CABT SHS Statistics & Probability - Sampling Distribution of Means
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This document is a presentation on sampling distributions of means for a Grade 11 Statistics and Probability lecture. It begins by defining populations and samples, and explaining how inferential statistics makes conclusions about populations based on sample data. It then discusses different sampling techniques like simple random sampling, systematic random sampling, stratified random sampling and cluster sampling. The key concepts of parameters, statistics, and sampling distributions are also introduced. Examples are provided to illustrate how to construct sampling distributions of means.
CABT SHS Statistics & Probability - Sampling Distribution of Means
- 1. CABT Statistics & Probability – Grade 11 Lecture Presentation
- 3. Sampling Distribution of Means A Grade 11 Statistics and Probability Lecture
- 4. Population and Sample CABT Statistics & Probability – Grade 11 Lecture Presentation Sampling Distribution of Means A population is the set of all items or individuals of interest. Examples: All registered voters Cell phones manufactured today Sales receipts for August A sample is a subset of the population. Examples: 1200 voters selected at random for a survey A few cell phones selected for testing Random receipts selected for audit
- 5. Inferential Statistics CABT Statistics & Probability – Grade 11 Lecture Presentation Sampling Distribution of Means Inferential statistics is concerned with drawing conclusions and/or making decisions concerning a population based only on sample data. http://www.gohomeworkhelp.com/admin/photos/what-is-inferential-statistics.jpg
- 6. Population inferences can be made...
- 7. ...by selecting a representative sample from the population
- 8. Why Sample?! CABT Statistics & Probability – Grade 11 Lecture Presentation Sampling Distribution of Means Researchers use sampling if taking a census of the entire population is impractical. It is less time-consuming and less costly than a census. It is possible that data from the sample can be used to estimate corresponding population measures.
- 9. Types of Sampling CABT Statistics & Probability – Grade 11 Lecture Presentation Sampling Distribution of Means Simple random sampling – every element of the population has the same probability of being selected for inclusion in the sample. Some simple random sampling techniques: 1. Lottery or drawing – with or without replacement 2. Using Random Numbers
- 10. Types of Sampling CABT Statistics & Probability – Grade 11 Lecture Presentation Sampling Distribution of Means Simple random sampling and Slovin’s FormulaSlovin’s Formula is a formula used to determine the sample size n from a population of size N and margin of error e is given by: 2 1 N n Ne Example: The number of samples needed for a population of 1,000 at a margin of error of 5% is 2 1000 286 1 1000 0.05 n
- 11. Types of Sampling CABT Statistics & Probability – Grade 11 Lecture Presentation Sampling Distribution of Means – a list of randomly-arranged elements of the population is used as a sampling frame, and the elements to be included in the sample are selected by skipping through the list at regular intervals. Systematic random sampling http://worldartsme.com/images/holiday-shopping-list-clipart-1.jpg
- 12. Types of Sampling CABT Statistics & Probability – Grade 11 Lecture Presentation Sampling Distribution of Means In doing systematic random sampling in a population of size N using a sample size n, the sampling interval k is given by Systematic random sampling N k n Example: To get a sample containing 8 students from a group of 40 students using systematic random sampling, the sampling interval is 40 5 8 k
- 13. CABT Statistics & Probability – Grade 11 Lecture Presentation Sampling Distribution of Means http://users.hist.umn.edu/~ruggles/hist5011/sampling.ppt
- 14. Types of Sampling CABT Statistics & Probability – Grade 11 Lecture Presentation Sampling Distribution of Means – the population is partitioned into several subgroups or strata, then samples are randomly selected separately from each stratum Stratified random sampling https://faculty.elgin.edu/dkernler/statistics/ch01/images/strata-sample.gif
- 15. Types of Sampling CABT Statistics & Probability – Grade 11 Lecture Presentation Sampling Distribution of Means Proportional allocation of sample size Stratified random sampling Population Strata Number of Students per Strata No. of Samples per Strata Male 340 20 Female 500 30 TOTAL 840 50 340 50 20.24 20 840 males 500 50 29.76 30 840 females Allocation of samples per stratum: Suppose that a sample of size 50 is to be obtained from a population of 840 students grouped according to gender:
- 16. Types of Sampling CABT Statistics & Probability – Grade 11 Lecture Presentation Sampling Distribution of Means – the population is broken into small groups or clusters, then some of the clusters are randomly selected. Cluster or area sampling http://research-methodology.net/wp-content/uploads/2013/07/Cluster-sampling.gif
- 17. Types of Sampling CABT Statistics & Probability – Grade 11 Lecture Presentation Sampling Distribution of Means This type of sampling is frequently used by Philippine survey companies like SWS and Pulse Asia. Cluster or area sampling
- 18. Types of Sampling CABT Statistics & Probability – Grade 11 Lecture Presentation Sampling Distribution of Means The difference between cluster sampling and stratified sampling is that the sample consists of elements from the selected clusters only, while in stratified sampling, the sample comes from all the strata. Cluster vs stratified sampling http://keydifferences.com/wp-content/uploads/2016/04/stratified-vs-cluster-sampling.jpg
- 19. CABT Statistics & Probability – Grade 11 Lecture Presentation Sampling Distribution of Means Your class would like to interview 50 Junior High School students to determine their opinion regarding the new school ID. Explain how you can choose your samples using each of the sampling techniques discussed. Check your understanding https://boonesunriserotary.files.wordpress.com/2016/05/interview.jpg
- 20. Do you have any QUESTIONs?
- 21. 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).
- 22. Parameter & Statistic CABT Statistics & Probability – Grade 11 Lecture Presentation Sampling Distribution of Means https://www.cliffsnotes.com/assets/267169.png
- 23. Parameter & Statistic CABT Statistics & Probability – Grade 11 Lecture Presentation Sampling Distribution of Means Parameter or statistic? 1. All CABT Senior High School students 2. A selected group of CABT Senior High School students 3. Average scores of Grade 10 NAT takers nationwide 4. Average of NAT scores taken from a group of Grade 10 students from CABT
- 24. 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 1. A group of 20 high school students are selected to join the Valenciana Festival. 2. Ten of the teachers in a SHS have Master’s degrees. 3. The Philippine Senate has 24 senators. 4. CABT currently has 18 teachers. 5. Eight teachers in CABT are chosen to attend a seminar.
- 25. 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
- 26. 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
- 27. 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
- 28. 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: For sampling without replacement: n N N n C
- 29. Constructing Sampling Distributions of Sample Means CABT Statistics & Probability – Grade 11 Lecture Presentation Sampling Distribution of Means 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.
- 30. Constructing Sampling Distributions of Sample Means CABT Statistics & Probability – Grade 11 Lecture Presentation Sampling Distribution of Means Solution – sampling with replacement Number of samples of size 2: 1st 2nd Observation Obs 18 20 22 24 18 18,18 18,20 18,22 18,24 20 20,18 20,20 20,22 20,24 22 22,18 22,20 22,22 22,24 24 24,18 24,20 24,22 24,24 1st 2nd Observation Obs 18 20 22 24 18 18 19 20 21 20 19 20 21 22 22 20 21 22 23 24 21 22 23 24 The 16 samples of size 2 The 16 sample means 2 4 16
- 31. Constructing Sampling Distributions of Sample Means CABT Statistics & Probability – Grade 11 Lecture Presentation Sampling Distribution of Means 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: 2 4 16
- 32. Constructing Sampling Distributions of Sample Means CABT Statistics & Probability – Grade 11 Lecture Presentation Sampling Distribution of Means Solution – sampling without replacement Number of samples of size 2: 4 2 6C 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
- 33. Constructing Sampling Distributions of Sample Means CABT Statistics & Probability – Grade 11 Lecture Presentation Sampling Distribution of Means Solution – sampling without replacement Number of samples of size 2: 4 2 6C 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
- 34. Constructing Sampling Distributions of Sample Means CABT Statistics & Probability – Grade 11 Lecture Presentation Sampling Distribution of Means Consider a population with values 1, 2, 3. a. List all possible samples of size 2 from the population if sampling is with replacement. b. Construct a sampling distribution of the sample means for this population.
- 35. Constructing Sampling Distributions of Sample Means CABT Statistics & Probability – Grade 11 Lecture Presentation Sampling Distribution of Means Consider a population with values 2, 5, 7, 8. a. List all possible samples of size 2 from the population if sampling is without replacement. b. Construct a sampling distribution of the sample means for this population.
- 36. Constructing Sampling Distributions of Sample Means CABT Statistics & Probability – Grade 11 Lecture Presentation Sampling Distribution of Means Consider a population with values 1, 1, 2, 3, 4. a. List all possible samples of size 3 from the population if sampling is without replacement. b. Construct a sampling distribution of the sample means for this population.
- 37. CABT Statistics & Probability – Grade 11 Lecture Presentation Sampling Distribution of Means Check your understandingA group of students got the following scores in a test: 6, 9, 12, 15, 18, and 21. Consider sample scores of size 3 that can be drawn without replacement from the given scores. a. List all the possible samples and the corresponding means. b. Construct the sampling distribution of the sample means.
- 38. Do you have any QUESTIONs?
- 40. Thankyou!