Example 4: Consider a population consisting of 1, 2, 3, 4, and 5. Suppose samples of size 2 are drawn from this population. Describe the sampling distribution of the sample means. 1. What is the mean and variance of the sampling distribution of the sample means? 2. Compare these values to the mean and variance of the population. 3. Draw a histogram of the sampling distribution of the population mean. Step 1: - Compute the mean of the population EX N 1+2+3+4+5 5 H = 3.00 So, the mean of the population is 3.00 Step 2: `Compute the variance of the population [o). X X - H (X- p)? Enrique Zobel Foundation | Senior High School 17 1 -3=-2 (-2)(-2) = 4 NJ 2-3=-1 (-1)(-1) =1 3- 3=0 (0)(0) = 0 4 - 3 =1 (1)(1) = 1 5- 3 =2 (2)(2) = 4 E(X - [) 2 4+1+0+1+4 =10 2-2(X-[1)2 N 10 5 2 = 2 So, the variance of the population is 2.4.00 4.50 10 - Total 10 1.00 Step 6: Compute the mean of the sampling distribution of the sample means ( x). Follow these steps. Multiply the sample mean by the corresponding probability- Add the results. Sample Mean Probability X . P (x ) 0.15 1.50 (Multiply the sample mean to the probability) 0.20 To 2.00 (Multiply the sample mean to the probability) 0.50 2.50 (Multiply the sample mean to the probability) 0.60 3.00 (Multiply the sample mean to the probability) 0.70 3.50 (Multiply the sample mean to the probability) 0.40 10 4.00 (Multiply the sample mean to the probability) Enrique Zobel Foundation | Senior High School 14Step 3: - Determine the number of possible samples of size n = 2. Use the formula .C. . Here N = 5 and n = 2 So, there are 10 possible samples of size 2 that can be drown. Step 4: - List all possible samples and their corresponding means. Samples Mean 1, 2 1.50 1 , 3 2,00 Enrique Zobel Foundation | Senior High School 12 1. 4 2.50 1,5 3,00 2. 3 2.50 2, 4 3,00 2, 5 3.50 3, 4 3.50 3, 5 4,00 4, 5 4.50 Stop 5: Construct the sampling distribution of the sample means. Sampling Distribution of Sample leans Sample Mean Frequency Probability 1.50 10 2.00 10 2.50 2 3.00 2 3.50 2Step 7: Compute the variance () of the sampling distribution of the sample means. Follow these steps: a. Subtract the population mean ( ) from each sample mean ( x ). Label this as x -. b.c. Square the difference. Label this asMultiply the results by the corresponding probability. Label this as P(u (x - 1) zx) d. .Add the results. ( x - 1) 2 P(1) ( x - 1) : P(x) . ( x - 1) 2 1.50 -1.50 2.25 0.225 (1.50 - 3.00) 2.00 10 1 -1.00 1.00 0.100 Enrique Zobel Foundation | Senior High School 15 2.50 -0.50 0.25 0.050 15 3.00 0.00 0.00 0.000 15 3.50 0.50 0.25 0.050 15 4.00 1.00 1.00 101 0.100 4.50 1.50 2.25 0.225 Total 1.00 0.750 So, the variance of the sampling distribution is 0.750oz= EP("'x2)= 0.75 . (x - H) 2Step 8: - Construct the histogram for the sampling distribution of the sample means. 0.25 0.2 0.15 Probability (X) 0.1 0.05 1.5 2 25 3 3.5 4.5 Sample Mean (X) Try This!Performance Task Rolling a Die! Directions: Do the task below. Write your answers in the space provided below. 1. Roll a die 30 times. 2. List all the outcomes of your experiment. 3. Find the mean and variance of the sampling distribution of the acquired data from the experiment. Note: You may use the back portion of this page for your answers. You will be graded according to the rubric: Grading Rubric Standards Outstanding Accomplished Developing Needs (3 Points) (2 Points) Improvement (4 Points) Rating (1 Point) Student Student are at Students do demonstrate Student ease with the not mastery of the demonstrate a mathematical demonstrate mathematical complete and concepts of complete Content concepts and comprehensive and the shows ability to comprehensi Knowledge understanding of presentation VE apply and the but lack a transfer learning understandin mathematical deep g of the with depth and concepts conceptual mathematica complexity understanding I concepts. Student presents Student presents Students presents information in information in Students does extremely well logical, information in logical not present Organization organized logical interesting the sequence form that flow sequence which which information smoothly from can audience audience can properly start to the end can follow follow All steps in solving the problem are All steps in Some steps in solving the meticulously solving the Student did Work Shown problem are problem are shown and not shown and not show any presented with shown with some answers work originality correct results are wrong All of the answers Few to none Accuracy are correct and All of the answers Some of the of the answers are completely are correct answers are correct accurate correct Enrique Zobel Foundation | Senior High School 20