greater than (>) less than (<)
H 0 always has a symbol with an equal in it. H a never has a symbol with an equal in it. The choice of symbol depends on the wording of the hypothesis test. However, be aware that many researchers (including one of the co-authors in research work) use = in the null hypothesis, even with > or < as the symbol in the alternative hypothesis. This practice is acceptable because we only make the decision to reject or not reject the null hypothesis.
H 0 : No more than 30% of the registered voters in Santa Clara County voted in the primary election. p ≤ 30
H a : More than 30% of the registered voters in Santa Clara County voted in the primary election. p > 30
A medical trial is conducted to test whether or not a new medicine reduces cholesterol by 25%. State the null and alternative hypotheses.
H 0 : The drug reduces cholesterol by 25%. p = 0.25
H a : The drug does not reduce cholesterol by 25%. p ≠ 0.25
We want to test whether the mean GPA of students in American colleges is different from 2.0 (out of 4.0). The null and alternative hypotheses are:
H 0 : μ = 2.0
H a : μ ≠ 2.0
We want to test whether the mean height of eighth graders is 66 inches. State the null and alternative hypotheses. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses. H 0 : μ __ 66 H a : μ __ 66
We want to test if college students take less than five years to graduate from college, on the average. The null and alternative hypotheses are:
H 0 : μ ≥ 5
H a : μ < 5
We want to test if it takes fewer than 45 minutes to teach a lesson plan. State the null and alternative hypotheses. Fill in the correct symbol ( =, ≠, ≥, <, ≤, >) for the null and alternative hypotheses. H 0 : μ __ 45 H a : μ __ 45
In an issue of U.S. News and World Report , an article on school standards stated that about half of all students in France, Germany, and Israel take advanced placement exams and a third pass. The same article stated that 6.6% of U.S. students take advanced placement exams and 4.4% pass. Test if the percentage of U.S. students who take advanced placement exams is more than 6.6%. State the null and alternative hypotheses.
H 0 : p ≤ 0.066
H a : p > 0.066
On a state driver’s test, about 40% pass the test on the first try. We want to test if more than 40% pass on the first try. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses. H 0 : p __ 0.40 H a : p __ 0.40
In a hypothesis test , sample data is evaluated in order to arrive at a decision about some type of claim. If certain conditions about the sample are satisfied, then the claim can be evaluated for a population. In a hypothesis test, we: Evaluate the null hypothesis , typically denoted with H 0 . The null is not rejected unless the hypothesis test shows otherwise. The null statement must always contain some form of equality (=, ≤ or ≥) Always write the alternative hypothesis , typically denoted with H a or H 1 , using less than, greater than, or not equals symbols, i.e., (≠, >, or <). If we reject the null hypothesis, then we can assume there is enough evidence to support the alternative hypothesis. Never state that a claim is proven true or false. Keep in mind the underlying fact that hypothesis testing is based on probability laws; therefore, we can talk only in terms of non-absolute certainties.
H 0 and H a are contradictory.
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Published on 5 October 2022 by Shaun Turney . Revised on 6 December 2022.
The null and alternative hypotheses are two competing claims that researchers weigh evidence for and against using a statistical test :
The effect is usually the effect of the independent variable on the dependent variable .
Answering your research question with hypotheses, what is a null hypothesis, what is an alternative hypothesis, differences between null and alternative hypotheses, how to write null and alternative hypotheses, frequently asked questions about null and alternative hypotheses.
The null and alternative hypotheses offer competing answers to your research question . When the research question asks “Does the independent variable affect the dependent variable?”, the null hypothesis (H 0 ) answers “No, there’s no effect in the population.” On the other hand, the alternative hypothesis (H A ) answers “Yes, there is an effect in the population.”
The null and alternative are always claims about the population. That’s because the goal of hypothesis testing is to make inferences about a population based on a sample . Often, we infer whether there’s an effect in the population by looking at differences between groups or relationships between variables in the sample.
You can use a statistical test to decide whether the evidence favors the null or alternative hypothesis. Each type of statistical test comes with a specific way of phrasing the null and alternative hypothesis. However, the hypotheses can also be phrased in a general way that applies to any test.
The null hypothesis is the claim that there’s no effect in the population.
If the sample provides enough evidence against the claim that there’s no effect in the population ( p ≤ α), then we can reject the null hypothesis . Otherwise, we fail to reject the null hypothesis.
Although “fail to reject” may sound awkward, it’s the only wording that statisticians accept. Be careful not to say you “prove” or “accept” the null hypothesis.
Null hypotheses often include phrases such as “no effect”, “no difference”, or “no relationship”. When written in mathematical terms, they always include an equality (usually =, but sometimes ≥ or ≤).
The table below gives examples of research questions and null hypotheses. There’s always more than one way to answer a research question, but these null hypotheses can help you get started.
( ) | ||
Does tooth flossing affect the number of cavities? | Tooth flossing has on the number of cavities. | test: The mean number of cavities per person does not differ between the flossing group (µ ) and the non-flossing group (µ ) in the population; µ = µ . |
Does the amount of text highlighted in the textbook affect exam scores? | The amount of text highlighted in the textbook has on exam scores. | : There is no relationship between the amount of text highlighted and exam scores in the population; β = 0. |
Does daily meditation decrease the incidence of depression? | Daily meditation the incidence of depression.* | test: The proportion of people with depression in the daily-meditation group ( ) is greater than or equal to the no-meditation group ( ) in the population; ≥ . |
*Note that some researchers prefer to always write the null hypothesis in terms of “no effect” and “=”. It would be fine to say that daily meditation has no effect on the incidence of depression and p 1 = p 2 .
The alternative hypothesis (H A ) is the other answer to your research question . It claims that there’s an effect in the population.
Often, your alternative hypothesis is the same as your research hypothesis. In other words, it’s the claim that you expect or hope will be true.
The alternative hypothesis is the complement to the null hypothesis. Null and alternative hypotheses are exhaustive, meaning that together they cover every possible outcome. They are also mutually exclusive, meaning that only one can be true at a time.
Alternative hypotheses often include phrases such as “an effect”, “a difference”, or “a relationship”. When alternative hypotheses are written in mathematical terms, they always include an inequality (usually ≠, but sometimes > or <). As with null hypotheses, there are many acceptable ways to phrase an alternative hypothesis.
The table below gives examples of research questions and alternative hypotheses to help you get started with formulating your own.
Does tooth flossing affect the number of cavities? | Tooth flossing has an on the number of cavities. | test: The mean number of cavities per person differs between the flossing group (µ ) and the non-flossing group (µ ) in the population; µ ≠ µ . |
Does the amount of text highlighted in a textbook affect exam scores? | The amount of text highlighted in the textbook has an on exam scores. | : There is a relationship between the amount of text highlighted and exam scores in the population; β ≠ 0. |
Does daily meditation decrease the incidence of depression? | Daily meditation the incidence of depression. | test: The proportion of people with depression in the daily-meditation group ( ) is less than the no-meditation group ( ) in the population; < . |
Null and alternative hypotheses are similar in some ways:
However, there are important differences between the two types of hypotheses, summarized in the following table.
A claim that there is in the population. | A claim that there is in the population. | |
| ||
Equality symbol (=, ≥, or ≤) | Inequality symbol (≠, <, or >) | |
Rejected | Supported | |
Failed to reject | Not supported |
To help you write your hypotheses, you can use the template sentences below. If you know which statistical test you’re going to use, you can use the test-specific template sentences. Otherwise, you can use the general template sentences.
The only thing you need to know to use these general template sentences are your dependent and independent variables. To write your research question, null hypothesis, and alternative hypothesis, fill in the following sentences with your variables:
Does independent variable affect dependent variable ?
Once you know the statistical test you’ll be using, you can write your hypotheses in a more precise and mathematical way specific to the test you chose. The table below provides template sentences for common statistical tests.
( ) | ||
test
with two groups | The mean dependent variable does not differ between group 1 (µ ) and group 2 (µ ) in the population; µ = µ . | The mean dependent variable differs between group 1 (µ ) and group 2 (µ ) in the population; µ ≠ µ . |
with three groups | The mean dependent variable does not differ between group 1 (µ ), group 2 (µ ), and group 3 (µ ) in the population; µ = µ = µ . | The mean dependent variable of group 1 (µ ), group 2 (µ ), and group 3 (µ ) are not all equal in the population. |
There is no correlation between independent variable and dependent variable in the population; ρ = 0. | There is a correlation between independent variable and dependent variable in the population; ρ ≠ 0. | |
There is no relationship between independent variable and dependent variable in the population; β = 0. | There is a relationship between independent variable and dependent variable in the population; β ≠ 0. | |
Two-proportions test | The dependent variable expressed as a proportion does not differ between group 1 ( ) and group 2 ( ) in the population; = . | The dependent variable expressed as a proportion differs between group 1 ( ) and group 2 ( ) in the population; ≠ . |
Note: The template sentences above assume that you’re performing one-tailed tests . One-tailed tests are appropriate for most studies.
The null hypothesis is often abbreviated as H 0 . When the null hypothesis is written using mathematical symbols, it always includes an equality symbol (usually =, but sometimes ≥ or ≤).
The alternative hypothesis is often abbreviated as H a or H 1 . When the alternative hypothesis is written using mathematical symbols, it always includes an inequality symbol (usually ≠, but sometimes < or >).
A research hypothesis is your proposed answer to your research question. The research hypothesis usually includes an explanation (‘ x affects y because …’).
A statistical hypothesis, on the other hand, is a mathematical statement about a population parameter. Statistical hypotheses always come in pairs: the null and alternative hypotheses. In a well-designed study , the statistical hypotheses correspond logically to the research hypothesis.
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Updated: July 5, 2023 by Ken Feldman
Hypothesis testing is a branch of statistics in which, using data from a sample, an inference is made about a population parameter or a population probability distribution .
First, a hypothesis statement and assumption is made about the population parameter or probability distribution. This initial statement is called the Null Hypothesis and is denoted by H o.
An alternative or alternate hypothesis (denoted Ha ) is then stated, which will be the opposite of the Null Hypothesis.
The hypothesis testing process and analysis involves using sample data to determine whether or not you can be statistically confident that you can reject or fail to reject the H o. If the H o is rejected, the statistical conclusion is that the alternative or alternate hypothesis Ha is true.
Hypothesis testing applies to all forms of statistical inquiry. For example, it can be used to determine whether there are differences between population parameters or an understanding about slopes of regression lines or equality of probability distributions.
In all cases, the first thing you do is state the Null Hypothesis. The word “null” in the context of hypothesis testing means “nothing” or “zero.”
If we wanted to test whether there was a difference in two population means based on the calculations from two samples, we would state the Null Hypothesis in the form of:
Ho: mu1 = mu2 or mu1 – mu2 = 0
In other words, there is no difference, or the difference is zero. Note that the notation is in the form of a population parameter, not a sample statistic.
The analysis of the Null Hypothesis is designed to test the Null, which will determine whether the Null should be rejected so that the Alternate Hypothesis is defaulted to and assumed to be true, or not to reject the Null so it is assumed to be the true condition.
A classic example is when you get the results back from your doctor after taking a blood test. The Null is written to state that there is no infection. Remember, the Null is always in the form of “nothingness.” The alternate hypothesis is that you have an infection. Once the test is done, the lab will determine whether the Null can be rejected or not. If the test shows an infection, the Null will be rejected, and the Alternate will be assumed to be true. If the test shows no infection, we cannot reject the Null.
While the Null can only be written in one form — “equal to” or “no difference” — the Alternate can be written for three conditions. For example, a marketing director wants to improve sales. She designs and launches a new social media campaign, collecting sample data for sales activity prior to the new campaign. After six months, sample sales data was collected to determine whether the campaign was successful. Hypothesis testing was used to statistically confirm whether the campaign was successful or not. The Null Hypothesis was written as: Ho: muBefore = muAfter, where the claim was that the population average sales before the campaign is the same as the population average sales after the campaign. In other words, the campaign had no effect on sales.
The Alternate Hypothesis can now be written in one of three forms:
The stating and testing of the Null and the default to the Alternate hypothesis is the foundation of hypothesis testing. By doing so, you set the parameters for your statistical inference.
Just looking at the mathematical difference between the means of two samples and making a decision is woefully inadequate. By statistically testing the Null hypothesis, you will have more confidence in any inferences you want to make about populations based on your samples. If you reject the Null, you’ll know which Alternate is most appropriate.
Many statistical tests require assumptions of specific distributions. Many of these tests assume that the population follows the normal distribution . If it doesn’t, the test may be invalid. Rejecting the Null will establish whether the estimated distribution fulfills any test assumption or not.
Hypothesis testing calculations will provide some relative strength to your decisions as to whether you reject or fail to reject the Null hypothesis and, therefore, the Alternate.
It is the interpretation of the statistics relative to the Null and Alternate hypotheses that is important.
The Alternate can take the form of “not equal,” “less than,” or “greater than.”
You don’t want to place too big of a hurdle (or burde)n on your decision-making relative to action on the Null hypothesis by selecting an alpha value that is too high or too low. The Alternate can really reflect the true condition of the population, so failing to reject the Null too often can mask the truth.
Since your decision relative to rejecting or not rejecting the Null impacts the Alternate Hypothesis, it’s important to understand how that decision works.
The director of manufacturing at a medium-size window manufacturer recently had an older machine retrofitted to increase run speed of the equipment. The supplier, after doing the retrofit, claimed that the machine was now running significantly faster. He showed a comparison of the sample average run speed before the retrofit and the sample average run speed after the retrofit. Looking at the two averages, it appeared that the supplier was correct and that the retrofit did indeed increase the speed.
However, having had some training in Lean Six Sigma, the director asked his local Black Belt for some help in doing a hypothesis test on the data to see if there was truly a statistically significant improvement of at least 100 rpm. The Null hypothesis was written as:
mu1-mu2= -100 rpms
That is the Before speed minus the After speed equaled -100 rpms. This was slightly different than the format he was used to seeing, where the Null would be reflective of “no difference” rather than a value of interest.
The Alternate hypothesis was written to reflect an increase of at least 100 rpms. That form was written as:
mu1 – mu2 greater -100
That would mean the After speed was more than 100 rpms faster than the Before speed. Fortunately, the results indicated that the Null hypothesis should be rejected and that in actuality, the difference was greater than the 100 rpms he wanted.
Using hypothesis testing to help make better data-driven decisions requires that you properly address the Null and Alternate Hypotheses.
The Alternate should be in the form of “not equal to,” “greater than,” “less than,” or “equal to some value of interest.”
The writing of the Alternate Hypothesis can vary, so be sure you understand exactly what condition you are testing against.
Being too cautious will lead you to not reject the Null enough so you will never learn anything about your population data.
What form should the Alternate Hypothesis be written in?
The Alternate Hypothesis should be in the form of “not equal to,” “greater than,” “less than,” or “equal to some value of interest.”
When do you default to the Alternate Hypothesis?
If your analysis of the sample data suggests that you should reject the Null Hypothesis, you will default to the statement of the Alternate Hypothesis.
Should I be disappointed if I default to the Alternate Hypothesis?
Usually no. Typically, you want the actions you took to have had some impact or effect. Rejecting the Null and defaulting to the Alternate signals that something did, in fact, happen — and that might be a good thing.
The Alternate Hypothesis is the default should you reject the Null. It is an indication that something has happened that is significant. Often, that is what you want to see if you’re comparing a before and after situation.
While the form of the Null Hypothesis is usually written in a single format, the format of the Alternate can be written a number of different ways. This provides more sensitivity to your interpretation of the population data and will therefore provide a richer insight for decision-making.
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Methodology
Published on May 6, 2022 by Shona McCombes . Revised on November 20, 2023.
A hypothesis is a statement that can be tested by scientific research. If you want to test a relationship between two or more variables, you need to write hypotheses before you start your experiment or data collection .
Daily apple consumption leads to fewer doctor’s visits.
What is a hypothesis, developing a hypothesis (with example), hypothesis examples, other interesting articles, frequently asked questions about writing hypotheses.
A hypothesis states your predictions about what your research will find. It is a tentative answer to your research question that has not yet been tested. For some research projects, you might have to write several hypotheses that address different aspects of your research question.
A hypothesis is not just a guess – it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).
Hypotheses propose a relationship between two or more types of variables .
If there are any control variables , extraneous variables , or confounding variables , be sure to jot those down as you go to minimize the chances that research bias will affect your results.
In this example, the independent variable is exposure to the sun – the assumed cause . The dependent variable is the level of happiness – the assumed effect .
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Writing a hypothesis begins with a research question that you want to answer. The question should be focused, specific, and researchable within the constraints of your project.
Your initial answer to the question should be based on what is already known about the topic. Look for theories and previous studies to help you form educated assumptions about what your research will find.
At this stage, you might construct a conceptual framework to ensure that you’re embarking on a relevant topic . This can also help you identify which variables you will study and what you think the relationships are between them. Sometimes, you’ll have to operationalize more complex constructs.
Now you should have some idea of what you expect to find. Write your initial answer to the question in a clear, concise sentence.
You need to make sure your hypothesis is specific and testable. There are various ways of phrasing a hypothesis, but all the terms you use should have clear definitions, and the hypothesis should contain:
To identify the variables, you can write a simple prediction in if…then form. The first part of the sentence states the independent variable and the second part states the dependent variable.
In academic research, hypotheses are more commonly phrased in terms of correlations or effects, where you directly state the predicted relationship between variables.
If you are comparing two groups, the hypothesis can state what difference you expect to find between them.
If your research involves statistical hypothesis testing , you will also have to write a null hypothesis . The null hypothesis is the default position that there is no association between the variables. The null hypothesis is written as H 0 , while the alternative hypothesis is H 1 or H a .
Research question | Hypothesis | Null hypothesis |
---|---|---|
What are the health benefits of eating an apple a day? | Increasing apple consumption in over-60s will result in decreasing frequency of doctor’s visits. | Increasing apple consumption in over-60s will have no effect on frequency of doctor’s visits. |
Which airlines have the most delays? | Low-cost airlines are more likely to have delays than premium airlines. | Low-cost and premium airlines are equally likely to have delays. |
Can flexible work arrangements improve job satisfaction? | Employees who have flexible working hours will report greater job satisfaction than employees who work fixed hours. | There is no relationship between working hour flexibility and job satisfaction. |
How effective is high school sex education at reducing teen pregnancies? | Teenagers who received sex education lessons throughout high school will have lower rates of unplanned pregnancy teenagers who did not receive any sex education. | High school sex education has no effect on teen pregnancy rates. |
What effect does daily use of social media have on the attention span of under-16s? | There is a negative between time spent on social media and attention span in under-16s. | There is no relationship between social media use and attention span in under-16s. |
If you want to know more about the research process , methodology , research bias , or statistics , make sure to check out some of our other articles with explanations and examples.
Statistics
Research bias
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A hypothesis is not just a guess — it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).
Null and alternative hypotheses are used in statistical hypothesis testing . The null hypothesis of a test always predicts no effect or no relationship between variables, while the alternative hypothesis states your research prediction of an effect or relationship.
Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.
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The actual test begins by considering two hypotheses . They are called the null hypothesis and the alternative hypothesis . These hypotheses contain opposing viewpoints.
H 0 , the — null hypothesis: a statement of no difference between sample means or proportions or no difference between a sample mean or proportion and a population mean or proportion. In other words, the difference equals 0.
H a —, the alternative hypothesis: a claim about the population that is contradictory to H 0 and what we conclude when we reject H 0 .
Since the null and alternative hypotheses are contradictory, you must examine evidence to decide if you have enough evidence to reject the null hypothesis or not. The evidence is in the form of sample data.
After you have determined which hypothesis the sample supports, you make a decision. There are two options for a decision. They are reject H 0 if the sample information favors the alternative hypothesis or do not reject H 0 or decline to reject H 0 if the sample information is insufficient to reject the null hypothesis.
Mathematical Symbols Used in H 0 and H a :
equal (=) | not equal (≠) greater than (>) less than (<) |
greater than or equal to (≥) | less than (<) |
less than or equal to (≤) | more than (>) |
H 0 always has a symbol with an equal in it. H a never has a symbol with an equal in it. The choice of symbol depends on the wording of the hypothesis test. However, be aware that many researchers use = in the null hypothesis, even with > or < as the symbol in the alternative hypothesis. This practice is acceptable because we only make the decision to reject or not reject the null hypothesis.
H 0 : No more than 30 percent of the registered voters in Santa Clara County voted in the primary election. p ≤ 30 H a : More than 30 percent of the registered voters in Santa Clara County voted in the primary election. p > 30
A medical trial is conducted to test whether or not a new medicine reduces cholesterol by 25 percent. State the null and alternative hypotheses.
We want to test whether the mean GPA of students in American colleges is different from 2.0 (out of 4.0). The null and alternative hypotheses are the following: H 0 : μ = 2.0 H a : μ ≠ 2.0
We want to test whether the mean height of eighth graders is 66 inches. State the null and alternative hypotheses. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.
We want to test if college students take fewer than five years to graduate from college, on the average. The null and alternative hypotheses are the following: H 0 : μ ≥ 5 H a : μ < 5
We want to test if it takes fewer than 45 minutes to teach a lesson plan. State the null and alternative hypotheses. Fill in the correct symbol ( =, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.
An article on school standards stated that about half of all students in France, Germany, and Israel take advanced placement exams and a third of the students pass. The same article stated that 6.6 percent of U.S. students take advanced placement exams and 4.4 percent pass. Test if the percentage of U.S. students who take advanced placement exams is more than 6.6 percent. State the null and alternative hypotheses. H 0 : p ≤ 0.066 H a : p > 0.066
On a state driver’s test, about 40 percent pass the test on the first try. We want to test if more than 40 percent pass on the first try. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.
Bring to class a newspaper, some news magazines, and some internet articles. In groups, find articles from which your group can write null and alternative hypotheses. Discuss your hypotheses with the rest of the class.
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ANOVA Testing Example
Excel Example for this ANOVA
See a HOW TO Video of this Example
A research study compared the ounces of coffee consumed daily between three groups. Group1 was Italians, Group 2 French, and Group 3 American. Determine if there is a significant difference among the groups using a 5% level (alpha is .05).
Group1: Italian
Group 2: French
Group 3: American
The Results of this study are in the following table:
Note that here in this example:
“n” is the sample size for each group
“M” is the “sample mean” for each group
“s^2” is the sample variance for each group
N = n1 + n2 + n3 = 70+ 70+70 = 210
NOTE that “N” is the combined sample size for all three groups.
Step 1: The null and alternative hypothesis Ho and Ha
Ho (the null) will represent that all the groups are statistically the same.
Ho: mean group 1 = mean group 2 = mean group 3
The Ha (alternative or research) hypothesis will represent that at least one of the groups is statistically significantly different.
Ha: mean group 1 ≠ mean group 2 ≠ mean group 3
Step 2: Determine the “degrees of freedom” also called df for each group and for the combination of groups:
The df BETWEEN is calculated by subtracting 1 from the number of groups you have. We have three groups here.
df BETWEEN = 3 – 1 = 2 (Used as the numerator or top df)
Next, the df WITHIN is calculated by first determining the individual df for each group and then adding them together:
df group1 = 70 – 1 = 69
df group 2 = 70 – 1 = 69
df group 3 = 70 – 1 = 69
df WITHIN = 69 + 69 + 69 = 207 (Used as the denominator or bottom df)
Step 3: Use the F-Table or a technology to get the “cut-off” values for this F-test ANOVA. Remember that all hypothesis tests have cut-off values that you use to determine if your F-test result is in the rejection region or not.
NOTE: The F-Table cannot contain all possible values. So most F Tables contain some values and you will choose the closet value to your df numbers.
IMPORTANT: Because of this, different textbooks or examples will have slightly different cut-off values for their F-tests. In this case, I am using the table below and my cut-off value is 3.07. However, a similar example might have a cut-off of 3.09 for example.
To get a more exact cut-off, use Excel to run the ANOVA. Excel will generate the p values for you.
Excel Example of ANOVA
Therefore, our cut-off value for the F-test is 3.07 here.
Step 4: Run the F-test to determine the F values. Then compare the F test value results to the cut-off values.
Running an F-test by hand has a few steps.
First Step:
Calculate the grand mean (GM) = (4.0 + 3.7 + 3.4) / 3 = 3.7
This grand mean is the sum of all of your individual means, divided by the total number of your groups. We have three groups. So we add the three group means together and divide by 3.
Second Step
Calculate the variance of the means. NOTE that 3.7 is the grand mean. Note that 4.0, 3.7, and 3.4 are individual group means. Note that “2” is the df BETWEEN.
s^2M = ( [ 4.0 – 3.7 ]^2 + [3.7 – 3.7 ]^2 + [3.4 – 3.7]^2 ) / dfBETWEEN
= ( [ 4.0 – 3.7 ]^2 + [3.7 – 3.7 ]^2 + [3.4 – 3.7]^2 ) / 2
= ( [ .3 ]^2 + [0 ]^2 + [-.3]^2 ) / 2
= ( .09 + 0 + .09 ) / 2
What did we do here?
The s^M is computed by subtracting the grand mean in this problem of 3.7 that we calculated in the first step from each individual group mean. Then, we squared each difference. Then, we added those results. Then, we divided by the dfBETWEEN which is 2 in this example.
Do not let formulas scare you.
Next, we can calculate the variance between the groups or s^2BETWEEN
s^2 BETWEEN = s^2M*n = .09 * 70 = 6.3
Remember, the s^2M was just calculated in the second step. The “n” is the size of each group sample. If your groups are different sizes, use the average size.
Fourth Step
Now we will calculate the WITHIN variance called s^2WITHIN
s^2WITHIN = (4.4 + 5.2 + 6.1)/3 = 5.233
Where did these numbers come from?
Here, these are the three individual variances of each group. See the table at the start of the example. You will see that Group 1 has a variance (s^2) of 4.4 and group 2 has a variance or 5.2, and group 3 has a variance of 6.1. To get the average of the three group variances, we add them together and divide by 3. This is what we did here in the fourth step.
Now we are ready to calculate F because we have calculated all the parts that we need for F.
F = s^2BETWEEN / s^2WITHIN
F = 6.3/5.233 = 1.20
The s^2 BETWEEN is what we calculated in the third step above and the s^2WITHIN is what we calculated in the fourth step above.
The F test is the between variance divided by the within variance.
Our F-test result here is 1.20
Step 5: Determine the final result and conclusion for your ANOVA F-test
The F-test is 1.20.
The cut-off value is 3.07
We can reject Ho only when our test value (1.20 in this case) is LARGER THAN our cut-off value (3.07 in this case).
However, in our example here, the F-test < cut-off
1.20 < 3.07
Therefore we CANNOT reject Ho
In conclusion, there is NOT a significant difference between our three groups.
We can say there is not enough evidence to conclude that people from the three groups of Italian, French, and American drink a sig diff amount of coffee per day. Therefore, statistically, we say that the amount of coffee consumed by these groups is NOT sig diff.
NOTE: These results are made up and are pretend. This is not a real test, nor a real conclusion. This is pretend and only for example purposes.
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The null and alternative hypotheses are two competing claims that researchers weigh evidence for and against using a statistical test: Null hypothesis (H0): There’s no effect in the population. Alternative hypothesis (Ha or H1): There’s an effect in the population.
Once you develop a research hypothesis, it’s important to state it is as a Null hypothesis (Ho) and an Alternative hypothesis (Ha) to test it statistically. A null hypothesis is a preferred choice as it provides the opportunity to test the theory.
Describe hypothesis testing in general and in practice. The actual test begins by considering two hypotheses. They are called the null hypothesis and the alternative hypothesis. These hypotheses contain opposing viewpoints.
When the research question asks “Does the independent variable affect the dependent variable?”, the null hypothesis (H 0) answers “No, there’s no effect in the population.” On the other hand, the alternative hypothesis (H A) answers “Yes, there is an effect in the population.”
An alternative or alternate hypothesis (denoted Ha) is then stated, which will be the opposite of the Null Hypothesis. The hypothesis testing process and analysis involves using sample data to determine whether or not you can be statistically confident that you can reject or fail to reject the H o.
Step 1: State your null and alternate hypothesis. After developing your initial research hypothesis (the prediction that you want to investigate), it is important to restate it as a null (H o) and alternate (H a) hypothesis so that you can test it mathematically.
In all three examples, our aim is to decide between two opposing points of view, Claim 1 and Claim 2. In hypothesis testing, Claim 1 is called the null hypothesis (denoted “ Ho “), and Claim 2 plays the role of the alternative hypothesis (denoted “ Ha “).
Step 1. Ask a question. Writing a hypothesis begins with a research question that you want to answer. The question should be focused, specific, and researchable within the constraints of your project.
H0 always has a symbol with an equal in it. Ha never has a symbol with an equal in it. The choice of symbol depends on the wording of the hypothesis test. However, be aware that many researchers use = in the null hypothesis, even with > or < as the symbol in the alternative hypothesis.
Step 1: The null and alternative hypothesis Ho and Ha. Ho (the null) will represent that all the groups are statistically the same. Ho: mean group 1 = mean group 2 = mean group 3 . The Ha (alternative or research) hypothesis will represent that at least one of the groups is statistically significantly different.