Estimating-parameters.pptxhajjanakkakwknss

Stat and probab

1. Understanding Estimation of Parameters

2. In Statistics, population parameters cannot be measured or there is no way in finding the exact value of population parameters. In inferential statistics, conclusions or descriptions can be made about the population using the samples derived from the population itself.

3. Estimation - is the process used to calculate these population parameters by analyzing only a small random sample from the population. - the value or range of values used to approximate a parameter is called estimate.

4. Estimation Point Estimation - refers to a single value that best determines the true parameter value of the population. Interval Estimation- gives the range of values within which the parameter value possibly falls.

5. - The confidence interval is defined to be the range of values used to estimate a parameter. It is also called interval estimate . - The confidence level of an interval estimate of a parameter is the probability that the interval estimate contains the parameter. It describes what percentage of intervals from many different samples contains the unknown population parameter.

6. INTERVAL ESTIMATE OF POPULATION MEAN Confidence Interval for Population Mean (𝜇) When Population Standard Deviation (𝜎) is Given . The formula for the confidence interval of the mean for the specific 𝛼 is

7. INTERVAL ESTIMATE OF POPULATION MEAN

8. INTERVAL ESTIMATE OF POPULATION MEAN    

9. INTERVAL ESTIMATE OF POPULATION MEAN The table shows the three commonly used confidence levels and their associated values of . For instance, if the confidence interval is 1 – 𝛼 = 0.90, then 𝛼= 0.10, a/2 = 0.05, and = 1.645. The resulting confidence interval estimator is called 90% confidence interval estimator of 𝜇.  

10. MAXIMUM ERROR ESTIMATE – is the maximum of a parameter and the actual value of the parameter. It is the term 𝐸=  

11. MAXIMUM ERROR ESTIMATE

12. Examples 1. From a normally distributed population, we took a simple random sample of 500 students with a mean score of 85 on the math section of the National Achievement Test (NAT). Suppose the standard deviation of the population is 2.8, what is the estimated population mean for 95% confidence level?

13. Solution: STEP 1. Given : n = 500 ; 𝜎 = 2.8 x̅ = 85 ; Confidence level = 0.95 STEP 2. Population is normal Sample is randomly selected STEP 3: Since the 95% confidence interval is to be used. α = 1 − 0.95 = 0.05 ; = = 0.025 . = 1.960   Two conditions that must be met

14. Solution: STEP 4.   85-1.960 ( ) 85- 0.245 84.755   85+1.960 ( ) 85+ 0.245 85.245  

15. Solution: STEP 4. 84.755   -2.365 2.365 0.025 0.025 84.755 85.245 95% Conclusion : We are 95% confident that the population mean score (𝜇) on the math section of the NAT lies between 84.755 points and 85.245 points.  

16. Examples 2. Find the 90% confidence interval of the population mean for the scores of Grade 11 students in Tarlac Province High School. A random sample of 30 Grade 11 students is shown. 20 21 23 14 11 22 22 12 11 12 11 19 18 14 15 16 24 24 15 21 22 23 20 20 20 18 18 13 19

17. Solution: STEP 1. Given : n = 30 ; 𝜎 = ? x̅ = ? ; Confidence level = 0.90 STEP 2. Find the mean and the standard deviation of the given data.     4.185  

18. Solution: STEP 3. Population is normal Sample is randomly selected STEP 4: Since the 90% confidence interval is to be used. α = 1 − 0.90 = 0.10 ; = = 0.05 . = 1.645   Two conditions that must be met

19. Solution: STEP 4.   17.933-1.645 ( ) 17.933-1.256 16.677   17.933+1.645 ( ) 17.933+1.256 19.189  

20. Solution: STEP 4. 16.677   -2.365 2.365 0.05 0.05 16.677 19.189 90% Conclusion : We are 90% confident that the population mean score (𝜇) of grade 11 students in Tarlac Province High School lies between 16.677 points and 19.189 points.  

21. Examples 3. Given: n=20; s= 3; = 82.5. The parents population is normally distributed. Use 95 % confidence interval to find the maximum error E and the estimated population mean.  

22. Solution: Step 1 :Identify the given n=20; s= 3; E= ? = 82.5 ; Confidence interval=95% Population mean=?   Step 2 : Identify the df df = n-1 df = 20-1 df = 19

23. Solution: 𝐸=   Step 4: Solve for the maximum error E 𝐸=   𝐸= 2.093   𝐸= 2.093   𝐸= 1.40 STEP 3: Since the 95% confidence interval is to be used. α = 1 − 0.95 = 0.05 ; = = 0.025 . = 2 .093  

24. Solution: STEP 5 .   82.5-2.093( ) 82.5-1.40 81.1   82.5-2.093( ) 82.5+1.40 83.9  

25. Solution: STEP 6. 8   -2.365 2.365 0.025 0.025 81.1 83.9 95% Conclusion : W e can say with 95% confidence that the interval between 81.1 and 83.9 contains the population mean based on the sample size of 20.  

26. ACTIVITY Identify the confidence interval for given the data below. Assume that the population is normally distributed. 2. Hazel wants to estimate the average blood pressure of adults in a certain city with a known standard deviation of 15. If she surveys a sample of 50 and determined a sample mean of 120, compute a 90% confidence interval for the average blood pressure of adults. 3. The average weight of 25 chocolate bars selected from a normally distributed population is 200 g with a standard deviation of 10 g. Find the maximum error E and interval estimates using 95% confidence level.