Monthly Archives: July 2014

Hypothesis Testing for Two Means: Large Independent Samples

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Hypothesis testing for two large samples examines again if there is a difference between the two means. We infer that there is a difference between the population means by seeing if there is a difference between the sample means. The assumptions for testing for the difference between two means are below.

  • Subjects are randomly selected and independently assigned to groups
  • Population is normally distributed
  • Sample size is greater than 30

The hypotheses can be stated as follows

  • Null hypothesis: There is no difference between the population means of the two groups
    • The technical way to say this is…  H0: μ1 = μ2
  • Alternative hypothesis: There is a difference between the population means of the two groups. One is greater or smaller than the other
    • The technical way to say this is… H1: μ1≠ μ2 or μ1> μ2 or         μ1< μ2

The process for conducting a z test for independent samples is provided below

  1. Develop your hypotheses
  2. Determine the level of significance (normally .1, .05, or .01)
  3. Decide if it is a one-tail or two tail test.
  4. Determine the critical value of z. This is found in chart in the back of most stat books common values include +1.64, +1.96, or +2.32
  5. Calculate the means and standard deviations of the two samples.
  6. Calculate the test for the two independent samples. Below is the formula

z = (sample mean 1 – sample  mean 2)

√[(variance of sample 1 squared/ sample population 1) +
(variance  of sample 2 squared/ sample population 2)]

7. If the computed z is less than the critical z then you do not reject your null hypothesis. This means there is no difference between the means. If the computed z is greater than the critical z then you reject the null hypothesis and this indicates that there is evidence that there is a difference.

Below is an example

A business man is comparing the price of buildings in two different provinces to see if there is a difference. Below are the results. Determine if the buildings in Bangkok cost more than the buildings in Saraburi.

Bangkok                                   Saraburi
average price     2,140,000                                1,970,000
variance                 226,000                                     243,000
sample size           47                                                  45

Now let us go through the steps

  1. Develop your hypotheses
    • Null hypothesis: There is no difference between the average price of buildings in Bangkok and Saraburi
      • In stat language, it would be
      • H0: μ1 ≠ μ2
    • Alternative hypothesis: The  average price of buildings in Bangkok is higher than in  Saraburi
      • In stat language, it would be
      • H1: μ1 > μ2
  2. Determine the level of significance (normally .1, .05, or .01)
    • We will select .05
  3. Decide if it is a one-tail or two tail test.
    • This is a one-tail test. We want to know if one mean is greater than another. Therefore, to reject the null we need a z computed that is positive and larger than our z critical.
  4. Determine the critical value of z. This is found in chart in the back of most stat books common values include +1.64, +1.96, or +2.32 when it is a two tailed test
    • Our z critical is + 1.64  since this is a one-tail test we only have one value so we do not split the probable and place have on one side and half on the other side. If this were two-tailed we would have -1.96 and +1.96 which indicates that the difference is greater or less
  5. Calculate the means and standard deviations of the two samples.
    • Already done in the table above
  6. Calculate the test for the two independent samples. Below is the formula.

(2,140,000 – 1,970,000)
√[((226,000)²)/47) + ((243,000)²)/45)]
our final answer for are z computed is 3.47

Since 3.47 is greater than our z critical of +1.64 we reject the null hypothesis and state that there is evidence that building prices are higher in Bangkok than in Saraburi.

What is a One Sample z Test?

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There are actually several different situations in which a researcher can use hypothesis testing. The first instance we will look at is the one sample z test. The one sample z test has the following assumptions that need to be met before employing it.

  • Sample size > 30
  • Subjects are randomly selected
  • Population is normally distributed
  • Cases within the sample are independent
  • One sample was taken

If your data collection meets the above assumptions one sample z test may be appropriate.

With the one sample z test, you are comparing your results to a known expected value. For example, if someone states that the average salaries for teachers are $63,000.00 you can assess this by collecting data from teachers to compare it to this known value. You collect some data and you find that the average salary for 35 teachers was $65,7000.00. The questions you have is who is right? Do teachers really make on average $63,000.00 like the report or do they make $65,700.00 as my data says? Before going further let us establish are hypotheses for this example.

  • Null hypothesis: the average salaries for my sample of teacher salaries will be the same as the average salary’s of the reported value of $63,000.00
    1. The mathematical shorthand for this is H0: μ = 63,000.00
  • Alternative hypothesis: the average salaries for my sample of teacher salaries will be different (greater or lesser) than the average salary’s of the reported value of $63,000.00
    1. The mathematical shorthand for this is H1: μ ≠ 63,000.00

Keep in mind that this is a two-tail level of significance. This is because our final value has the option of being either greater or lesser than $63,000.00. Two-tail means two options, greater or lesser than the expected value while one-tail means only one option either we expect greater or we expected lesser but not both. This is why we will have two z critical values to think about in the near future.

We also need two more pieces of information before we put our numbers into the equation. The two items we need to know are the standard deviation of the sample and the level of statistical significance. For the sample data, we collected we will say the standard deviation is $5,250.00 and the level of statistical significance is α = 0.01. When we convert this alpha value to the z critical value we get 2.32 and -2.32 because we are using a two-tail or two option approach. Do not get distracted by the z critical value it is the same as the alpha value but translated for the numbers set to the normal distribution. It is similar to switching from one language to another, same meaning but different language.

If our final value is greater than 2.32 or less than -2.32 we will reject the null hypothesis that average teacher salaries are $63,000.00. Now we can take a look at the equation

z critical value = sample data – expected value                                                                                           Sample standard deviation / square root of the                                                    number of those in the sample population

In simple English

z critical value  = 65,700 – 63,000                                                                                                                         5,250 / square root of 35

Z critical value = 3.04

Our answer is 3.04, which is greater than +2.32. This indicates that we can reject the null hypothesis that the average salary teachers are $63,000.00 as our data indicate that there is evidence that teachers make more on average.

We don’t want to get too excited here. We found evidence that teachers make more but further testing would be needed to validate these claims. As more data confirms our findings we can confidently state that teachers make more.

I would like to thank andydevil12 for the question and suggestion. If there are any other questions please send them to me as they help me to understand research and statistics much better as well.

What is Hypothesis Testing?

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Hypothesis testing is a statistical approach used in making decisions about data.  In hypothesis testing, there are two hypotheses that are posed by the researcher and they are…

  1. Null hypothesis-There is no difference between the sample population and the statistical population in relation to the mean or some other parameter that is being assessed
  2. Alternative hypothesis-There is a difference between the sample population and the statistical population in relation to the mean or some other parameter that is being assessed

Generally, researchers often hope to reject the null hypothesis which indicates that the alternative hypothesis is correct.  However, strictly speaking, a researcher never accepts any hypothesis. Instead, you reject or you do not reject the null hypothesis. This is because further testing will always be needed to confirm the results.

How to know whether to reject or not reject the null depends on the results of the analysis. A researcher needs to select a level of statistical significance which is usually 1%, 5%, or 10%. The significance level changes the size of the rejection region at the tails of the normal distribution. The lower the significance level the smaller the rejection region which influences the interpretation of the results.  To reject a null hypothesis, the results of the analysis must fall within the rejection region.

After determining the level of significance a researcher analyzes the data to determine the results. The results then need to be interpreted by stating them in simple English.  From this, the researcher can develop a conclusion about what the results mean.

Direct Instruction

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Direct instruction is probably one of the most common ways of teaching in the classroom. This model is useful for delivering large amounts of information to students. It is particularly useful for stimulating lower-level thinking and can serve as a foundation for going into more complex thinking in the future. There are six steps to direct instruction which are…

  1. Review of prior knowledge needed for the current lesson
    • Connect yesterday’s learning with today’s
  2. Presentation of new knowledge
    • Give an overview of what is going on and move into the topic
  3. Guided practice
    • Whatever you teach them they need to practice it without being formally evaluated
  4. Feedback
    • Tell them how they are doing during the guided practice
  5. Independent practice
    • Students work separately from the teacher. It is an opportunity to demonstrate mastery
  6. Weekly review
    • Reteach content occasionally to deepen understanding.

Direct instruction has a bad reputation in a world that is focused on student-centered learning and application. The student is very passive in direct instruction which is the main compliment of this teaching model. However, it is just one of many tools that a teacher has at their discretion. Leaning on any approach exclusively has its drawbacks. As such, occasional use of direct instruction can be beneficial just as the occasional use of any approach.

Types of Objectives

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Within education,  the majority of objectives used in curriculum are behavioral. Behavioral objectives are actions the student performs that are measurable and observable.  As shared previously, objectives need to have an action, condition, and proficiency in order to meet the general criteria of being an objective.

In order to develop various objectives that are able to assess different aspects of the educational experience, researchers have developed three domains of learning. The domains are cognitive, affective, and psychomotor.  Each domain has several levels of behavior from simple to complex and within each level, there are many verbs that can be used to develop the active component of an objective.

Cognitive Domain (Bloom’s Taxonomy)

The cognitive domain has six levels that address cognitive learning. Verbs for each level can be found on the internet

  1. Knowledge-Recall specific facts
  2. Comprehension-Interpreting or summarizing information
  3. Application-Using knowledge in a different setting
  4. Analysis-Breaking a whole into parts to identify relationships
  5. Synthesis-Combining ideas into a new concept
  6. Evaluation-Making judgments based on criteria

This is the most famous domain of the three and the majority of objectives for many curricula are derived from it.

Affective Domain

The affective domain looks at the values, beliefs, and attitudes of students. This domain has 5 levels.

  1. Receiving-Awareness of a stimulus
  2. Responding-Paying attention to a stimulus
  3. Valuing-Showing preference for something
  4. Organization-Developing a system of values
  5. Characterization-Consistency of internal beliefs with behavior

Psychomotor Domain

This domain focuses on movement and mastery of action. It is divided into five levels.

  1. Imitation-Seeing a behavior and duplicating it
  2. Manipulation-Performing an action by hearing or reading about it but not seeing it
  3. Precision-Performing an action without seeing, hearing or reading about it. Takes practice and precision
  4. Articulation-Performing a series of actions accurately
  5. Naturalization-Action is complex yet performed automatically with little effort

Each of these domains is appropriate in the classroom depending on the needs of the student and teacher’s interest.

Creating Objectives

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Objectives are more specific than goals or aims in terms of indicating exactly what the students will do. They guide the instruction of the teacher and help to maintain consistency within the curriculum. Objectives are derived from behaviorism and they must be observable and measurable.

There are three components to objectives, which are

  • action-what the student will do
  • condition-in what context the student will do it (optional)
  • proficiency-the minimum level of mastery expected

An example of an objective is provided below

Using a calculator, the student will solve the mathematical word problems with at least 80% accuracy.

An analysis of this objective indicates that it has all three components. Below is an analysis of the objective.

  • Action-The student will solve mathematical word problems
  • Condition-using a calculator
  • Proficiency-At least 80% accuracy

It does not matter what order you put these three components in as long as they are present. For example,

  • The student will solve the mathematical word problems with at least 80% accuracy using a calculator.
  •  With at least 80% accuracy, the student will solve the mathematical word problems using a calculator.
  • etc.

Along with these three components. Objectives need to be clear, appropriate and logically to ensure student success.

Generating Goals

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After aims are developed the next step is to develop goals. The difference between aims and goals is how specific they are. Aims are the broadest statements about the philosophy of the school while goals provide a vision of the destination or results of learning.

We are going to develop a goal from the aims of the previous post. Below are the aims of the previous post in a philosophy statement

  • School A supports that students need to be provided with the tools necessary to learn continuously (intellectual aim) through a stimulating social environment that encourages collaboration (social-personal aim), which prepares students to be active members of the workplace and society (productive aim)

Now we will see one potential goal derived from the intellectual aim of the philosophy statement.

  • Students will develop fluency in their language

The question to ask yourself is whether developing fluency in one’s language is a tool for continuous learning.  The purpose is not to agree on the appropriateness of the goal but to see that it was inspired by the intellectual aim of the philosophy statement. The aim provides a general direction while the goal provides a way to achieve the aim. This process of deriving goals from aims helps in maintaining consistency within the curriculum.

The next question is how will the students develop fluency in their language? This question is answered when objectives are developed, which will be the discussion of the next post.

Developing a Philosophy

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One of the first steps in curriculum development, regardless of the type of approach one takes, is the development of a philosophy or mission statement. A school’s philosophy/mission provides a framework for the purpose of the school and what the stakeholders believe is important.

The philosophy comes from the stakeholders in the local community. One way to develop a concise philosophy is to develop several different aims for the school. Aims are in many ways statements that provide direction and reflect the values of the stakeholders. Several aims in a paragraph can be used to develop a philosophy/mission statement of a school.

There are several types of aims such as intellectual, social-personal, productive, physical, moral, and spiritual.   Intellectual aims focus on the development of the mind. Social-personal aims focus on relationships. Productive aims center on functioning in the workplace. Physical aims are about the development of the body. Moral aims are about deciphering right and wrong.  Lastly, spiritual aims relate to relating to God.

Which types of aims to use to develop a philosophy depends on the local context. Aims should be exceedingly broad and vague intentionally as the details of the curriculum come at the goals and objectives level. An example of an aim is the following.

  • Provide the tools needed for continuous learning (intellectual aim)

What this means would be hashed out in further details in another part of the curriculum.

An example of a philosophy statement would be the following

School A supports that students need to be provided with the tools necessary to learn continuously (intellectual aim) through a stimulating social environment that encourages collaboration (social-personal aim), which prepares students to be active members of the workplace and society (productive aim)

In this statement, you can see three aims spliced together in one statement on what the school values. This is not the only way to approach this process but it serves as an example of how this could be done

Curriculum Development: Non Scientific Approach

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The emphasis so far has been on scientific approaches to curriculum development, which is considered rational, universal, and objective. However, a minority of educators support a non-scientific approach to curriculum development, which is seen as personal, subjective, transactional, and aesthetic.  Supporters of non-scientific models see learning as a holistic process rather than as segmented subjects.

One model under the non-scientific approach is the Deliberation model. This model has six steps.

  1. Public sharing
  2. Highlighting agreement and disagreement
  3. Explaining position
  4. Highlighting change in position
  5. Negotiating points of agreement
  6. Adopting a decision

Step one is focused on sharing ideas about the curriculum to be developed. For example, if there is a need for a new English curriculum someone may suggest that public speaking should be a part of the new curriculum.

Step two is where people discuss agreement and disagreement. Should public speaking be a part of the new curriculum? If so, how should it be taught, what evidence should the students provide, and how much public speaking should the students do? These are some of the questions and objectives discuss here. People support or attack the ideas developed.

Step three is where people provide support for the position. If I am a supporter of public speaking I might show that companies are now looking for people who are articulate and can express themselves in front of a group. Such evidence builds credibility for change. Here the Deliberation model is showing traits of rationalism. This is why it is not simple to put any model in one approach or another.

Step four is where consensus takes place. People have presented their arguments and evidence. Now decisions are made about what to do. The group decides if public speaking is going to be a part of the new English curriculum.

Step 5 is when the group works at the details of the agreement reached in step 4. How will we teach public speaking and other questions are now answered in detail.  In other words, the curriculum is now formally developed.

Step 6 the curriculum is finalized and ready for use.

Non-scientific approaches are not common. However, it is necessary to provide some idea to alternatives to scientific approaches. Teachers need to decide for themselves what is the most appropriate form of curriculum development for their students.

Curriculum Development: Backward Design

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The Tyler model is the “way” of developing curriculum for most of the past 70 years. One variation of the Tyler model is the Backward Design model by Wiggins and McTighe. The model is backward because it changes the order of the steps in the Tyler model.

Let’s review the four steps of the Tyler model.

  1.  Develop objectives
  2. Identify experiences related to objectives
  3. Organize experiences
  4. Evaluate objectives

The Backward Design model simplifies the already simple Tyler model. The steps are below.

  1. Develop objectives (aka outcomes)
  2. Determine evidence that objectives are met (this is step four in Tyler’s model)
  3. Identify learning experiences related to objectives (this is step two in Tyler’s model)

The changes are as follows. Step four and two in Tyler’s model have switch places. Step Four jump to step two in the Backward Design and step two moves to step three. The original step three in the Tyler model is removed because organization is assumed in Backward Design. Backward Design is backward because developing assessment now comes before determining learning experiences. This is backward from the perspective of the Tyler model

The reason for developing assessment before learning experiences is that by creating your assessment first, it helps to make sure that your learning experiences are consistent with the assessment you developed in advanced. Many times, teachers beginning teaching and create their assessment at the last minute. By doing this, sometimes the assessment addresses concepts that were not taught. By developing the assessment first, it helps the teacher to know what they need to cover in their learning experiences with the students.

The three steps of Backward Design are a small improvement in the Tyler model. Most accredited K-12 schools in the US use this model for developing curriculum.  These simple steps of developing objectives, determining your assessment, and planning learning experiences, is a practical model that is used extensively.

Curriculum Development: The Tyler Model

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The Tyler Model, developed by Ralph Tyler in the 1940s, is the quintessential prototype of curriculum development in the scientific approach. One could almost dare say that every certified teacher in America and maybe beyond has developed a curriculum directly or indirectly using this model or one of the many variations.

Tyler did not intend for his contribution to the curriculum to be a lockstep model for development. Originally, he wrote down his ideas in the book Basic Principles of Curriculum and Instruction for his students to give them an idea about principles for making a curriculum. The brilliance of Tyler’s model is that it was one of the first models and it was and still is a highly simple model consisting of four steps.

  1. Determine the school’s purposes (aka objectives)
  2. Identify educational experiences related to purpose
  3. Organize the experiences
  4. Evaluate the purposes


Basic Principles of Curriculum and Instruction

Step one is determining the objectives of the school or class. In other words, what do the students need to do to be successful? Each subject has natural objectives that are indicators of mastery. All objectives need to be consistent with the philosophy of the school and this is often neglected in curriculum development. For example, a school that is developing an English curriculum may create an objective that students will write essays. This would be one of many objectives within the curriculum.

Step two is developing learning experiences that help the students to achieve step one. For example, if students need to meet the objective of writing an essay. The learning experience might be a demonstration by the teacher of writing an essay. The students then might practice writing essays. The experience (essay demonstration and writing) is consistent with the objective (Student will write an essay).

Step three is organizing the experiences. Should the teacher demonstrate first or should the students learn by writing immediately? Either way could work and preference is determined by the teacher’s philosophy and the students’ needs. The point is that the teacher needs to determine a logical order of experiences for the students.

Lastly, step four is the evaluation of the objectives. Now the teacher assesses the students’ ability to write an essay. There are many ways to do this. For example, the teacher could have the students write an essay without assistance. If they can do this, it is evidence that the students have achieved the objective of the lesson.

There are variations on this model. However, the Tyler model is still considered by many to be the strongest model for curriculum development.

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