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.
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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Researchers are often disappointed when their work yields "negative" results, meaning that the null hypothesis cannot be rejected. However, negative results are essential for research to progress. Negative results tell researchers that they are on the wrong path, or that their current techniques are ineffective. This is a natural and necessary part of discovering something that was previously unknown. Solving problems that lead to negative results is an integral part of being an effective researcher. Publishing negative results that are the result of rigorous research contributes to scientific progress.
There are three main reasons for negative results:
Here, we will discuss how to write about negative results, first focusing on the most common reason: technical problems.
Technical problems might include faulty reagents, inappropriate study design, and insufficient statistical power. Most researchers would prefer to resolve technical problems before presenting their work, and focus instead on their convincing results. In reality, researchers often need to present their work at a conference or to a thesis committee before some problems can be resolved.
When presenting at a conference, the objective should be to clearly describe your overall research goal and why it is important, your preliminary results, the current problem, and how previously published work is informing the steps you are taking to resolve the problem. Here, you want to take advantage of the collective expertise at the conference. By being straightforward about your difficulties, you increase the chance that someone can help you find a solution.
When presenting to a thesis committee, much of what you discuss will be the same (overall research goal and why it is important, results, problem(s) and possible solutions). Your primarily goal is to show that you are well prepared to move forward in your research career, despite the recent difficulties. The thesis defense is a defined stopping point, so most thesis students should write about solutions they would pursue if they were to continue the work. For example, "To resolve this problem, it would be advisable to increase the survey area by a factor of 4, and then…" In contrast, researchers who will be continuing their work should write about possible solutions using present and future tense. For example, "To resolve this problem, we are currently testing a wider variety of standards, and will then conduct preliminary experiments to determine…"
Whether you are presenting at a conference, defending a thesis, applying for funding, or simply trying to make progress in your research, you will often need to search through the academic literature to determine the best path forward. This is especially true when you get unexpected results—either positive or negative. When trying to resolve a technical problem, you should often find yourself carefully reading the materials and methods sections of papers that address similar research questions, or that used similar techniques to explore very different problems. For example, a single computer algorithm might be adapted to address research questions in many different fields.
In searching through published papers and less formal methods of communication—such as conference abstracts—you may come to appreciate the important details that good researchers will include when discussing technical problems or other negative results. For example, "We found that participants were more likely to complete the process when light refreshments were provided between the two sessions." By including this information, the authors may help other researchers save time and resources.
Thus, you are advised to be as thorough as possible in reviewing the relevant literature, to find the most promising solutions for technical problems. When presenting your work, show that you have carefully considered the possibilities, and have developed a realistic plan for moving forward. This will help a thesis committee view your efforts favorably, and can also convince possible collaborators or advisors to invest time in helping you.
Negative results due to technical problems may be acceptable for a conference presentation or a thesis at the undergraduate or master's degree level. Negative results due to technical problems are not sufficient for publication, a Ph.D. dissertation, or tenure. In those situations, you will need to resolve the technical problem and generate high quality results (either positive or negative) that stand up to rigorous analysis. Depending on the research field, high quality negative results might include multiple readouts and narrow confidence intervals.
Researchers are often reluctant to publish negative results, especially if their data don't support an interesting alternative hypothesis. Traditionally, journals have been reluctant to publish negative results that are not paired with positive results, even if the study is well designed and the results have sufficient statistical power. This is starting to change— especially for medical research —but publishing negative results can still be an uphill battle.
Not publishing high quality negative results is a disservice to the scientific community and the people who support it (including tax payers), since other scientists may need to repeat the work. For studies involving animal research or human tissue samples, not publishing would squander significant sacrifices. For research involving medical treatments—especially studies that contradict a published report—not publishing negative results leads to an inaccurate understanding of treatment efficacy.
So how can researchers write about negative results in a way that reflects its importance? Let's consider a common reason for negative results: the original hypothesis was incorrect.
Researchers should be comfortable with being wrong some of the time, such as when results don't support an initial hypothesis. After all, research wouldn't be necessary if we already knew the answer to every possible question. The next step is usually to revise the hypothesis after reconsidering the available data, reading through the relevant literature, and consulting with colleagues.
Ideally, a revised hypothesis will lead to results that allow you to reject a (revised) null hypothesis. The negative results can then be reported alongside the positive results, possibly bolstering the significance of both. For example, "The DNA mutations in region A had a significant effect on gene expression, while the mutations outside of domain A had no effect. Don't forget to include important details about how you overcame technical problems, so that other researchers don't need to reinvent the wheel.
Unfortunately, it isn't always possible to pair negative results with related positive results. For example, imagine a year-long study on the effect of COVID-19 shelter-in-place orders on the mental health of avid video game players compared to people who don't play video games. Despite using well-established tools for measuring mental health, having a large sample size, and comparing multiple subpopulations (e.g. gamers who live alone vs. gamers who live with others), no significant differences were identified. There is no way to modify and repeat this study because the same shelter-in-place conditions no longer exist. So how can this research be presented effectively?
When you write a scientific paper to report negative results, the sections will be the same as for any other paper: Introduction, Materials and Methods, Results and Discussion. In the introduction, you should prepare your reader for the possibility of negative results. You can highlight gaps or inconsistencies in past research, and point to data that could indicate an incomplete understanding of the situation.
In the example about video game players, you might highlight data showing that gamers are statistically very similar to large chunks of the population in terms of age, education, marital status, etc. You might discuss how the stigma associated with playing video games might be unfair and harmful to people in certain situations. You could discuss research showing the benefits of playing video games, and contrast gaming with engaging in social media, which is another modern hobby. Putting a positive spin on negative results can make the difference between a published manuscript and rejection.
In a paper that focuses on negative results—especially one that contradicts published findings—the research design and data analysis must be impeccable. You may need to collaborate with other researchers to ensure that your methods are sound, and apply multiple methods of data analysis.
As long as the research is rigorous, negative results should be used to inform and guide future experiments. This is how science improves our understanding of the world.
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Published on May 23, 2022 by Shaun Turney . Revised on June 22, 2023.
A Pearson’s chi-square test is a statistical test for categorical data. It is used to determine whether your data are significantly different from what you expected. There are two types of Pearson’s chi-square tests:
What is a chi-square test, the chi-square formula, when to use a chi-square test, types of chi-square tests, how to perform a chi-square test, how to report a chi-square test, practice questions, other interesting articles, frequently asked questions about chi-square tests.
Pearson’s chi-square (Χ 2 ) tests, often referred to simply as chi-square tests, are among the most common nonparametric tests . Nonparametric tests are used for data that don’t follow the assumptions of parametric tests , especially the assumption of a normal distribution .
If you want to test a hypothesis about the distribution of a categorical variable you’ll need to use a chi-square test or another nonparametric test. Categorical variables can be nominal or ordinal and represent groupings such as species or nationalities. Because they can only have a few specific values, they can’t have a normal distribution.
There are two types of Pearson’s chi-square tests, but they both test whether the observed frequency distribution of a categorical variable is significantly different from its expected frequency distribution. A frequency distribution describes how observations are distributed between different groups.
Frequency distributions are often displayed using frequency distribution tables . A frequency distribution table shows the number of observations in each group. When there are two categorical variables, you can use a specific type of frequency distribution table called a contingency table to show the number of observations in each combination of groups.
Bird species | Frequency |
---|---|
House sparrow | 15 |
House finch | 12 |
Black-capped chickadee | 9 |
Common grackle | 8 |
European starling | 8 |
Mourning dove | 6 |
Right-handed | Left-handed | |
---|---|---|
American | 236 | 19 |
Canadian | 157 | 16 |
Both of Pearson’s chi-square tests use the same formula to calculate the test statistic , chi-square (Χ 2 ):
The larger the difference between the observations and the expectations ( O − E in the equation), the bigger the chi-square will be. To decide whether the difference is big enough to be statistically significant , you compare the chi-square value to a critical value.
A Pearson’s chi-square test may be an appropriate option for your data if all of the following are true:
The two types of Pearson’s chi-square tests are:
Chi-square test of independence.
Mathematically, these are actually the same test. However, we often think of them as different tests because they’re used for different purposes.
You can use a chi-square goodness of fit test when you have one categorical variable. It allows you to test whether the frequency distribution of the categorical variable is significantly different from your expectations. Often, but not always, the expectation is that the categories will have equal proportions.
Expectation of different proportions
You can use a chi-square test of independence when you have two categorical variables. It allows you to test whether the two variables are related to each other. If two variables are independent (unrelated), the probability of belonging to a certain group of one variable isn’t affected by the other variable .
Some consider the chi-square test of homogeneity to be another variety of Pearson’s chi-square test. It tests whether two populations come from the same distribution by determining whether the two populations have the same proportions as each other. You can consider it simply a different way of thinking about the chi-square test of independence.
McNemar’s test is a test that uses the chi-square test statistic. It isn’t a variety of Pearson’s chi-square test, but it’s closely related. You can conduct this test when you have a related pair of categorical variables that each have two groups. It allows you to determine whether the proportions of the variables are equal.
Like chocolate | Dislike chocolate | |
---|---|---|
Like vanilla | 47 | 32 |
Dislike vanilla | 8 | 13 |
There are several other types of chi-square tests that are not Pearson’s chi-square tests, including the test of a single variance and the likelihood ratio chi-square test .
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The exact procedure for performing a Pearson’s chi-square test depends on which test you’re using, but it generally follows these steps:
If you decide to include a Pearson’s chi-square test in your research paper , dissertation or thesis , you should report it in your results section . You can follow these rules if you want to report statistics in APA Style :
If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.
Methodology
Research bias
The two main chi-square tests are the chi-square goodness of fit test and the chi-square test of independence .
Both chi-square tests and t tests can test for differences between two groups. However, a t test is used when you have a dependent quantitative variable and an independent categorical variable (with two groups). A chi-square test of independence is used when you have two categorical variables.
Both correlations and chi-square tests can test for relationships between two variables. However, a correlation is used when you have two quantitative variables and a chi-square test of independence is used when you have two categorical variables.
Quantitative variables are any variables where the data represent amounts (e.g. height, weight, or age).
Categorical variables are any variables where the data represent groups. This includes rankings (e.g. finishing places in a race), classifications (e.g. brands of cereal), and binary outcomes (e.g. coin flips).
You need to know what type of variables you are working with to choose the right statistical test for your data and interpret your results .
If you want to cite this source, you can copy and paste the citation or click the “Cite this Scribbr article” button to automatically add the citation to our free Citation Generator.
Turney, S. (2023, June 22). Chi-Square (Χ²) Tests | Types, Formula & Examples. Scribbr. Retrieved September 22, 2024, from https://www.scribbr.com/statistics/chi-square-tests/
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H 0 (Null Hypothesis): Population parameter =, ≤, ≥ some value. H A (Alternative Hypothesis): Population parameter <, >, ≠ some value. Note that the null hypothesis always contains the equal sign. We interpret the hypotheses as follows: Null hypothesis: The sample data provides no evidence to support some claim being made by an individual.
The null hypothesis (H0) answers "No, there's no effect in the population.". The alternative hypothesis (Ha) 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.
To distinguish it from other hypotheses, the null hypothesis is written as H 0 (which is read as "H-nought," "H-null," or "H-zero"). A significance test is used to determine the likelihood that the results supporting the null hypothesis are not due to chance. A confidence level of 95% or 99% is common. Keep in mind, even if the confidence level is high, there is still a small chance the ...
The null hypothesis is a statement that assumes there is no significant effect or relationship between the variables being studied. It represents the status quo or the assumption of no effect until proven otherwise. It's the hypothesis that researchers typically aim to test against and is denoted as H0.
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): There's an effect in the population. The effect is usually the effect of the independent variable on the dependent ...
Developing a hypothesis (with example) 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. Example: Research question.
When your sample contains sufficient evidence, you can reject the null and conclude that the effect is statistically significant. Statisticians often denote the null hypothesis as H 0 or H A.. Null Hypothesis H 0: No effect exists in the population.; Alternative Hypothesis H A: The effect exists in the population.; In every study or experiment, researchers assess an effect or relationship.
A null hypothesis (H0) is a statement that there is no effect or no difference, and it serves as the starting point for statistical testing. Formulating a null hypothesis involves defining a clear and concise research question, stating the hypothesis in a way that allows for empirical testing, and considering the potential for Type I errors.
Write a statistical null hypothesis as a mathematical equation, such as. μ 1 = μ 2 {\displaystyle \mu _ {1}=\mu _ {2}} if you're comparing group means. Adjust the format of your null hypothesis to match the statistical method you used to test it, such as using "mean" if you're comparing the mean between 2 groups.
Complex Hypothesis Examples. A complex hypothesis involves more than two variables. An example could be, "If students sleep for at least 8 hours and eat a healthy breakfast, then their test scores and overall well-being will improve." This type of hypothesis examines multiple factors and their combined effects.
Step 5: Present your findings. The results of hypothesis testing will be presented in the results and discussion sections of your research paper, dissertation or thesis.. In the results section you should give a brief summary of the data and a summary of the results of your statistical test (for example, the estimated difference between group means and associated p-value).
Simple hypothesis. A simple hypothesis is a statement made to reflect the relation between exactly two variables. One independent and one dependent. Consider the example, "Smoking is a prominent cause of lung cancer." The dependent variable, lung cancer, is dependent on the independent variable, smoking. 4.
Example Null Hypothesis: Daily consumption of green tea has no effect on weight loss. 4. Ensure the Statement is Testable: The null hypothesis should be clear and specific enough that it can be tested using scientific methods. 5. Avoid Words of Judgement: Words like "good", "better", or "improves" are subjective and can be ambiguous ...
Null Hypothesis Examples. "Hyperactivity is unrelated to eating sugar " is an example of a null hypothesis. If the hypothesis is tested and found to be false, using statistics, then a connection between hyperactivity and sugar ingestion may be indicated. A significance test is the most common statistical test used to establish confidence in a ...
Below are the primary purposes of the null hypothesis: 1. Baseline for Comparison. The null hypothesis provides a baseline or a default position that indicates no effect, no difference, or no relationship between variables. It is the statement that researchers aim to test against an alternative hypothesis.
10.1 - Setting the Hypotheses: Examples. A significance test examines whether the null hypothesis provides a plausible explanation of the data. The null hypothesis itself does not involve the data. It is a statement about a parameter (a numerical characteristic of the population). These population values might be proportions or means or ...
The null hypothesis and the alternative hypothesis are types of conjectures used in statistical tests to make statistical inferences, which are formal methods of reaching conclusions and separating scientific claims from statistical noise.. The statement being tested in a test of statistical significance is called the null hypothesis. The test of significance is designed to assess the strength ...
Review. 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 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: It is a statement about the population that either is believed to be true or is used to put forth an argument unless it can be shown to be incorrect beyond a reasonable doubt.
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.
Researchers are often disappointed when their work yields "negative" results, meaning that the null hypothesis cannot be rejected. However, negative results are essential for research to progress. Negative results tell researchers that they are on the wrong path, or that their current techniques are ineffective. This is a natural and necessary part of discovering something that was previously ...
Null hypothesis (H 0): ... Example: Chi-square test of independence. Null hypothesis (H 0): The proportion of people who are left-handed is the same for Americans and Canadians. ... If you decide to include a Pearson's chi-square test in your research paper, dissertation or thesis, ...