1. Section 9.1 Samples and Central Tendency Section 9.1 Samples and Central Tendency

2. Objectives: 1.To distinguish population parameters from sample statistics. 2.To find measures of central tendency. 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 a sample, to approximate information about a larger group, called the population.

4. A population is the complete collection of elements (scores, people, measurements) to be studied. A sample is a subset of a population. DefinitionDefinition

5. The larger group of interest is called the population, and the selected subset is a sample.

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

7. A stratified random sample is a random sample within certain groups of a population.

8. A stratified random sample is a sample obtained by separating the population elements into nonoverlapping groups, called strata, and then selecting a simple random sample within each stratum.

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

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

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 is N, and the data values are numbered with subscripts x 1, x 2, x 3,... x n. These values can be referred to as x i for i = 1, 2, 3,... n.

13. The letter i is a counter variable or index. The symbol  is used to represent the addition of data values because it is the capital Greek letter sigma that corresponds to our letter s as an abbreviation for sum.

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 n x x x x n n i = 1 i i   = = DefinitionDefinition

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

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

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

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

20. Practice: The following test scores were recorded: 85, 93, 96, 74, 65, 88, 87, 88 Practice: The following test scores were recorded: 85, 93, 96, 74, 65, 88, 87, 88 x1,x2,x3,x4,x5,x6,x7,x8x1,x2,x3,x4,x5,x6,x7,x8 x1,x2,x3,x4,x5,x6,x7,x8x1,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. Practice: The following test scores were recorded: 85, 93, 96, 74, 65, 88, 87, 88 Find the sample size. n =8

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

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

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

26. 5 5.. 80 = = 2 2 96 65 + + Practice: The following test scores were recorded: 85, 93, 96, 74, 65, 88, 87, 88 Find the midrange. 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 i =1 n n n n  (x i + y i ) =  x i +  y i n n i =1  k = nk for any k  R 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 sample size n =8

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

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

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

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

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

34. Homework: pp. 451-453 Homework: pp. 451-453

35. ■ Cumulative Review 27.Solve  ABC if b = 29, c = 21, and  C = 42°. ■ 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. ■ 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? ■ Cumulative Review 29.Which elementary row operation changes the sign of the determinant?

38. ■ Cumulative Review 30.Find 5 – 2 ■ Cumulative Review 30.Find 5 – 2

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