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Estimation of Parameters

1. CABT Statistics & Probability – Grade 11 Lecture Presentation

2. The session shall begin shortly… CABT Statistics & Probability – Grade 11 Lecture Presentation

3. Let’s Start! CABT Statistics & Probability – Grade 11 Lecture Presentation

4. Estimation of Parameters A CABT Grade 11 Statistics and Probability Lecture

5. Inferential Statistics CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Inferential statistics is concerned with drawing conclusions and/or making decisions concerning a population based only on sample data. Main functions of inferential statistics: estimate population parameters test statistical hypotheses

6. http://www.gohomeworkhelp.com/admin/photos/what-is-inferential-statistics.jpg Inferential Statistics CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters

7. CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Parameter & Statistic A parameter is a descriptive measure that describes a population . A statistic is a descriptive measure that describes a sample . Usually , parameters are denoted by lower-case GREEK letters (e.g.  or  ), while statistics use lower-case ROMAN letter (e.g. x and s).

8. Estimation of Population Parameters CABT Statistics & Probability – Grade 11 Lecture Presentation An estimator of a population parameter is a random variable that depends on sample information whose value provides an approximation to this unknown parameter. A specific value of that random variable is called an estimate . CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters

9. Properties of Good Estimators UNBIASED. The expected value or the mean of the estimates obtained from samples of a given size is equal to the parameter being estimated. CONSISTENT. As sample size increases, the value of the estimator approaches the value of the parameter being estimated. RELATIVELY EFFICIENT. Of all the statistics that can be used to estimate a parameter, the relatively efficient estimator has the smallest variance. CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Estimation of Population Parameters

10. There are two types of estimates: Point estimate : It is a specific numerical value used to approximate a population parameter. Interval estimate : It is a range of values used to approximate a population parameter . It’s also called a confidence interval . CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Estimation of Population Parameters

11. Note that: a point estimate is a single number. a confidence interval provides additional information about variability. Point Estimate Lower Confidence Limit Upper Confidence Limit Width of confidence interval CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Estimation of Population Parameters

12. Point Estimation Point estimation is the process of finding a point estimate from a random sample of a population to approximate a parameter value. The statistic value that approximates a parameter value is called a point estimate. CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters

13. The point estimate is the BEST GUESS or the BEST ESTIMATE of an unknown (fixed or random) population parameter. Point Estimation CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters

14. MEASURE Population Value (PARAMETER) Sample Statistic (POINT ESTIMATE) Mean  Standard deviation  s Proportion p Point Estimation CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters

15. Notes: Don’t expect that the point estimate is exactly equal to the population parameter. Any point estimate used should be as close as possible to the true parameter. Sampling should be done at random, using a sample size that is as large as possible. Point Estimation CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters

16. CABT Statistics & Probability – Grade 11 Lecture Presentation The following are some situations that use point estimates: CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Point Estimation 1 (estimating a mean) A sample of 50 households is used to determine the average number of children in a household in a barangay . (estimating a proportion) A sample of 50 households is used to determine the percentage of households in a barangay watching a particular teleserye .

17. CABT Statistics & Probability – Grade 11 Lecture Presentation The SAMPLE MEAN is used to estimate the population mean  . The following are the lengths of seedlings in a plant box. We want to estimate the mean length of the seedlings. CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Point Estimation 2 (Exercise 2 of the textbook)

18. Estimate the mean length using the following: average of the row averages average of the column averages using the average of the first row using the average of the last two columns Point Estimation 2 CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters (Exercise 2 of the textbook)

19. To determine the average monthly income of factory workers of a CEPZ company, ten workers were randomly sampled. Their monthly incomes (in thousand pesos) are shown in the table. Calculate the point estimate for the average monthly income. CABT Statistics & Probability – Grade 11 Lecture Presentation CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Point Estimation 3 Worker Monthly Income (thousand pesos ) Worker Monthly Income (thousand pesos ) 1 11.5 6 11.5 2 10 7 12 3 9.5 8 10.5 4 9 9 11.5 5 10 10 9

20. CABT Statistics & Probability – Grade 11 Lecture Presentation Find the point estimate of the proportion of private school teachers who are LET passers in a city given that 480 out of a sample of 600 randomly selected teachers passed the LET. CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Point Estimation 4

21. CABT Statistics & Probability – Grade 11 Lecture Presentation Find the point estimate of the proportion of the number of junior high school students who owns at least one cell phone given the following sample: CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Point Estimation 5 Grade Number of students surveyed Number of students surveyed with at least one cell phone [phone] 10 20 14

22. CABT Statistics & Probability – Grade 11 Lecture Presentation An interval estimate is a range of values used to approximate a population parameter . This estimate may or may not contain the actual value of the parameter being estimated. CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Interval Estimation An interval estimate has two components: a range or interval of values an associated level of confidence

23. CABT Statistics & Probability – Grade 11 Lecture Presentation CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Interval Estimation Why use an interval estimate instead? Using a point estimate, while unbiased, poses a degree of uncertainty . There is no way of expressing the degree of accuracy of a point estimate. An interval estimate provides more information about a population characteristic than does a point estimate.

24. CABT Statistics & Probability – Grade 11 Lecture Presentation confidence n. a feeling or belief that you can do something well or succeed at something CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters ( http://www.merriam-webster.com/dictionary/confidence ) Confidence Levels and Intervals

25. CABT Statistics & Probability – Grade 11 Lecture Presentation The confidence level c of an interval estimate is the probability that the parameter is contained in the interval estimate . CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Confidence Levels

26. CABT Statistics & Probability – Grade 11 Lecture Presentation The value of c is given by CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Confidence Levels where  represents a level of significance , which indicates the long-run percentage of confidence intervals which would include the parameter being estimated. T he value of the level of significance  is always between 0 and 1.

27. CABT Statistics & Probability – Grade 11 Lecture Presentation The significance of the level of significance CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Confidence Levels The level of significance  represents a probability of lack of confidence ; that is, the probability of NOT capturing the value of a population parameter in the interval estimate. The confidence level c = 1   , meanwhile represents the probability of confidence that the population parameter lies within the interval estimate.

28. CABT Statistics & Probability – Grade 11 Lecture Presentation The significance of the level of significance CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Confidence Levels probability that  lies in the interval estimate probability that  does NOT lie in the interval estimate

29. CABT Statistics & Probability – Grade 11 Lecture Presentation A confidence interval is a specific interval estimate of a parameter determined by using data obtained from a sample and by using the specific confidence level of the estimate. CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Confidence Intervals http://blog.minitab.com/blog/adventures-in-statistics/understanding-hypothesis-tests:-confidence-intervals-and-confidence-levels

30. CABT Statistics & Probability – Grade 11 Lecture Presentation Notes: CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters For a parameter , if P ( a <  < b ) = 1   , then the interval a <  < b is called a 100(1  )% confidence interval of  . In repeated samples of the population, the true value of the parameter  would be contained in 100(1  )% of intervals calculated this way. Confidence Intervals

31. CABT Statistics & Probability – Grade 11 Lecture Presentation CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Confidence Intervals REGION OF CONFIDENCE 100(1 - )% of all intervals contain the value of  Distribution of

32. CABT Statistics & Probability – Grade 11 Lecture Presentation Illustration: CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters A 95% confidence interval of a population mean  means that 95% of the samples from the same population will produce the same confidence intervals that contain the value of . http://www.statistica.com.au/confidence_interval.html Also, this means that so is the level of significance. Confidence Intervals

33. CABT Statistics & Probability – Grade 11 Lecture Presentation Illustration: CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters a 95% confidence interval for the mean  in a normally-distributed population Confidence Intervals

34. CABT Statistics & Probability – Grade 11 Lecture Presentation CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Confidence Intervals 6 Determine the confidence level for the following levels of significance: Level of Significance Confidence Level

35. CABT Statistics & Probability – Grade 11 Lecture Presentation CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Confidence Intervals 7 Determine the levels of significance for the following confidence levels: Confidence Level Level of Significance

36. CABT Statistics & Probability – Grade 11 Lecture Presentation CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Point Estimate Lower Confidence Limit Upper Confidence Limit Margin of Error Margin of Error Width of confidence interval Important parts of a confidence interval Confidence Intervals

37. CABT Statistics & Probability – Grade 11 Lecture Presentation General Formula for Confidence Intervals CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Confidence Intervals The general formula for all confidence intervals is given by: The value of the reliability factor depends on the desired level of confidence. Wow!

38. CABT Statistics & Probability – Grade 11 Lecture Presentation General Formula for Confidence Intervals CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Confidence Intervals https://onlinecourses.science.psu.edu/stat504/sites/onlinecourses.science.psu.edu.stat504/files/lesson01/simple_expres_CI.gif

39. CABT Statistics & Probability – Grade 11 Lecture Presentation General Formula for Confidence Intervals CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Confidence Intervals Usually, the general formula for a confidence interval is written as where is the estimate of the parameter  and E is the margin of error. In INEQUALITY FORM, the confidence interval of a parameter  is given by

40. Estimation of Parameters Population Mean  Unknown Confidence Intervals Population Proportion  Known Confidence Intervals Wow!

41. Do you have any QUESTIONs?

42. Confidence Intervals for the Population Mean for Known and Unknown Variances

43. CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Confidence Intervals for the Population Mean To construct an interval estimate for the population mean, we use a point estimate for the mean. a margin of error.

44. CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Confidence Intervals for the Population Mean The confidence interval for the population mean  is given by where E is the margin of error dependent on a given confidence level. Wow!

45. CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Confidence Intervals for the Population Mean In the confidence interval = lower confidence limit = upper confidence limit = width of the confidence interval = margin of error

46. CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Confidence Intervals for the Population Mean Population Mean Lower Confidence Limit Upper Confidence Limit Margin of Error Margin of Error Width of Confidence Interval

47. CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Confidence Intervals for the Population Mean REGION OF CONFIDENCE 100(1 - )% of all intervals contain the value of the population mean 

48. CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Confidence Intervals for the Population Mean Suppose that a sample is taken from a normally-distributed population. If the sample mean is 10, the confidence interval for the population mean  at a margin of error of 2 is or From the confidence interval, we have: Lower confidence limit: 8 Upper confidence limit: 12 Width of confidence interval: 2 E = 4 or 12 – 8 = 4 1

49. CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Confidence Intervals for the Population Mean Find the margin of error and the width of the following confidence intervals: 2 Confidence Interval Width of Confidence Interval Margin of Error

50. CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Confidence Intervals for the Population Mean In constructing an interval estimate for the population mean, we consider two cases: CASE 1 – the standard deviation  of the population is known CASE 2 – the standard deviation  of the population is not known

51. CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Confidence Intervals for the Population Mean The standard deviation  of the population is known. 1 CASE A confidence interval for a population mean  with a known standard deviation  is based on the fact that the sample means follow an approximately normal distribution . 

52. CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Confidence Intervals For the Population Mean 1 CASE The Central Limit Theorem – A Throwback: The mean and standard deviation of the distribution are, respectively, If random samples of size n are drawn from a population with replacement , then as n becomes larger, the sampling distribution of the mean approaches the normal distribution , regardless of the shape of the population distribution .

53. CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Confidence Intervals For the Population Mean 1 CASE Because of the Central Limit Theorem, we can think of the confidence level c = 1 –  as the area under the standard normal curve between two CRITICAL VALUES and .

54. CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Confidence Intervals For the Population Mean 1 CASE To get a 100(1 –  )%  confidence interval for a given level of significance  , we must include the central (1 –  ) of the probability of the normal distribution , leaving a total area of   in both tails, or /2 in each tail, of the normal distribution.

55. Confidence Intervals Intervals extend from 100(1   )% of intervals constructed contain μ ; 100(  )% do not . Sampling Distribution of the Mean CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Confidence Intervals For the Population Mean 1 CASE to

56. CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Confidence Intervals For the Population Mean 1 CASE Sampling Error The difference between the point estimate and the actual parameter value is called the SAMPLING ERROR. For the sampling distribution of sample means, the sampling error is equal to

57. CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Confidence Intervals For the Population Mean 1 CASE Margin of Error The margin of error E is the maximum error of estimate given by Wow! or where  is the level of significance,  is the population standard deviation, and n is the sample size.

58. CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Confidence Intervals For the Population Mean 1 CASE Steps in Constructing a Confidence Interval for a Population Mean if the Standard Deviation is Known STEP 1 – Calculate the sample mean. This is the point estimate for the population mean . STEP 2 – Find the z -score (critical value) that corresponds to the confidence level . STEP 3 – Calculate the margin of error E . STEP 4 – Construct the confidence interval for : Interpret the result.

59. CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Confidence Intervals For the Population Mean 1 CASE Common Confidence Levels and the Corresponding z Values Confidence Level Confidence Coefficient c = 1 –  Level of Significance  Value of z -Value 80% 0.80 0.20 0.10 1.28 90% 0.90 0.10 0.05 1.645 95% 0.95 0.05 0.025 1.96 98% 0.98 0.02 0.01 2.33 99% 0.99 0.01 0.005 2.575 99.8% 0.998 0.002 0.001 3.08 99.9% 0.999 0.001 0.0005 3.27

60. CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Confidence Intervals For the Population Mean 1 CASE A normally distributed population has standard deviation 1.5. A sample of size 36 is obtained from the population with sample mean 4. Find the margin of error for a 99% confidence interval for the population mean. 3

61. CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Confidence Intervals For the Population Mean 1 CASE 3 Solution Given: Value of : Value of z : Value of E :

62. CABT Statistics & Probability – Grade 11 Lecture Presentation Check your understanding Compute the margin of error for the estimation of the population mean  for a 90% confidence with a sample of size 400 and population standard deviation of 58. Mean and Variance of Sampling Distributions of Sample Means Estimation of Parameters

63. CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Confidence Intervals For the Population Mean 1 CASE A normally distributed population has standard deviation 2. A sample of size 25 is obtained from the population with sample mean 10. Construct a confidence interval for the mean  of the population using 90% confidence 95% confidence 4

64. CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Confidence Intervals For the Population Mean 1 CASE 4 a. Solution Given: Value of : Value of z : Value of E : Confidence limits: Confidence interval:

65. CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Confidence Intervals For the Population Mean 1 CASE 4 What does our answer mean? We are 90% confident that the true population mean  lies between 9.34 and 10.66. Wow!

66. CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Confidence Intervals For the Population Mean 1 CASE 4 b. Solution Given: Value of : Value of z : Value of E : Confidence limits: Confidence interval:

67. CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Confidence Intervals For the Population Mean 1 CASE 4 What does our answer mean? We are 95% confident that the true population mean  lies between 9.22 and 10.78. Wow!

68. CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Confidence Intervals For the Population Mean 1 CASE To determine the average amount of purchase of its customers, a convenience store samples 150 of its customers. The average purchase of the group is P 125. If the store knew that the standard deviation of all purchases is P 50, what is the 95% confidence interval for the average purchase in the store? 5

69. CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Confidence Intervals For the Population Mean 1 CASE 5 Solution Given: Value of : Value of z : Value of E :

70. CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Confidence Intervals For the Population Mean 1 CASE 5 Solution Confidence limits: Confidence interval: We are 95% confident that the actual average purchase is between P 117 and P 133. Conclusion:

71. CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Confidence Intervals For the Population Mean 1 CASE A study of 400 kindergarten pupils showed that they spend on average 5,000 hours watching TV. The standard deviation of the population is 900. Find the 95% confidence level of the mean TV time for all pupils. If a parent claimed that his children watched 4,000 hours of TV, would the claim be valid? Why? 6

72. CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Confidence Intervals For the Population Mean 1 CASE 6 a. Solution Given: Value of : Value of z : Value of E :

73. CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Confidence Intervals For the Population Mean 1 CASE 6 a. Solution (continued) Confidence limits: Confidence interval: We are 95% confident that the actual average TV time is between 4,911.8 and 5,088.2 hours. Conclusion:

74. CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Confidence Intervals For the Population Mean 1 CASE 6 b. Question: Is the claim of the parent valid? Answer: NO, the claim of the parent is NOT valid because the average is NOT in the confidence interval.

75. CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Confidence Intervals For the Population Mean 1 CASE In a nutshell: Steps in Finding the Confidence Interval for  Given: Value of : Value of z : Value of E : Confidence limits: Confidence interval:

76. CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Confidence Intervals For the Population Mean 1 CASE In a nutshell: Steps in Finding the Confidence Interval for  Conclusion : We are _____% confident that the true mean / average _____ is between _____ and _____. Okay!

77. CABT Statistics & Probability – Grade 11 Lecture Presentation Check your understanding Solve Exercise 7(a) and 8(a) on page 166 of your textbook. Mean and Variance of Sampling Distributions of Sample Means Estimation of Parameters

78. CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Confidence Intervals For the Population Mean 1 CASE Sample Size Determination The MINIMUM sample size n needed to estimate the population mean  is where  is the level of significance,  is the population standard deviation and E is the margin of error. Okay!

79. CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Confidence Intervals For the Population Mean 1 CASE Sample Size Determination Since the confidence interval widens as the confidence level increases, the precision of the interval estimate decreases. One way to increase the precision without changing c is to increase the sample size. The larger the sample size, the better. Why compute the sample size?

80. CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Confidence Intervals For the Population Mean 1 CASE Determine the minimum sample size needed to estimate the population mean  with 95% confidence using a margin of error of 4. It is known that the population standard deviation is 8. 7

81. CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Confidence Intervals For the Population Mean 1 CASE Solution: 7 Value of : Value of z : Minimum sample size: Note : ROUND UP your answer Given:

82. CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Confidence Intervals For the Population Mean 1 CASE If the variance of a national accounting examination is 900, how large a sample is needed to estimate the true mean score within 5 points with 99% confidence? 8

83. CABT Statistics & Probability – Grade 11 Lecture Presentation Estimation of Parameters Confidence Intervals For the Population Mean 1 CASE Solution: 8 Value of : Value of z : Minimum sample size: Given:

84. CABT Statistics & Probability – Grade 11 Lecture Presentation Check your understanding Ehljie wants to conduct a study on the average number of hours a Grade 11 student spends in studying Statistics and Probability in a school week with 98% confidence and a margin of error of 2 hours. What sample size should Ehljie use for her study if the population standard deviation is 1.5 hours? Mean and Variance of Sampling Distributions of Sample Means Estimation of Parameters Okay! Gamitin mo ‘ yung formula na ibinigay ni Sir!

85. Do you have any QUESTIONs?

86. Summing it up!