Comparing Two Samples Part 2

Principles of inferential statistics The Normal distributions The normal distribution is a symmetrical curve shape that has one vertex and depends on probability distributions. The area under the curve represents the percentages of people or cases who got certain degrees on any variable (phenomenon) we are studying. natural curve shape The middle of the curve represents most degrees under the curve and the ends (tails) represent the least or outliers and they extend to infinity. Example: Intelligence, where the middle represents people of average intelligence, and the more it turns to the sides, the proportions of people decrease, so that it turns to highly intelligent (gifted) people in the positive direction or to people with limited intelligence (special groups or mental retardation) in the negative direction. One of the characteristics of the normal curve (normal distribution) is that the area under the curve equals 1 or 100%, and thus this normal distribution depends on the principle of probability, and this is the essence and basis of hypothesis testing and inferential statistics. Example of calculating the area of a normal curve Many human characteristics, such as height, weight, age, and others, are located close to the normal curve, so we often assume that the degrees are normally distributed when we perform statistical operations. The standard normal distribution It is also a normal distribution but is calibrated so that its mean is 0 = 0 and its standard deviation is one 1 = Its scores are called Z scores. That is, all its degrees are calibrated to have the same standard units, so we convert the measurement degrees into standard degrees (Z degrees). Standard scores (Z scores) aim to determine the location of the score and if it is above or below the mean (positive standard scores lie above the mean, and negative standard scores lie below the mean) It also makes it easier for us to compare individuals and their performance, so there are so-called standardized tests. It also facilitates the process of calculating probabilities, ratios and percentiles. Since the normal distribution and the standard normal distribution have the same properties mentioned above (a symmetric curve with area equal to 1 or 100%), they share the empirical law of normal distribution, which is: 1- 68% of the data lie between one standard deviation above and below the mean 2- 95% of the data lie between 2 standard deviations above and below the mean 3- 99% of the data lie between 3 standard deviations above and below the mean Example: Not standardized degrees for length and the same degrees when converted to standardized degrees Normal distribution in detail and between each standard degree Normal distribution and known standard score values There are pre-calculated tables for calculating the areas of the normal curve for standard degrees (Z degrees) and they are found in most statistics books, or they can be easily calculated on specialized sites on the Internet or statistical programs such as Excel. How do we calculate Z scores? We calculate Z by the following equation: where X is the degree, is the mean, and is the standard deviation Example: A female student gets a score of 115 on an intelligence test, and the test average is 100 and its standard deviation is 15. What is her standard score? If the student's grade is above the average, one standard degree Also, the value of any probability can be calculated for any variable that follows the normal distribution. As mentioned previously, the values of these possibilities are taken from the table of the normal distribution (Z-score table) An example of calculating the probabilities using standard scores: If the income of 1000 families in a city follows a normal distribution, the average income is 1800 riyals, and its standard deviation is 300 riyals. What is it: 1- The possibility of obtaining a family income greater than 2,400 riyals. 2- The possibility of obtaining a family income of less than 1200 riyals. 3- The probability of obtaining a family income between 1650 riyals and 2250 riyals. The solution: 1We then find the ratio that corresponds to Z=2 from the normal distribution table and the value is 0.9772 We draw the curve and determine the required area, which is greater than 2400 riyals or after the value of 0.9772 (note: you can draw the curve with standard degrees or non-standard degrees) And because the required area or ratio lies after 0.9772, we subtract it from 1 1- 0.9772 = 0.0228 Then we multiply it by 100 to convert it to a ratio 0.0228 x 100 = 2.28 If the percentage or probability of obtaining a family income greater than 2,400 riyals is equal to 2.28% 2We then find the ratio that corresponds to Z= -2 from the normal distribution table and the value is 0.0228 We draw the curve and determine the required area which is less than 1200 riyals or before the value of 0.0228 And because the required area or ratio falls before 0.0228, we do not subtract it from 1 Then we multiply it by 100 to convert it to a ratio 0.0228 x 100 = 2.28 If the percentage or probability of obtaining a family income of less than 1200 riyals is equal to 2.28% Important notes: 1- If a greater probability is requested, we subtract it from the ratio taken from the table, and if a smaller probability is required, write the percentage taken from the table as it is. 2- Multiply the probability value that we get from the table by 100 to convert it to a ratio 3We calculate each value separately We then find the ratio that corresponds to Z= - .5 from the normal distribution table and the value is 0.3085 And find the ratio that corresponds to Z = 1.5 from the normal distribution table and the value is 0.9332 We draw the curve and determine the required area, which is between 0.3085 and 0.9332 Then we subtract the largest percentage from the smallest percentage that we got from the table = 0.9332 - 0.3085 .6247 Then we multiply it by 100 to get the percentage If the percentage or probability of obtaining a family income between 1650 riyals and 2250 riyals is 62.47% Percentiles . Percentiles The singular percentile The percentile or the percentile degree is a point on the distribution of its frequencies as percentages of a total sum (dividing the distribution into one hundred equal parts). for the individual in his group). Simply, it can be said that a student in the third grade of secondary B got 752 points out of the total score (1000 degrees) and by comparing him with the students of his class (25 students) his rank was on the sixteenth grade (16) and here we have the value 752 and it represents the student’s total grades, the number 16 Indicating the order of the student in the class, i.e. he is better than 15 students in the class, or that there are 9 students better than him, i.e. they got a total of more than 752 degrees, we would like to say that we must distinguish between the value of 752 as degrees and the value of 16 as order or we must differentiate between the value of the percentile And the order of the percentile and the number 16 indicative of the order here means 16 x 100 ÷ 25 = 64% means that the value 752 corresponds to the 64th percentile, which means that 64% of the students got a score of 752 or less than the class students, or it means that 36% [(1 - 64) %] got a score greater than 752 (9 x 100 ÷ 25 = 36%). And the previous explanation of the standard degrees is to find the percentile rank for each degree, i.e. its arrangement as a percentage, but how do we know the degree that corresponds to a certain percentage, i.e. we know the degree corresponding to the percentage To calculate the degree corresponding to a specific ratio, we use the following equation: where X is the degree and is unknown, is the mean, is the standard deviation and is the standard degree Example: What is the 75th percentile of the income of 1,000 families in the previous example? X= 1800 + 300 × Z We do the previous steps that were previously explained to calculate the standard degree, but here we convert the required percentage (75) to a standard degree by finding the percentage from the table (75) and then finding the corresponding standard degree The closest value from the table was 0.7517 and the corresponding standard score is .68 and we plug it into the equation to become X= 1800 + 300 × .68 = 2004 The 75th percentile of the income of 1,000 families is 2004 riyals, meaning that 75% of 1,000 families have income of 2004 riyals or less, and 25% of families have incomes of more than 2004 riyals. Preview distributions Sampling distributions are also called probability distributions. The population mean is and its standard deviation is. Instead of studying all the vocabulary of the community, we choose a representative random sample of the community and its mean and standard deviation is and is used to estimate the corresponding parameter (the community from which the sample was taken). In sampling distributions (probability distributions), we take a set of random samples and each group has a certain number and an arithmetic mean that may differ from other groups (for example: the average of the first group, the average of the second group, the average of the third group, ..... etc.). Thus, we have many mean values for these groups, and it is called the community of arithmetic averages or the sampling distribution of the arithmetic mean, which is a probability distribution. Then we calculate the arithmetic mean of the groups' averages. Then we calculate the arithmetic mean of the groups' averages. These principles of sampling distributions fall under the so-called central limit theorem, where the arithmetic mean of these groups (group averages) is equal to the population mean as this theory assumes. But its standard deviation varies and is calculated by the following equation: Where is the sample number and is the population standard deviation. The standard deviation of a community of arithmetic averages calculates the deviation of the means and groups and their variance or difference from one group to another. The standard deviation of the samples is always less than the value of the population standard deviation. When the sample number is large enough, we can judge that the sampling distribution of the means is a normal distribution. How many samples should be to be able to judge that the distribution of samples is a normal distribution? Most books and studies see that when the sample number is 30 or more, we can judge that the distribution of samples is a normal distribution. The previous equation is an estimate of the standard deviation of the population, and when we know the value of the standard deviation of the population, but when it is unknown (and this is mostly) and we want to estimate the standard deviation of the population, the equation differs slightly and is calculated by the following equation It is also known as the standard error And because we use samples that represent the population and not all the vocabulary of the community, we must have an error rate or percentage because we cannot obtain a representative sample of the population absolutely, completely, completely or ideally if statistical operations in general are a process of applying a statistical model (such as T-test or analysis Variance, correlation coefficient, etc.) + error rate or percentage It is an important concept when we calculate the degree of confidence and the level of statistical significance when we perform inferential statistics, which will be explained later. The higher the sample number, the lower the standard error level and the closer it is to the value of the standard deviation of the population and the characteristics of the population in general. Types of statistical samples Statistics is one of the most important sciences that helps to count data and information in an easy and correct way in order to reach the results of various research and studies carried out by researchers and scientists in various scientific fields. The results of statistics depend on the nature of the samples collected. Define the statistical sample and its types The sample that is relied upon in the research is a part of each, provided that this part expresses the same characteristics and characteristics of the whole, that is, it is a subset taken from the statistical community and has the same characteristics as this original community, and the statistical sample helps in identifying many of the The information and data associated with this statistical community, which is the subject of research and study, is done by selecting a number of elements and vocabulary that represent this community and thus conducting the study on this sample, and then generalizing the results to the entire research community. The types of statistical samples are divided into: Random Statistical Sample The random sample is a sample consisting of a specific group that is selected from the research community and has the same chance of selection, and this means that the choice can fall on any member of the community without being bound by any conditions, and the random sample is relied on in homogeneous communities that carry each He singled out all the characteristics and characteristics of this society without difference, for example conducting an experiment on skin cells or liver cells and others. stratified statistical sample This sample is relied upon if the researcher is able to identify the relative structure of this society, and here the research community consists of several layers and there are clear differences between each layer and the other in the characteristics and characteristics, and researchers resort to the stratified sample in order to represent all layers of society in the sample, and always What individuals of that sample are internally dissimilar, and externally different. An example of this is conducting an experiment on blood cells, as the blood cells may be externally similar, but in fact they are internally different; Where there are red balls and white cells and others. Cluster Statistical Sample This sample differs from the stratified sample because it follows the principle of clusters, as those elements within the sample, which are called clusters, are internally differentiated and homogeneous among themselves, and thus are in complete contrast to the stratified sample, and if this sample is applied to a practical model; We find, for example, that the clothing market has no sections, that is, all kinds of clothes are in one place only without distinguishing between women's, men's and children's clothes, without dividing the market into sections, and therefore the sample is differentiated within it, and this is exactly what happens when collecting the sample cluster; As one element or cluster carries all the characteristics of the research community, and therefore one item or one cluster is sufficient to express this community without having to choose from all clusters. Regular Statistical Sample This sample is based mainly on the number; Where the total number of the research community is divided according to the number of the sample to be collected, thus distributing the vocabulary of this community equally, and illustrative examples of this are the following: If the total number of male and female students in the College of Science is 5000, and the sample to be collected is equal to 200 students Here, students are divided into groups by dividing the total number by the sample number 5000 / 200 = 25. Hence, the numbers of the elements and vocabulary of the sample are determined, and therefore the name of the first student must be a number less than (25), for example (5), and here the sample is distributed to the rest of the students, provided that the first number is (5) and the second number (25 + 5 = 30), the third number (30 + 25 = 55), the fourth number (25 + 55 = 80) and so on until 25 students complete, and thus the researcher ensures that each of the members of the research community (students in the College of Science) is given an opportunity to be one of the sample members Statistics on a regular basis. Types of samples and questionnaire: Sample selection methods Sampling methods are classified into non-random methods and random or probabilistic methods. Non-random sampling methods The samples in this method are selective and do not represent the community correctly, but are made according to the researcher’s choice, and therefore there is no equal opportunity for the members of the community to appear in the sample, and these samples are used with the aim of obtaining exploratory results because the selection of random samples requires time, cost or efforts In these samples, it is not possible to use analytical statistics methods, which are limited to random samples. Non-random samples include the following: 1. Accidental samples: They occur when data is collected from citizens or workers in a large factory who encounter them about their attitudes towards certain commodities or towards managing a factory or its control systems in order to obtain some information and indicators at the lowest possible cost or effort. 2. Quota sampling: It occurs, for example, when the study population in a factory is divided into the administrative class and the workers class, or into females and males, and thus the proportion of subgroups in the study is taken into account. But the sample from each layer is not taken randomly, but rather the researcher chooses those he encounters. 3. Purposive sampling: It is used when studying the costs of an industry, for example, which requires cooperation from the interviewer to provide information. Random sampling methods Random sampling methods allow obtaining representative samples of the population, and the probability of withdrawing any item is known, equal and can be calculated, and therefore it is called a probability sample. Definition of random sample: It is the sample in which the probability of selecting all items is equal and known and can be calculated. There are different methods of sample selection, the most important of which are: 1- Sample random sampling A simple random sample is characterized as a subset of the original population of a certain size that has the same chance (probability) to be selected as a sample from that population. Simple random samples can be obtained using random number tables. We will illustrate the example of selecting a random sample using the tables in the lecture. 2- Systematic sampling In essence, many see the systematic sampling method as a form of simple random sampling. A regular sample is defined as a sample that is taken so that a certain number is added regularly from a randomly arranged complete list of members of the population. A regular sampling is considered a substitute for a simple random sampling for the following reasons: (a) A regular sample is easier to implement than a simple random sample. (b) A random sample can be assigned to an untrained person. Example: If we want to choose a sample of size n = 200 from a set of registration cards in a university in which N = 3000 students are registered, let us study the cards that have errors. Solution: The systematic sampling method requires that the length of the period from which the first item will be drawn in a random manner is 3000/200 = 15. So we choose a random number from 1 to 15, let it be 8. We choose the number 8 and then add 15 to the number 8 and thus withdraw the number 23, then add the number 15 to the number 23 to withdraw the number 38, and so on .... and the last card drawn is the number 2993. We note here that if the length of the period is not an integer, we round the answer to an integer. 3- Stratified random sampling This method is used when the population is divided into natural classes and we want to represent all these classes in the sample. We define a regular sample as: Definition of systematic random sample: It is the sample that is taken by dividing the population units into homogeneous strata and selecting a simple or regular random sample from each of them. The method is summarized by determining the proportional partial samples size from each stratum on the basis of the equation Stratified sample size = (class size ÷ population size) x sample size Example: If the layers of a community contain the elements as in the following table: The first layer 500 The second layer 400 The third layer 280 The fourth layer 200 The fifth layer 220 A researcher wanted to choose a sample size of 150 from this population, so what is the sample size in each stratum? Solution: Total population size = 500 + 400 + 280 + 200 + 220 = 1600 Sample size of the first stratum = 160/1600 * 500 = 50 Sample size of the second layer = 160/1600 * 400 = 40 Sample size of the third layer = 28 Fourth stratum sample size = 20 The sample size from the fifth layer = 22 Second: Data collection: Collecting data There are several ways to collect data, including: 1- Personal Interview It is that you interview the sample members and talk to them about the subject in which the research is being conducted. Thus, the amount of information that we will collect will be accurate to some extent, but its analysis will be difficult, and you should pay attention to recording the data during the interview because any mistake in recording this data leads to to an error in the results. 2- Direct Observation When there are no individuals for the sample, you use this method, that is, direct observation, and examples of it are to stop at an intersection, and prepare the cars that pass from this intersection from eight to nine in order to limit the traffic intensity at the time the employees go to work, or to By observing the behavior of a group of children while playing and taking notes in order to identify the behavior of children in some situations. 3- Questionnaire Definition: A form that includes questions about one of the topics that the researcher studies and is filled out by the respondent. Questionnaire types: 1- Closed Questionnaire: • Includes specific answers/pre-options such as yes/no, agree/disagree/neutral/etc... His problems: he delegates an opinion to the respondent/the inability to express his opinion and viewpoints/determine the answer. • Pros: Easy to analyze information. 2- Open questionnaire: • Leave the answer to the respondent without specifying prior options. • Problems: Difficulty analyzing information. • Pros: An abundance of information that we get. 3- Closed/open questionnaire: • It includes closed questions with specific options and open questions without options. • It avoids the negatives of the first and second types and takes advantage of their positives. ** Questionnaire design: 1- The importance of having experience and knowledge of how to formulate questions, their clarity, their arrangement, and their relation to the hypotheses of the study. 2- The importance of avoiding repeating words when formulating questions / avoiding long / subjective questions due to the difficulty of analyzing them. ** Design stages: • Determining the quality of information: facts/opinions/beliefs. • Questionnaire type: closed / open / direct or indirect questions. • A first draft of the questionnaire: defining the main objective, then dividing it into parts (hypotheses) and setting questions for each part/hypothesis covering this part. • Re-examination and modification of questions/consultation with those who have experience in the subject of the research. • Initial test: distributing it to a specific number of individuals and knowing their opinions and observations on the questionnaire. • Check the questionnaire and clarify how to use it. - The importance of paying attention to: - questionnaire form / formulation and order of questions / instructions for its use and how to distribute it. Recently, the possibility of using E-Mail to collect data by means of a questionnaire. • Benefits: Low cost / Distribution to a large number of people / A specific skill to complete it / Greater confidence in answering without fear / Not putting pressure on the respondent because there is a period of thought / Sending him to hard-to-reach groups / The possibility of obtaining sensitive information through him / He does not need a large number of people to distribute it Standardization of its results. • Disadvantages: Low response rate, especially for the postal sender/ Not used in an illiterate society/ Possibility of wrong answers/ Not noticing emotions/ Difficulty putting many questions/ Possibility of neglecting some options/ Unwillingness of some to answer in writing. Student mistakes: - Use the questionnaire to obtain information that can be accessed in another way. - Not to conduct a preliminary experiment to find out the observations about the questionnaire before distributing it. Putting too many questions that put pressure on the respondent and lead to him not completing the answer. Failure to verify the possibility of bias by distributing the questionnaire to people other than the specified sample and comparing the answer. the interview Definition: A purposeful conversation, directed towards a specific goal and not simply a desire to talk, including oral conversation / facial expression, movements, general behavior and tone of voice during the interview. - Its use: used to obtain broad information about a particular phenomenon, including: individuals' feelings/beliefs/future plans, their past actions. Its types, there are 3 types:1- Structured interview: It includes predetermined questions and answers in the same order, and the questions are either open or closed. A- Closed-question interview: includes yes/no answers, agree/disagree, easy and quick to analyze, accurate results, specific answers, in-depth information. B - Interview with open questions: greater freedom for the respondent, especially in complex topics such as feelings / motives, the difficulty of analyzing them. C - open and then closed questions or vice versa: to take advantage of the advantages of the two methods and avoid their disadvantages. 2- Free interview: - unstructured/has a goal, but the formulation of questions is left to the interviewer. By which we obtain information about trends/motivations/perceptions that the structured interview does not provide. It has types, including: focused interview / clinical interview / unguided interview. 2- Telephone interview: - It is done by phone. - Easy to perform in terms of cost / time / effort. - Do not cause embarrassment. The importance of distributing the sample appropriately in large geographical areas. - Good for female students to use in light of the social situation in some countries and not to embarrass them. Her problems: - Not noticing and knowing facial and body expressions. - Possibility to cut off the call suddenly. Difficulty verifying the accuracy of the information. Questioning and Interview Arrangements: - Setting questions: determining their type and clarity / respondent knowledge of the subject / keeping embarrassing questions to the end / not putting questions known to answer / directing questions and answering as they were prepared in advance with putting the same expressions / problems of using recorders and people's sensitivity to that. - The sample and how to contact it and not bias in that like choosing people we are comfortable with The content of the questions and their identification: do they include facts/actions/belief about a fact/feeling/behaviours, past and present? Interview arrangements: The researcher’s impartiality / Appearance appropriate to the group you are interviewing with / Full knowledge of the questions / How to ask questions and record the answer / Create a comfortable atmosphere to answer questions / Present himself and then enter into the subject of the interview / Neutral so that it does not affect the interpretation of the questions. Benefits of the interview: - Used with all groups / Better sample of the community / Collaboration during conversation / Obtaining emotional and complex information / Confidence in information / Making sure questions are understood and answered. its difficulties: Individual questioning for each person/ The difficulty of reaching some groups/ The extent of the person’s desire to speak/ The researcher’s tendency to obtain answers in support of his point of view, Students' mistakes when conducting interviews: Use of unclear or incomprehensible expressions. Unclear business plan for the interview. The possibility of bias of the interrogator. No trial interviews. Asking questions about information he does not know or is not known to the respondent. Observation Definition: recording and regulating behaviors related to the purpose of the study and its hypotheses. Its uses: first in the natural sciences, then moved to social sciences, used in evaluating programs prepared by government institutions, observing the performance and behavior of employees in government and private institutions. The importance of planning it in an organized manner / and that it is linked to a hypothesis that can be checked. At the present time, the natural behavior of the research subject is being monitored. There are two types of observation: 1- Simple Observation: Describing what happens in front of the observer as it happens without developing a prior plan for actions and their repetition/describing the participants/event/place/and the behaviors, the importance of impartiality of the observer, and the importance of the influence of the observer’s bias to describe his feeling instead of what is happening in front of him Example: An announcer’s description of a match running in front of him. 2- Structured observation: we predetermine the behaviors, their types, and their frequency / we test hypotheses / the observer chooses what he wants to watch so that it relates to the hypotheses / goal. - The importance of recording information accurately and correctly / Adjusting the information and writing it in a clear language and reducing errors / Establishing a system for classifying information / The importance of paying attention to overestimation. - The importance of clarifying the procedural definitions of the variables and then specifying the behaviors that you want to observe. - Example: the match referee determines the behaviors that he must observe and the extent of their recurrence (number of goals, infractions, expulsions, etc...) Ethical aspects of the note: - Hiding the personality of the researcher: so as not to affect the results of the research / the ability to reach groups that you do not want to notice. Disadvantages: Possibility of bias and deception Intervention of the researcher to influence the behavior of the researchers, whether intentionally or unintentionally. The problem of concealing the researcher's personality in whole or in part - Methods and technology observed: - Multiple means have been developed to increase the accuracy of observation and record what is happening with cameras or microphones / record conversations remotely. The extent of commitment to the ethical aspects of the observation process and not exceeding the limits so that it becomes a violation of the privacy of individuals and the need for the researcher and whoever helps him in the observation process to pay attention to these matters.