Estimating population mean

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1. REVIEW

2. Basic Concept of ESTIMATION

3. ESTIMATION is the process used to calculate these population parameters by analyzing only a small random sample from the population Basic Concepts of Estimation ESTIMATE the value or range of values used to approximate a parameter.

4. TWO TYPES OF PARAMETER ESTIMATES: Point Estimate Interval Estimate Basic Concepts of Estimation

5. TWO TYPES OF PARAMETER ESTIMATES: Point Estimate Refers to a single value that best determines the true parameter value of the population Interval Estimate Gives a range of values within which parameter values possibly falls. Basic Concepts of Estimation

6. Basic Concepts of Estimation Point Estimate Method a population parameter is estimated simply with its corresponding sample statistics the sample mean estimates the population mean, the sample variance estimates the population variance x _ =   s 2 =   (x i – x ) 2 _ Point Estimator of Population MEAN μ Point Estimator of Population VARIANCE σ²

7. Basic Concepts of Estimation Interval Estimation To construct an interval around the point estimate and to state the degree of certainty that the population mean is within that interval. Thus, a < μ < b, where and b are the ends points of the interval between which the population mean lies. The interval of values that predicts where the true population parameter belongs is called INTERVAL ESTIMATE (Confidence Interval) The degree of certainty that the true population parameter falls within the constructed confidence interval is referred to as CONFIDENCE LEVEL.

8. Interval Estimation Example 2 A 95% confidence interval covers 95% of the normal curve. C = represent the confidence level of the interval estimate a = the area outside the boundaries for of interval estimate Interval estimate z a /2 is simply the z-score corresponding to the probability of found in the z-distribution table   - z < < z   2 a __ 2 a __

9. ESTIMATING POPULATION MEANS

10. Objectives: M11/12SP-IIIf-2 The learner identifies the appropriate form of the confidence interval estimator for the population mean when: (a) the population variance is known, (b) the population variance is unknown, (c) the Central Limit Theorem is to be used.

11. ESTIMATING POPULATION MEANS Guess the Word

12. ESTIMATING POPULATION MEANS FIRST WORD

13. Guess the Word ESTIMATING POPULATION MEANS N __ R __ __ L M O A

14. ESTIMATING POPULATION MEANS SECOND WORD

15. P O P __ L __T__O__ A N Guess the Word U I ESTIMATING POPULATION MEANS

16. ESTIMATING POPULATION MEANS THIRD WORDS

17. K N __ W N V A __ __ A N __ E I C Guess the Word R O ESTIMATING POPULATION MEANS

18. NORMAL Answers to “ Guess the Word ” Game POPULATION ESTIMATING POPULATION MEANS KNOW VARIANCE

19. ESTIMATING POPULATION MEANS Estimating the Mean of a Normal Population with Known Variance

20. ESTIMATING POPULATION MEANS with KNOWN VARIANCE Estimating the Mean of a Normal Population with Known Variance z-score transformation formula z =   Where: = sample mean = population mean = standard deviation   As a Result of Central Limit Theorem z formula for sample mean z =  

21. z-score transformation formula z =   As a Result of Central Limit Theorem z formula for sample mean z =   = + z   Population Mean formula ESTIMATING POPULATION MEANS with KNOWN VARIANCE

22. z = + z   Population Mean formula Using the Property of Symmetry and equation above we can generalize the formula for Confidence Interval of the population from a given Confidence level Sample mean Population standard deviation - z ( ) < < + z ( )   2 a __ 2 a __ ESTIMATING POPULATION MEANS with KNOWN VARIANCE

23. Example Consider a random sample of 36 items take from a normally distributed population with a sample mean of 211. Compute a 95% confidence interval for if the population standard deviation is 23.   Given: C = .095 = z 0.025 = ±1.96 = 211 = 23 n = 36   2 a __ z ESTIMATING POPULATION MEANS with KNOWN VARIANCE

24. 2 a __ z - z ( ) < < + z ( )   2 a __ 2 a __ Formula: Given: C = .095 = z 0.025 = ±1.96 = 211 = 23 n = 36   -1.96 ( ) < < + 1.96( )   < < 218.51   This mean that the mean of given population lies between 203.55 and 218.51 on a 95% confidence interval ESTIMATING POPULATION MEANS with KNOWN VARIANCE

25. ESTIMATING POPULATION MEANS with KNOWN VARIANCE

26. ESTIMATING POPULATION MEANS with KNOWN VARIANCE 1. The shape of the sampling distribution depends on the shape of the population. Two Problems will exist when the given sample is small 2. The population standard deviation is almost unknown

27. Activity

28. Basic Concepts of Estimation Exercise 1 Suppose that systolic blood pressures of a certain population are normally distributed with = 8 and a sample with size 25 is taken from this population. The average of the systolic blood pressures of these is 25 individuals is found to be 122. Find the 99% confidence interval of the average systolic pressure for all the members of the population.   Exercise 2 A random sample with size 9 is selected from a certain population with standard deviation of 3.2 the mean is 15. Estimate the mean for the population at 90% confidence level.

29. QUIZ

30. Basic Concepts of Estimation Problem 1 The ages of the member of a given population have standard deviation of 10. If the mean age of the sample with size 9 is 33.5 and considering that the given population follows a normal distribution, compute the 99% confidence interval of the mean age of the members of the population. Problem 2 The average height of the students in a famous college in the city is unknown but the standard deviation is approximated to be 0.8 ft. If a random sample of 49 students is chosen and the mean of their heights is 6.1 ft, estimate the average height of all the students in the college using 95% confidence level.

31. Assignment: 1.T-distribution 2.Estimating the Mean of a Normal Population with unknown Variance Read and Study: