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1. Sampling Distribution of Means A Grade 11 Statistics and Probability Lecture

2. Parameter & Statistic CABT Statistics & Probability – Grade 11 Lecture Presentation Sampling Distribution of Means A parameter is a descriptive measure that describes a population . A statistic is a descriptive measure that describes a sample . Usually , parameters are denoted by lower-case GREEK letters (e.g.  or  ), while statistics use lower-case ROMAN letter (e.g. x and s).

3. Parameter & Statistic CABT Statistics & Probability – Grade 11 Lecture Presentation Sampling Distribution of Means https://www.cliffsnotes.com/assets/267169.png

4. Parameter & Statistic CABT Statistics & Probability – Grade 11 Lecture Presentation Sampling Distribution of Means Parameter or statistic? All CABT Senior High School students A selected group of CABT Senior High School students Average scores of Grade 10 NAT takers nationwide Average of NAT scores taken from a group of Grade 10 students from CABT

5. CABT Statistics & Probability – Grade 11 Lecture Presentation Sampling Distribution of Means Determine if the given situation describes a parameter or a statistic Check your understanding A group of 20 high school students are selected to join the Valenciana Festival. Ten of the teachers in a SHS have Master’s degrees. The Philippine Senate has 24 senators. CABT currently has 18 teachers. Eight teachers in CABT are chosen to attend a seminar.

6. Sampling Distributions CABT Statistics & Probability – Grade 11 Lecture Presentation Sampling Distribution of Means A sampling distribution is a probability distribution of all of the possible values of a statistic for a given sample size selected from a population. http://labs.geog.uvic.ca/geog226/images/Lab3/img_samp_dist.gif

7. Sampling Distribution of the Mean CABT Statistics & Probability – Grade 11 Lecture Presentation Sampling Distribution of Means 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, with the sample mean as the random variable. http://www.philender.com/courses/intro/notes2/sam.gif

8. Constructing Sampling Distributions of Sample Means CABT Statistics & Probability – Grade 11 Lecture Presentation Sampling Distribution of Means How to Construct a Sampling Distribution of Sample Means from a Given Population STEP 1 – Determine the number of samples of size n from the population of size N . STEP 2 – List all the possible samples and compute the mean of each sample. STEP 3 – Construct a frequency distribution of the sample means obtained in Step 2. STEP 4 – Construct the probability distribution for the sample means X .

9. Constructing Sampling Distributions of Sample Means CABT Statistics & Probability – Grade 11 Lecture Presentation Sampling Distribution of Means How to Construct a Sampling Distribution of Sample Means from a Given Population The number of samples of size n from the population of size N are given as follows: For sampling with replacement: Ibig sabihin pede maulit ang combination. For sampling without replacement: Dito naman bawal maulit combination Dito ay gagamitan ng calculator ung nCr ang nakalagay

10. Constructing Sampling Distributions of Sample Means CABT Statistics & Probability – Grade 11 Lecture Presentation Sampling Distribution of Means 1 Consider a group of N = 4 people with the following ages: 18, 20, 22, 24. Consider samples of size n = 2 from the group. If X is the age of one person in the group and X is the average age of the two people in a sample, construct the sampling distribution of X .

11. Constructing Sampling Distributions of Sample Means CABT Statistics & Probability – Grade 11 Lecture Presentation Sampling Distribution of Means 1 Solution – sampling with replacement Number of samples of size 2: The 16 samples of size 2 The 16 sample means

12. Constructing Sampling Distributions of Sample Means CABT Statistics & Probability – Grade 11 Lecture Presentation Sampling Distribution of Means 1 Sample Mean Frequency Probability 18 1 1/16 19 2 1/8 20 3 3/16 21 4 1/4 22 3 3/16 23 2 1/8 24 1 1/16 TOTAL 16 1 X P( X ) 18 1/16 19 1/8 20 3/16 21 1/4 22 3/16 23 1/8 24 1/16 Solution – sampling with replacement Number of samples of size 2:

13. Constructing Sampling Distributions of Sample Means CABT Statistics & Probability – Grade 11 Lecture Presentation Sampling Distribution of Means 1 Solution – sampling without replacement Number of samples of size 2: Sample Sample Mean 18, 20 19 18, 22 20 18, 24 21 20, 22 21 20, 24 22 22, 24 23 Sample Mean Frequency Probability 19 1 1/6 20 1 1/6 21 2 1/3 22 1 1/6 23 1 1/6 TOTAL 6 1

14. Constructing Sampling Distributions of Sample Means CABT Statistics & Probability – Grade 11 Lecture Presentation Sampling Distribution of Means 1 Solution – sampling without replacement Number of samples of size 2: Sample Mean Frequency Probability 19 1 1/6 20 1 1/6 21 2 1/3 22 1 1/6 23 1 1/6 TOTAL 6 1 X P( X ) 19 1/6 20 1/6 21 1/3 22 1/6 23 1/6

15. Constructing Sampling Distributions of Sample Means CABT Statistics & Probability – Grade 11 Lecture Presentation Sampling Distribution of Means 2 Consider a population with values 1, 2, 3. List all possible samples of size 2 from the population if sampling is with replacement. Construct a sampling distribution of the sample means for this population.

16. Constructing Sampling Distributions of Sample Means CABT Statistics & Probability – Grade 11 Lecture Presentation Sampling Distribution of Means 1 Solution – sampling with replacement Number of samples of size 2: The 9 samples of size 2 The 16 sample means  

17. Constructing Sampling Distributions of Sample Means CABT Statistics & Probability – Grade 11 Lecture Presentation Sampling Distribution of Means 1 Sample Mean Frequency Probability 1 1 1/9 1.5 2 2/9 2 3 3/9 or 1/3 2.5 2 2/9 3 1 1/9 TOTAL 9 1 X P( X ) 1 1/9 1.5 2/9 2 3/9 or 1/3 2.5 2/9 3 1/9 Solution – sampling with replacement Number of samples of size 2:  

18. Constructing Sampling Distributions of Sample Means CABT Statistics & Probability – Grade 11 Lecture Presentation Sampling Distribution of Means 3 Consider a population with values 2, 5, 7, 8. List all possible samples of size 2 from the population if sampling is without replacement. Construct a sampling distribution of the sample means for this population.

19. Constructing Sampling Distributions of Sample Means CABT Statistics & Probability – Grade 11 Lecture Presentation Sampling Distribution of Means 1 Solution – sampling without replacement Number of samples of size 2: Sample Sample Mean 2,5 3.5 2, 7 4.5 2, 8 5 5, 7 6 5, 8 6.5 7, 8 7.5 Sample Mean Frequency Probability 3.5 1 1/6 4.5 1 1/6 5 1 1/6 6 1 1/6 6.5 1 1/6 7.5 total 1 6 1/6 1

20. Constructing Sampling Distributions of Sample Means CABT Statistics & Probability – Grade 11 Lecture Presentation Sampling Distribution of Means 1 Solution – sampling without replacement Number of samples of size 2: X P( X ) 3.5 1/6 4.5 1/6 5 1/6 6 1/6 6.5 7.5 1/6 1/6 Sample Mean Frequency Probability 3.5 1 1/6 4.5 1 1/6 5 1 1/6 6 1 1/6 6.5 1 1/6 7.5 total 1 6 1/6 1

21. CABT Statistics & Probability – Grade 11 Lecture Presentation Sampling Distribution of Means Check your understanding 1. A group of students got the following scores in a test: 6, 9, 12, 15, 18, and 21. Consider sample s cores of size 3 that can be drawn without replacement from th e given scores . List all the possible samples and the corresponding means . Construct the sampling distribution of the sample means.

22. Constructing Sampling Distributions of Sample Means CABT Statistics & Probability – Grade 11 Lecture Presentation Sampling Distribution of Means 2 Consider a population with values 1, 2, 3, 4. List all possible samples of size 3 from the population if sampling is without replacement. Construct a sampling distribution of the sample means for this population.