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Chapter 10

Chapter 10. Introduction to Statistics. Instructor: Nguyen Ngoc Trung nguyenngoctrung.dhsp@gmail.com. Contents. Frequency distributions and measure of central tendency Measure of Variation Normal distributions Normal approximation to the binomial distribution. Section 10.1.

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Chapter 10

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  1. Chapter 10 Introduction to Statistics Instructor: Nguyen Ngoc Trung nguyenngoctrung.dhsp@gmail.com

  2. Contents • Frequency distributions and measure of central tendency • Measure of Variation • Normal distributions • Normal approximation to the binomial distribution. Finite Maths

  3. Section 10.1 Frequency Distributions and Measures of Central Tendency Instructor: Nguyen Ngoc Trung nguyenngoctrung.dhsp@gmail.com

  4. Example 1. • Consider a survey asked s random sample of 30 business executives for their recommendations as to the number of college units in management that a business major should obtain • Identify the population and the variable, group the data into intervals, and find the frequency of each interval 3 25 22 16 0 9 14 8 34 21 15 12 9 3 8 15 20 12 28 19 17 16 23 19 12 14 29 13 24 18 Finite Maths

  5. Example 1. (cont) • Solution. • The population of interest is all business executives • The variable of interest is number of college units should be obtained. • Group data is in intervals of size 5 3 25 22 16 0 9 14 8 34 21 15 12 9 3 8 15 20 12 28 19 17 16 23 19 12 14 29 13 24 18 Finite Maths

  6. 4.5 4.5 9.5 9.5 14.5 14.5 19.5 19.5 24.5 24.5 29.5 29.5 34.5 34.5 Picturing data • Histogram is a graphical display of the information • A frequency polygon is another way of graphical display which is formed by joining consecutive midpoints of the tops of the histogram bars Histogram Frequency Polygon Finite Maths

  7. Stem-and-leaf plots • This is a way of organize the data into a distribution without losing information • In this way, we separate the digits in each data point into 2 parts consisting of the first one or two digits (the stem) and the remaining digits (the leaf) • We also provide a key to show reader the unit of data that was recorded Finite Maths

  8. Example 3. 3 25 22 16 0 9 14 8 34 21 15 12 9 3 8 15 20 12 28 19 17 16 23 19 12 14 29 13 24 18 Finite Maths

  9. Example 4. • List the origin data for the following stem and leaf plot of resting pulses taken on the first day of class for 36 students Finite Maths

  10. Mean • Summation notation (sigma notation) x1 + x2 + x3+ ….+ xn = • The mean of the n numbers x1, x2, x3, … xn is Finite Maths

  11. Examples • Example 5. Find mean number: Total: 20 Finite Maths

  12. Examples (cont) • Example 6. Find the mean of lengths Finite Maths

  13. Examples (cont) • Example 7. Consider again data in example 1 Finite Maths

  14. MEAN OF A GROUPED DISTRIBUTION • The mean of a distribution where x represents the midpoints, f the frequencies, and n= • Median: The middle entry in a set of data arranged in either increasing or decreasing order. If there is an even number of entries, the median is defined to be the mean of the two center entries. • Mode: the most frequent entry. If each entry has the same frequency, there is no mode. Finite Maths

  15. Examples • Example 8. Find the median: • 0, 7, 10, 20, 22, 25, 30 • 20, 0, 20, 30, 35, 30, 20, 23, 16, 38, 25 • 25, 18, 25, 20, 16, 12, 10, 0, 35, 32 Finite Maths

  16. Examples (cont) • Example 9. Find the mode for following data: • Ages of retirement: 55, 60, 63, 63, 70, 55, 60, 65, 68, 65, 65, 71, 65, 65 • Total cholesterol score: 180, 200, 220, 260, 220, 205, 255, 240, 190, 300, 240 • Prices of new cars: $25,789, $43,231, $33,456, $19,432, $22,971, $29,876 Finite Maths

  17. Examples (cont) • Example 10. Find the mean, median and mode for following data: Finite Maths

  18. Section 10.2 Measures of Variation Instructor: Nguyen Ngoc Trung nguyenngoctrung.dhsp@gmail.com

  19. Examples • Example 1. Finite Maths

  20. MEASURES OF VARIATION • Range of a list of numbers: max – min • Deviations from the mean of a sample of n numbers x1, x2 , x3, … xn, with mean is: x1 – x2 – … xn – Example 2. Finite Maths

  21. MEASURES OF VARIATION SAMPLE VARIANCE • The variance of a sample of n numbers x1, x2 , x3, … xn, with mean , is s2 = Population variance: s2 = Finite Maths

  22. MEASURES OF VARIATION SAMPLE STANDARD DEVIATION • The standard deviation of n numbers x1, x2 , x3, …, xn, with mean , is Example 3. Finite Maths

  23. MEASURES OF VARIATION STANDARD DEVIATION FOR A GROUP DISTRIBUTION • The standard deviation for a distribution with mean , where x is an interval midpoint with frequency f, and n = • Example 4. Finite Maths

  24. Section 10.3 Normal Distributions Instructor: Nguyen Ngoc Trung nguyenngoctrung.dhsp@gmail.com

  25. Example Continuous distribution Finite Maths

  26. Frequency 55 60 65 70 75 Heights of college women (in inches) Frequency 70,000 Income in the United States (in dollars) Example (cont) Finite Maths

  27. Frequency 55 60 65 70 75 Heights of college women (in inches) Normal Distributions Bell – Shaped Curves Finite Maths

  28. Normal Distributions (cont) • The peak occurs directly above the mean • The curve is symetric about the vertical line through the mean. • The curve never touches the x-axis • The area under the curve is 1 The mean: m Standard deviation: s Standard normal curve: m=0, s = 1 Finite Maths

  29. Frequency 55 60 65 70 75 Heights of college women (in inches) Normal Distributions (cont) • The area of the shaded region under the normal curve from a to b is the probability that an observed data value will be between a and b Probability that a women is from 62 to 66 inches Finite Maths

  30. Standard Normal Curve • Example 1. Consider standard normal curve ( = 0,  = 1). • Find the area between z = 0 and z = 1. • Find the area between z = - 2.43 and z = 0. • Example 2. Consider standard normal curve ( = 0,  = 1). • Find the area between .88 standard deviations below the mean and 2.35 standard deviations above the mean. • Find the area between .58 standard deviations above the mean and 2.35 standard deviations above the mean. • Find the area to the right of 2.09 standard deviations above the mean Finite Maths

  31. Examples (cont) • Example 3.  = 60,  = 5 • Find z-score for x = 65 • Find z-score for x = 52.5 If a normal distribution has mean  and standard deviation , then the z-score for the number x is: z = Finite Maths

  32. Area under a normal curve The area under normal curve between x=a and x=b is the same as the area under the standard normal curve between the z-score for a and the z-score for b. • Example 4. Average: 1200 miles/month, standard deviation: 150 miles.  = 1200,  = 150 • Find the probability that a sales person drives between 1200 and 1600 miles per month • Find the probability that a sales person drives between 1000 and 1500 miles per month Finite Maths

  33. Example 5. • National Health and Nutrional Examination Survey (2001 – 2002) •  = 187 mg/dL,  = 43 mg/dL • Find the probability that a American randomly chosen has cholesterol level higher than 250 • IF 200 Americans are rondomy chosen, how many can we expect to have total cholesterol higher than 250? Finite Maths

  34. Section 10.4 Normal Approximation to the Binomial Distribution Instructor: Nguyen Ngoc Trung nguyenngoctrung.dhsp@gmail.com

  35. BINOMIAL DISTRIBUTIONS • The same experiment is repeated several times. • There are only two possible outcomes, success or failure. • The repeated trials are independent so that the probability of each outcome remains the same for each trial Finite Maths

  36. Standard deviation • The expected number of successes in n binomial trials is np, where p is the probability of success in a single trial.  = E(x) = np • Variance and standard deviation 2 = np(1 – p) and  = Finite Maths

  37. Binomial propability. • Suppose an experiment is a series of n independent trials, where the probability of a success in a single trial is always p. Let x be the number of successes in the n trials. Then the probability that exactly x success will occur in n trials is given by: Finite Maths

  38. Example 1. • Tossing a fair coin 15 times. Finding the propability distribution for number of head showed. Finite Maths

  39. Example 1. (cont) • = np = 7.5,  = 1.94 Finite Maths

  40. Example 1 (cont) Approximately • Find probability that 9 heads is showed • P(9) = 9C15.(0.5)9(0.5)6 = 0.15274 • Area of red bar??? • x = 8.5  z = .52  Area: 0.1985 (from Table 2.) • x = 9.5  z = 1.03  Area: 0.3485 (from Table 2.) • P(9) = 0.3485 – 0.1985 = 0.15 • = np = 7.5 •  = 1.94 Finite Maths

  41. Example 2. • The Gallup Organization (8/2004) • Question: “Work outside or taking care of house?” • In general, 54% answer “Work outside” • Ask 100 persons: • Find the probability that at least 65 persons answer “Work Outside” • Find the probability that between 55 and 64 persons answer “Work Outside” Finite Maths

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