1 MOTIVATION Decision Making: List down 5 best decisions you made in your life and how these decisions made you a wise person? Sample answer for the activity: Making wise decision is very important in our daily lives and one of the reasons why we should study statistics is to help us make a wise decision. We cannot avoid making decision in every minute of our daily life. This decision we do are based on the information. we gathered. > For instance, before getting into marriage you need first to gather information on your partner like, how he treats the people surround him, ifhe has work and if the both of you is financially stable. } Before doctors give prescribe medicine to a patient, pieces of information were presented first and lastly, before you choose a university to study, there's a need for you first to have ‘a background check on the university you want to go to. These are some example decisions that we do in our life that will eventually have a huge Impact in our life. ll, LESSON PROPER Sampling Techniques Commonly Used in Research | WAYASSIPYHE | How do we determine samples? ‘This Is likely the question on mind when you plan to conduct a simple survey or study. Sample determination is considered to be the ultimate concem especially that the study will not move without the respondents. This method is used to determine which element is to be included in the sample. In order to oblain a genuine or unbiased sample. Each member should have an equal chance of being included or being selected to become a sample. Itis very important that each researcher have ‘a complete list of the population, so that every member is ready to be included is the sample.There are two classification of sampling techniques. It may be Probabillly Sampling or Non- Probability Sampling |. Probability Sampling - each member of the population has en equal chance of being selected as members of the sample Fivo Probability Sampling Techniquos 1. Simple Random Sampling is a part of the sampling technique in which each sample has an equal probability of being chosen. A sample chosen randomly is meant to be an unbiased representation of the total population. There are two ways to do a random sampling. > Lottery Sampling / Raffie. Process: ‘a. Each member of the population is numbered b. The pieces of paper shall be rolled evenly ¢. The desired number of samples are drawn one after the other. > Table of random number- A table of random numbers invented by statistician is used to draw the numbers for the sample. Example: Suppose in a group of 25 girls Process: a. 8 will be selected at random to join a competition. b. Then, if the first 2 digit number of the random numbers is 20, The number of 20 among the girls is the first sample, and: c. The process is continued until the 8" number is chosen. 2. Systematic Sampling is done systematically and itis done by numbering each member of the population and successively drawn the elements from the population. Example: There are 440 people in group. A researcher needs 20 samples from the population. Process: a, Divide the population by the number of the 440 4 desired sample. 20 n= 22 b, Select a random starting point For example, we start at the ‘st element. then We successively choose the member of our sample. 3" ( 4** member of our sample) 23 +22 = 45" (2 member of our sample) 45 + 22 = 67" ( 3" member of our sample)The process will continue until we will obtain the n® member of the sample. 4 nth member of the sample On the same given. If we will start on the 3” sample. This is the process 3+ 22= 25" ( 1°" member of our sample) 25 + 22 = 47" (2% member of our sample) 47 + 22 = 69" (3" member of our sample) until he nth member of our sample. 3, Stratified Random Sampling- this method will obtain its sample by dividing the population into its categories, strata (groups) or sub-population, then we obtain the sample proportionately from each stratum. Example: The researcher wants to equally obtain a total sample of 520 residents from the whole population in Barangay D. Data: Streets (strata) Total population of students on each grade level StreetA 350 Street B 433 Street C 324 Street D 293 Total population of Barangay D 1400 Process: a. To obtain a proportional Total number of populaton in strata number we must divide the Total number of population in Barangay D total population of strata and x the total population of Total number of desired samples residents then equate it to x divided by the total number of x represents the number of desired number of data per stratum. desired sampleFor Street 1: Total population of Barangay D= 1400 Total population of Street 1 =350 Desired number of samples= 520 350 x Cross multiply 1400 520 (350)(520) = 1400(x) Simplify 182000 = 1400x divide both sides by 1400 x= 130 ‘The researcher will get 130 residents in Street 1 For Street 2: Total population of Barangay D= 1400 Total population of Street 2 =433 Desired number of samples= 520 433 ox Cross multiply 1400520 (433)(520) = 14006) Simplify 225160 = 1400x divide both sides by 1400 x = 160.83 Round off x= 161 ‘The researcher will get 161 residents in Street 2. For Street 3: Total population of Barangay D= 1402 ‘Total population of Street 3 =325 Desired number of samples= 520 324 x Cross multiply 1402520 (324)(520) = 1400(x) Simplify 168480 = 1400x divide both sides by 1400 x= 12034 Round of f x 20 ‘The researcher will get 120 residents in Street 3.For Street 4: Total population of Barangay D= 1402 Total population of Street 3 =293 Desired number of samples= 520 203 Cross multiply a0 (293)(520) simplify 152360 = 1400x divide both sides by 1400 x = 100.01 Round off 109 ‘The researcher will get 109 residents in Street 4. ‘Summary of Data Grade Level (strata) Total population of students on Total number of samples from each grade level each stratum Street A 351 130 Street B 433 161 Street C 325 120 Street D 293 109 Total population of Barangay D 1402 residents Total: 520 residents 4. Cluster Sampling- Also called as area sampling because itis used on large population. We select members of the sample by area and individuals are randomly chosen. Process: a. Members of the sample are selected by group or per cluster b. Sample is selected randomly from each group or cluster randomly 5. Multi- Stage Sampiing- Combination of several sampling techniques, usually used by researchers who are interested n studying a very large population. This is done by; a. Diving the whole population by area b. Dividing each area into strata c. From each stratum, we get the sample by using random sampling technique. Il. Non- Probability Sampling This is a sampling technique where the researcher draws the sample based on his own judgement, therefore, the result is biased and not reliableA. Convenience Sampling - This is being used for its convenience to the researcher. The research conducts the study at his own convenient time, preferred place or Venue. He specifies the place and time. For example: A Researcher wants to find out which detergent is the most popular in household, he may just make a phone call using the phone number he found on the telephone, B. Quota Sampling - A method which the researcher limits the number of his samples based on the required number of the subject under investigation. For example: A Researcher limits his samples into 200 policemen only. C. Purposive Sampling- A non-sampling method that the way researchers choose their samples based on certain criteria and rules that were set by the researchers on their own. For example: The study needs Teacher respondents. The researcher wants a sample from Mathematics Teachers only, D. Snowball Sampling- This method will be useful when a member of the sample is chosen through referral of the other member of the sample. For example: A boy who is a part of a sample introduced or referred his friend to the researcher to be a sample. E. Modal Instance Sampling — is a method of non-probability sampling where the members of the sample are selected based n the typical, most frequent observation and modal cases. For example: In a certain group of students, most of them are girls, then the samples are girls, present in the group.How do you identify sampling distribution of the sample means? What is the importance of sampling distribution of sample means? What are the steps in constructing the sampling Distribution of means? ‘Steps in Constructing the Sampling Distribution of the Means Step 1. Determine the number of possible samples that can be drawn trom the population using the formula Nn where: N = size of the population n= size of the sample Step 2. List all the possible samples and compute the mean of each sample. ‘Step 3. Construct a frequency distribution of the sample means oblain in Step 2. Recall that a variable is a characteristic or attribute that can assume different values. A population consists of the numbers 2, 4, 9, 10, and 5. Let us list all possible samples of size 3 from this population and compute the mean of each sample. 2.4.9 5.00 2,4, 10 5.33 245 3.67 2,9, 10 7.00 2,95 5.33 2,10,5 5.67 4,9, 10 7.67 49,5 6.00 4,105 6.33 9, 10,5 8.00 There are 10 possible semples of size 3 that can be crawn from the given population.3.67 1 5.00 7 5.33 zi 5.67 1 6.00 1 1 1 1 1 6.33 7.00 1.67 8.00 Total n=10 ‘Observe that the means vary from sample to semple. Thus, any mean based on the sample drawn, from a population is expected to assume different values for the samples. So, this leads us to a conclusion that sample mean is random variable, which depends on a particular sample. Being a random variable, it has a probability distribution. The probability distribution of the sample means is also called the sampling distribution of the sample means. ‘Sampling Distribution of Sample Means 3.67 1 5.00 7 5.33 2 5.67 1 6.00 1 1 1 1 1 633 7.00 787 8.00 Total n=10 10 Observe that the means of the samples could be less than, greater than, or equal to the mean of the population. The difference between the sample mean and the population mean is called the sampling error.Mean and Variance of the Sampling Distribution of Sample Means In this lesson, you will lear how to describe the sampling distribution of the sample means by ‘computing its mean and variance. You will aiso make a general conclusion regarding the mean, vatiance, and shape of the sampling distribution of the sample means. Example 1 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. a. What is the mean and variance of the sampling distribution of the sample means? b. Compare these values of the mean and variance of the population ‘Step 1. Compute the mean of the population (j:). _=x Tv 142434445 - 5 ‘So, the mean of the population is 3,00 ‘Step 2. Compute the variance of the population («), x Xn (x-w)? 1 2 4 2 a 1 3 0 0 4 1 1 5 z 4 Ta-m7=10 os So, the variance of the population is 2 Step 3. Determine the number of possible samples of size n= 3 Use the formula wen. Here, N=5 and n=2, 52 = 10So, there are 10 possible samples of size n= 2 that can be drawn. ‘Step 4. List all possible samples and their comesponding means. ‘Samples Mean 1,2 1.50 1,3 2.00 1.4 2.50 1,5 3.00 2,3 2.50 2.4 3.00 2,5 3.50 34 3.50 35 4.00 45 4.50 ‘Step 5. Construct the sampling distribution of the sample means. ‘Sampling Distribution of Sample Means Follow these steps: b. Add the results. a, Multiply the sample mean by the corresponding probability. ‘Sample Mean P(X) X+P(X) x 7.50 a 0.45 10 ‘Sample Mean Frequency Probability x x 1.50 i 1 10 2.00 1 1 | 10 2.50 2 1 4 5 3.00 2 1 3.50 2 1 4.00 i t 10 4.50 1 ze rn} Total 10 1.00 Step 6. Compute the mean of the sampling distribution of the sample means (1g).2.00 = 020 To 250 a 0.50 3.00 i 0.60 5 3.50 1 70 5 4.00 1 0.40 16 450 4 0.45 10 Total 1.00 3.00 He = LF © PR) =3.00 ‘So, the mean of the sampling distribution of the sample means is 3. 00 ‘Step 7. Compute the variance («?z) of tne sample means. Follow these steps: a. Subtract the population mean (12) from each the sample mean (®). Label this as. — 1. b. Square the difference. Label this as (X — y2)*. cc. Multiply the results by the comesponding probability. Label this as P() + (7 — p)?. . Add the results, x PQ) Xu GW? P(X)* (Xp)? 1.50 = =1.50 225 0.225 2.00 + -1.00 1.00 0.100 250 ? -0.50 025 0.050 3.00 ZL 0.00 0.00 0.000 3.50 t 0.50 0.25 0.050 4.00 9 1.00 1.00 0.100 450 < 150 225 0.225 Total 490 0.750 ate = SP U)* = 0.75 So, the variance of the sampling distribution is 0.75.Try to answer these questions: 1. How do you compare mean of the sample means and the mean of the population? 2. How do you compare variance of the sample means and the variance of the population? Example 1. The average time it takes a group of college students to complete a certain examination is 46.2 minutes. The standard deviation is 8 minutes. Assume that the Variable is normally distributed. a. What is the probability that a randomly selected college student will complete the examination in less than 43 minutes? ‘Solution: Given 46.2 P (X< 43) = P(2<-0.40) 0.5000~ 0. 1584 0.3446 So, the probability that a randomly selected college student will complete the examination in less. than 43 minutes is 0. 3446 or 34.46%. . If 50 randomly selected college students take the examination, what is the probability that the mean time it takes the group to complete the test will be less than 43 minutes? Solution Given2= 2.83 P (X< 43) = P (z<-2.83) 0.5000 — 0. 4977 0.0023 So, the probability that the mean time it will take the 50 randomly selected college students to complete the test in less than 43 minutes is 0.0023 or 0.23%. Example 2. The average number of milligrams (mg) of cholesterol in a cup of a certain brand of ice cream Is 660 mg, and the standard deviation is 35 mg. Assume the variable is normally distributed. a. If a cup of ice cream is selected, what is the probability that the cholesterol content will be more than 670 mg? Solution: Given y= 660 o=35 X= 670 P (x> 670) 2= 0.29 P(X > 670) =P (z> 0.29) 0.5000-0. 1141 0.3859 So, the probability that the cholesterol content will be more than 670 mg is 0.3859 or 38.59%b. Ifa sample of 10 cups of ice cream is selected, what is the probability that the mean of the sample will be larger than 670 mg? Given z- 0.90 P (X> 670) = P (z > 0.90) 5000 - 0.3159 = 0.1841 So, the probability of the mean of the sample larger than 670 mg is 0.1841 or 18.41%. MI Generalization «SAMPLING TECHNIQUES |. Probability Sampling — each member of the population has an equal chance of being selected as members of the sample Five Probability Sampling Techniques Simple Random Sampling is a part of the sampling technique in which each sample has en ‘equal probabilty of being chosen. A sample chosen randomly is meant to be an unbiased representation of the total population. There are two ways to do a random sampling, > Lottery Sampling / Raffle. » Table of random number ‘Systematic Sampling is done systematically and it is done by numbering each member of the population and successively drawn the elements from the population