measures of central tendency.

  3. Researchers use a small group of people, called asample, to approximate information about a larger group, called thepopulation.

  4. Definition Apopulationis the complete collection of elements (scores, people, measurements) to be studied. A sampleis a subset of a population.

  5. The larger group of interest is called thepopulation, and the selected subset is asample.

  6. If every member of the population has an equal chance of being included in the sample, then it is arandom sample.

  7. Astratified random sampleis a random sample within certain groups of a population.

  8. Astratified random sampleis a sample obtained by separating the population elements into nonoverlapping groups, calledstrata, and then selecting a simple random sample within each stratum.

  9. Definition Parameter The actual value of a quantity for the population, usually known only to God. *Usually represented with Greek letters

  10. Definition Statistic An estimate of the population parameter based on a sample. *Usually represented with English letters

  11. Greek letters usually represent parameters, while English letters represent statistics. For example, μ represents the population mean while x (x bar) represents the sample mean. The bar over the x distinguishes the sample mean from an individual value of the variable x.

  12. The number(n)of values is the sample size, the number in the population isN, and the data values are numbered with subscripts x1, x2, x3, . . . xn. These values can be referred to as xi for i = 1, 2, 3, . . . n.

  13. The letteriis a counter variable orindex. The symbolis used to represent the addition of data values because it is the capital Greek lettersigmathat corresponds to our letter s as an abbreviation forsum.

  14. The starting value of the index appears below the  and the ending value above the . The summation in the following definition is read“summation of x sub i as i goes from 1 to n.”

  15. Mean where n is the sample size n  x i x = i= 1 n Definition

  16. The mean is one of several statistics that are calledmeasures of central tendency.

  17. Definition Median The middle value (or average of the middle two values) after listing the data in order of size.

  18. Definition Mode The most frequent value(s) (if any).

  19. Definition Midrange The average of the highest and lowest value.

  20. Practice: The following test scores were recorded: 85, 93, 96, 74, 65, 88, 87, 88 x1, x2, x3, x4, x5, x6, x7, x8

  21. Practice: The following test scores were recorded: 85, 93, 96, 74, 65, 88, 87, 88 Find the sample size. n = 8

  22. 8  x i i = 1 Practice: The following test scores were recorded: 85, 93, 96, 74, 65, 88, 87, 88 Find the mean. = 85 + 93 + 96 + 74 + 65 + 88 + 87 + 88 = 676

  23. n  xi i =1 676 8 x = = n Practice: The following test scores were recorded: 85, 93, 96, 74, 65, 88, 87, 88 Find the mean. = 84.5

  24. 87 + 88 87 . 5 = 2 Practice: The following test scores were recorded: 85, 93, 96, 74, 65, 88, 87, 88 Find the median. 65, 74, 85, 87, 88, 88, 93, 96

  25. Practice: The following test scores were recorded: 85, 93, 96, 74, 65, 88, 87, 88 Find the mode. 65, 74, 85, 87, 88, 88, 93, 96

  26. 65 + 96 80 . 5 = 2 Practice: The following test scores were recorded: 85, 93, 96, 74, 65, 88, 87, 88 Find the midrange. 65, 74, 85, 87, 88, 88, 93, 96

  27. n n n (xi + yi) = xi + yi i =1 i =1 i =1 n k = nk for any k  R i =1 Summation Rules:

  28. Practice: Give the four measures of central tendency for the following scores: 76, 86, 78, 83, 90, 88, 94, 90. sample size n = 8

  29. 8  xi = 76 + 86 + 78 + 83 + 90 + i =1 Practice: Give the four measures of central tendency for the following scores: 76, 86, 78, 83, 90, 88, 94, 90. mean 88 + 94 + 90 = 685

  30. n  xi 685 i =1 x = = = 85.625 ≈ 85.6 n 8 Practice: Give the four measures of central tendency for the following scores: 76, 86, 78, 83, 90, 88, 94, 90. mean

  31. 86 + 88 = 87 2 Practice: Give the four measures of central tendency for the following scores: 76, 86, 78, 83, 90, 88, 94, 90. median 76, 78, 83, 86, 88, 90, 90, 94

  32. Practice: Give the four measures of central tendency for the following scores: 76, 86, 78, 83, 90, 88, 94, 90. mode 76, 78, 83, 86, 88, 90, 90, 94

  33. 76 + 94 = 85 2 Practice: Give the four measures of central tendency for the following scores: 76, 86, 78, 83, 90, 88, 94, 90. midrange 76, 78, 83, 86, 88, 90, 90, 94

  34. Homework: pp. 451-453

  35. ■ Cumulative Review 27. Solve ABC if b = 29, c = 21, and C = 42°.

  36. ■ Cumulative Review 28. Find the central angle of a circle of radius 10.2 m if the angle intercepts an arc of length 47.9 m. Give the answer in both radians and degrees.

  37. ■ Cumulative Review 29. Which elementary row operation changes the sign of the determinant?

  38. ■ Cumulative Review 30. Find 5<2, 7> – 2<1, -6>

  39. ■ Cumulative Review 31. Write the equation of an ellipse with center (5, -2), horizontal major axis of 10, and eccentricity of 0.75.

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