10-Sampling-Distribution-of-the-Sample-Means.pptx

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1. SAMPLING DISTRIBUTION OF SAMPLE MEANS DOMINIC DALTON L. CALING Statistics and Probability | Grade 11

2. LESSON OBJECTIVES At the end of this lesson, you are expected to: illustrate random sampling; distinguish between parameter and statistic; and construct sampling distribution of sample means.

3. Pre-Assessment A

4. Pre-Assessment A BLOOD TYPE FREQUENCY A B AB O 5 N = 25 7 4 9

5. Pre-Assessment B 5.40 7.67 7.00 15.57 22.125

6. Pre-Assessment C = 10   where and = total number of objects in the set = number of choosing objects from the set   = 70 = 84 = 120 = 495          

7. Lesson Introduction Researchers use sampling if taking a census of the entire population is impractical. Data from the sample are used to calculate statistics, which are estimates of the corresponding population parameters. For instance, a sample might be drawn from the population, its mean is calculated, and this value is used as a statistic or an estimate for the population mean. Thus, descriptive measures computed from a population are called parameters while descriptive measures computed from a sample are called statistics . We say that the sample mean is an estimate of the population mean .  

8. Discussion Points Sampling Distribution of Sample Means The number of samples of size n that can be drawn from a population of size N is given by N C n . A sampling distribution of sample means is a frequency distribution using the means computed from all possible random samples of a specific size taken from a population. The probability distribution of the sample means is also called the sampling distribution of the sample means.

9. Discussion Points Steps in Constructing the Sampling Distribution of the Means Determine the number of possible samples that can be drawn from the population using the formula: N C n where N = size of the population n = size of the sample List all the possible samples and compute the mean of each sample. Construct a frequency distribution of the sample means obtained in Step 2.

10. EXAMPLE 1 A population consists of the numbers 2, 4, 9, 10, and 5. List all possible samples of size 3 from this population . C ompute the mean of each sample. Prepare a sampling distribution of the sample means.

11. Solution to Example 1 a. The p ossible samples of size 3 from 2, 4, 9, 10, and 5 are…

12. Solution to Example 1 b. The mean of each sample are as follows:

13. Solution to Example 1 c. The sampling distribution of the sample means

14. EXERCISE 1 Samples of seven cards are drawn at random from a population of eight cards numbered from 1 to 8. How many possible samples can be drawn? ( 8 C 7 ) Construct the sampling distribution of sample means.      

15. SOLUTION Sample Mean Sample Mean Frequency Probability P(X) TOTAL 1, 2, 3, 4, 5, 6, 7 1, 2, 3, 4, 5, 6, 8 1, 2, 3, 4, 5, 7, 8 1, 2, 3, 4, 6, 7, 8 1, 2, 3, 5, 6, 7, 8 1, 2, 4, 5, 6, 7, 8 1, 3, 4, 5, 6, 7, 8 2, 3, 4, 5, 6, 7, 8 4.00 4.14 5.00 4.86 4.71 4.57 4.43 4.29 4.00 4.14 5.00 4.86 4.71 4.57 4.43 4.29 1 1 1 1 1 1 1 1                 1.00 n = 8

16. EXERCISE 2 A group of students got the following scores in a test: 6, 9, 12, and 15. Consider samples of size 3 that can be drawn from this population. List all the possible samples and the corresponding mean. Construct the sampling distribution of the sample means.      

17. SOLUTION Sample Mean Sample Mean Frequency Probability P(X) TOTAL 6, 9, 12 6, 9, 15 6, 12, 15 9, 12, 15 9.00 10.00 12.00 11.00 9.00 10.00 12.00 11.00 1 1 1 1         1.00 n = 4

18. EXERCISE 3 A finite population consists of 4 elements. 10 , 12 , 18 , 40 How many samples of size n = 2 can be drawn from this population? List all the possible samples and the corresponding means. Construct the sampling distribution of the sample means. 4 C 2 = 6

19. SOLUTION 10, 12, 18, 40 Sample Mean Sample Mean Frequency Probability P(X) TOTAL 10, 12 10, 18 10, 40 12, 18 11.00 14.00 15.00 25.00 11.00 14.00 25.00 15.00 1 1 1 1         1.02 1.00   n = 6 12, 40 18, 40 26.00 29.00 26.00 29.00 1 1    

20. Summary A random variable is a function that associates a real number to each element in the sample space. It is a variable whose values are determined by chance.

21. Summary The number of samples of size n that can be drawn from a population of size N is given by N C n . A sampling distribution of sample means is a frequency distribution using the means computed from all possible random samples of a specific size taken from a population. The probability distribution of the sample means is also called the sampling distribution of the sample means.

22. Summary Steps in Constructing the Sampling Distribution of the Means Determine the number of possible samples that can be drawn from the population using the formula: N C n where N = size of the population n = size of the sample List all the possible samples and compute the mean of each sample. Construct a frequency distribution of the sample means obtained in Step 2.

23. THANK YOU!