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The standard deviation of the sampling distribution of the sample means (o,), also known as the standard error of the mean, measures the degree of accuracy of the sample mean as an estimate of the population means. The standard deviation of the sampling distribution of the sample means is the square root of the variance () and solved using the formula of = . (_) or ax = Jok- Illustrative Example 3: A group of Grade 11 learners got the following scores in their Statistics and Probability test: 2, 4, 9, 10, and 5. List all possible random samples of size 3 from this population and do the following: a. construct the sampling distribution of the sample means, b. solve the mean, variance, and standard deviation of the sampling distribution. Steps Solution 1. Solve the population mean using the formula H = Ex 2+4+9+10+5 H = 42 N 5 - = 6.00 Thus, the population mean (u) is 6.00 2. Solve the variance of the 2 - (2 - 6)- + (4-6)2+ (9-6)+ (10 -6)+(5-6)2 population using the formula = 2(x- W) (-4)2 + (-2)2 + (3)2 + (4)3 + (-1)2 16 +4+9+16+1 46 = 9.20 Thus, the population variance (or) is 9.20. 3. Determine the number of possible samples of size n formula for 5! using the C(5,3) = (5-3)13! 5.4.8-2-2 - 20 = 10 (2.1)8-2.1 2 combination C(n, r) = = n! (n-r)tri' 4. List all possible samples 2 . 2, 4, 9. 10 , and solve their corresponding Samples 10 5 means. Mean 5.00 5.33 3.67 7.00 5.33 5.67 7.67 6.00 6.33 8.00 Sampling Distribution of the Sample Means 5. Construct the sampling Sample Mean Frequency Probability distribution of the sample (x) (1) P(X) means. 3.67 1 1/10 = 0.10 5.00 1 1/10 = 0.10 5.33 2 2/10 = 0.20 5.67 1 1/10 = 0.10 6.00 1 1/10 = 0.10 6.33 1 1/10 = 0.10 7.00 1 1/10 = 0.10 7.67 1 1/10 = 0.10 8.00 1/10 = 0.10 Total 10 10/10 = 1.00 3 GSC-CID-LRMS-ESSLM, v.r. 03.00, Effective June 14, 2021 6. Determine the mean of the Since Hy= H, then He = 6.0 sampling distribution of the So, the mean of the sampling distribution of the sample means (My) sample means (ly). is 6.00. 7. Solve the variance of the sampling distribution of the Given: N = 5, n = 3, o' = 9.2 sample means (o}) using the .= . ( = ".=153 formula of = . () Thus, the variance of the sampling distribution of the sample means (o;) is 1.53. 8. Solve the standard deviation of the sampling Of = 6} = V1.53 = 1.24 distribution of the sample Therefore, the standard deviation of the sampling distribution of the means (ox) using the sample means (62) is 1.24. formulas, = voz.Activity 2: Identify Me! Direction: Identify the parameter and statistic on the given situation. 1. A researcher wants to estimate the average age of 244 Grade 11 self-supporting learners. From a sample of 180 learners, he obtained a sample mean age of 18. a. Parameter: b. Statistic. 2. A researcher wants to estimate the average death age of Filipino women in the last decade and from a sample of 100 deaths, he obtained a sample mean age of 73. a. Parameter: b. Statistic. GSC-CID-LRMS-ESSLM, v.r. 03.00, Effective June 14, 2021 Let Us Do Activity 3. Fill Me! Direction: Do what is being asked on the given problem. A population consists of the numbers 9, 5, 6, 12, and 15. Random samples of size n= 2 are to be taken from this population. Construct the sampling distribution of the sample means. Solve the mean, variance, and standard deviation of the sampling distribution of the sample means by completing the table below. Steps Solution 1. Solve the population mean using the formula 9+5+6+ 12 + 15 5 The population mean is 2. Solve the variance of the population using the formula The population variance is 3. Determine the number of possible samples of size n C(n, r) = = using the formula for (n-r)tri combination There are possible random samples that can be drawn from the population. C(n, r) = n! (n-r)ir!" 4. List all the possible samples and solve the Samples corresponding mean. Mean Sampling Distribution of the Sample Means Sample Mean Frequency Probability (f) P(X) 5. Construct a sampling distribution of the sample means. Total 6. Determine the mean of the Since My= H, then Ha = sampling distribution of the So, the mean of the sampling distribution of the sample means (,) sample means (H,). is 7. Solve the variance of the Given: N = . n= sampling distribution of the sample means (o;) using the Thus, the variance of the sampling distribution of the sample means formula o} = =. ( ) (of) is 5 GSC-CID-LRMS-ESSLM, v.r. 03.00, Effective June 14, 2021 B. Solve the standard deviation of the sampling distribution of the sample The standard deviation of the sampling distribution of the sample means (ox) using the means is formula Of = Vo