hypothesis vs conclusion math

Understanding the Role of Hypotheses and Conclusions in Mathematical Reasoning

Hypothesis and conclusion.

In the context of mathematics and logic, a hypothesis is a statement or proposition that is assumed to be true for the purpose of a logical argument or investigation. It is usually denoted by “H” or “P” and is the starting point for many mathematical proofs.

For example, let’s consider the hypothesis: “If it is raining outside, then the ground is wet.” This statement assumes that whenever it rains, the ground will be wet.

The conclusion, on the other hand, is the statement or proposition that is inferred or reached by logical reasoning, based on the hypothesis or given information. It is typically denoted by “C” or “Q”.

Using the same example, the conclusion derived from the hypothesis could be: “It is currently raining outside, so the ground is wet.” This conclusion is based on the assumption that the given condition of rain implies a wet ground.

In mathematics, hypotheses and conclusions are commonly used in proofs and logical arguments. By stating a hypothesis and then deducing a conclusion from it, mathematicians can demonstrate the validity of certain mathematical concepts, theorems, or formulas.

It’s important to note that in mathematics, a hypothesis is not the same as a guess or a prediction. It is a statement that is assumed to be true and serves as the basis for logical reasoning, while the conclusion is the logical consequence or outcome that is drawn from the hypothesis.

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$Department of Mathematics$

Logic and Mathematical Statements

Worked examples, if...then... statements, mini-lecture..

(i)	Determine the hypotheses/assumptions and the conclusion.
(ii)	Rewrite this statement explicitly in the form "If A, then B" using Part (i).
(iii)	Is this statement true or false?

(i)	The hypothesis we are making is that it is raining. The conclusion we are making is that there must be a cloud in the sky.
(ii)	"If it's raining, then there must be a cloud in the sky."
(iii)	This statement is true. (Based on all that is currently known about how rain works!)

Example. Consider the statement "$x > 0 \Rightarrow x+1>0$". Is this statement true or false?

Example. consider the statement "if $x$ is a positive integer or a solution to $x+3>4$, then $x>0$ and $x> \frac{1}{2}$." is this statement true, example. consider the statement "$0>1 \rightarrow \sin x = 2$". is this statement true or false.

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5.2 - writing hypotheses.

The first step in conducting a hypothesis test is to write the hypothesis statements that are going to be tested. For each test you will have a null hypothesis ($H_0$) and an alternative hypothesis ($H_a$).

When writing hypotheses there are three things that we need to know: (1) the parameter that we are testing (2) the direction of the test (non-directional, right-tailed or left-tailed), and (3) the value of the hypothesized parameter.

At this point we can write hypotheses for a single mean ($\mu$), paired means($\mu_d$), a single proportion ($p$), the difference between two independent means ($\mu_1-\mu_2$), the difference between two proportions ($p_1-p_2$), a simple linear regression slope ($\beta$), and a correlation ($\rho$).
The research question will give us the information necessary to determine if the test is two-tailed (e.g., "different from," "not equal to"), right-tailed (e.g., "greater than," "more than"), or left-tailed (e.g., "less than," "fewer than").
The research question will also give us the hypothesized parameter value. This is the number that goes in the hypothesis statements (i.e., $\mu_0$ and $p_0$). For the difference between two groups, regression, and correlation, this value is typically 0.

Hypotheses are always written in terms of population parameters (e.g., $p$ and $\mu$). The tables below display all of the possible hypotheses for the parameters that we have learned thus far. Note that the null hypothesis always includes the equality (i.e., =).

One Group Mean
Research Question	Is the population mean different from $ \mu_{0} $?	Is the population mean greater than $\mu_{0}$?	Is the population mean less than $\mu_{0}$?
Null Hypothesis, $H_{0}$	$\mu=\mu_{0} $	$\mu=\mu_{0} $	$\mu=\mu_{0} $
Alternative Hypothesis, $H_{a}$	$\mu\neq \mu_{0} $	$\mu> \mu_{0} $	$\mu<\mu_{0} $
Type of Hypothesis Test	Two-tailed, non-directional	Right-tailed, directional	Left-tailed, directional

Paired Means
Research Question	Is there a difference in the population?	Is there a mean increase in the population?	Is there a mean decrease in the population?
Null Hypothesis, $H_{0}$	$\mu_d=0 $	$\mu_d =0 $	$\mu_d=0 $
Alternative Hypothesis, $H_{a}$	$\mu_d \neq 0 $	$\mu_d> 0 $	$\mu_d<0 $
Type of Hypothesis Test	Two-tailed, non-directional	Right-tailed, directional	Left-tailed, directional

One Group Proportion
Research Question	Is the population proportion different from $p_0$?	Is the population proportion greater than $p_0$?	Is the population proportion less than $p_0$?
Null Hypothesis, $H_{0}$	$p=p_0$	$p= p_0$	$p= p_0$
Alternative Hypothesis, $H_{a}$	$p\neq p_0$	$p> p_0$	$p< p_0$
Type of Hypothesis Test	Two-tailed, non-directional	Right-tailed, directional	Left-tailed, directional

Difference between Two Independent Means
Research Question	Are the population means different?	Is the population mean in group 1 greater than the population mean in group 2?	Is the population mean in group 1 less than the population mean in groups 2?
Null Hypothesis, $H_{0}$	$\mu_1=\mu_2$	$\mu_1 = \mu_2 $	$\mu_1 = \mu_2 $
Alternative Hypothesis, $H_{a}$	$\mu_1 \ne \mu_2 $	$\mu_1 \gt \mu_2 $	$\mu_1 \lt \mu_2$
Type of Hypothesis Test	Two-tailed, non-directional	Right-tailed, directional	Left-tailed, directional

Difference between Two Proportions
Research Question	Are the population proportions different?	Is the population proportion in group 1 greater than the population proportion in groups 2?	Is the population proportion in group 1 less than the population proportion in group 2?
Null Hypothesis, $H_{0}$	$p_1 = p_2 $	$p_1 = p_2 $	$p_1 = p_2 $
Alternative Hypothesis, $H_{a}$	$p_1 \ne p_2$	$p_1 \gt p_2 $	$p_1 \lt p_2$
Type of Hypothesis Test	Two-tailed, non-directional	Right-tailed, directional	Left-tailed, directional

Simple Linear Regression: Slope
Research Question	Is the slope in the population different from 0?	Is the slope in the population positive?	Is the slope in the population negative?
Null Hypothesis, $H_{0}$	$\beta =0$	$\beta= 0$	$\beta = 0$
Alternative Hypothesis, $H_{a}$	$\beta\neq 0$	$\beta> 0$	$\beta< 0$
Type of Hypothesis Test	Two-tailed, non-directional	Right-tailed, directional	Left-tailed, directional

Correlation (Pearson's )
Research Question	Is the correlation in the population different from 0?	Is the correlation in the population positive?	Is the correlation in the population negative?
Null Hypothesis, $H_{0}$	$\rho=0$	$\rho= 0$	$\rho = 0$
Alternative Hypothesis, $H_{a}$	$\rho \neq 0$	$\rho > 0$	$\rho< 0$
Type of Hypothesis Test	Two-tailed, non-directional	Right-tailed, directional	Left-tailed, directional

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Methodology

How to Write a Strong Hypothesis | Steps & Examples

How to Write a Strong Hypothesis | Steps & Examples

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 .

Example: Hypothesis

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).

Variables in hypotheses

Hypotheses propose a relationship between two or more types of variables .

An independent variable is something the researcher changes or controls.
A dependent variable is something the researcher observes and measures.

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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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.

Step 2. Do some preliminary research

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.

Step 3. Formulate your hypothesis

Now you should have some idea of what you expect to find. Write your initial answer to the question in a clear, concise sentence.

4. Refine your hypothesis

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:

The relevant variables
The specific group being studied
The predicted outcome of the experiment or analysis

5. Phrase your hypothesis in three ways

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.

6. Write a null hypothesis

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 .

H 0 : The number of lectures attended by first-year students has no effect on their final exam scores.
H 1 : The number of lectures attended by first-year students has a positive effect on their final exam scores.

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.

Sampling methods
Simple random sampling
Stratified sampling
Cluster sampling
Likert scales
Reproducibility

Statistics

Null hypothesis
Statistical power
Probability distribution
Effect size
Poisson distribution

Research bias

Optimism bias
Cognitive bias
Implicit bias
Hawthorne effect
Anchoring bias
Explicit 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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Featured Answers

What is the difference between a hypothesis and a conclusion in the scientific method?

Explanation:

#"...and a conclusion is whether that proposal was justified."#

From a dictionary of philosophy (that I happened to pick up last week for £2-00 at a second hand bookstore),

an hypothesis is a proposition made as a basis for reasoning, without any assumption of its truth.

I suppose the more rigorous scientific definition of hypothesis would include a proposal made on a limited number of experiments, that includes a basis for further experiments.....the which test whether that hypothesis holds.

And after the experiments have been performed, we can test whether the hypothesis was reasonable. If NO, then we throw the hypothesis out. If YES, then we continue to test the truth and extent of the hypothesis with further experiments....

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Fallacy of denying the hypothesis

I need help with this question Im not sure what a fallacy of denying the hypothesis is.

Use a truth table to show that $p\to q$ and $\neg p$, $\therefore \neg q$ is not a valid rule of inference. It is called the fallacy of denying the hypothesis.

discrete-mathematics
statistical-inference

2 Answers 2

It is called a fallacy because it is not a valid argument . What is a valid argument, then?

A valid argument is an argument where the conclusion can never be false when the premises are true .

Let's now build the truth table of your argument: \begin{array}{cc|ccc}p&q&p\to q&\neg p&\neg q\\\hline T&T&T&F&F\\T&F&F&F&T\\F&T&T&T&F\\F&F&T&T&T\end{array}

As you can see, there are two assignments of truth values to the sentences letters that make the premises true (last two lines of the table), yet only one of them also makes the conclusion true. Therefore, there is a case where the premises are true and the conclusion is false which, by definition, means that the argument is invalid .

Take a simple example, where $p$ means "it has rained" and $q$ means "the ground is wet". Assuming the first premise is true, i.e. "if it has rained, then the ground is wet", knowing $\neg p$, i.e. knowing that "it hasn't rained", isn't enough to conclude about whether "the ground is wet or not" (the ground might be wet for some other reason). This is the core idea of the fallacy.

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COMMENTS

Understanding the Role of Hypotheses and Conclusions in Mathematical
Hypothesis and conclusion. In the context of mathematics and logic, a hypothesis is a statement or proposition that is assumed to be true for the purpose of a logical argument or investigation. It is usually denoted by "H" or "P" and is the starting point for many mathematical proofs. For example, let's consider the hypothesis: "If ...
What is the difference between a hypothesis and a conclusion?
1 Answer. Well, an hypothesis is something that is proposed.... And the mark of a good hypothesis is its testability. That is there exist a few simple experiments whose results would confirm or deny the original hypothesis. And a conclusion is drawn AFTER the experiment is performed, and reports whether or not the results of the experiment ...
1.1: Statements and Conditional Statements
1.1: Statements and Conditional Statements
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Logic and Mathematical Statements - Worked Examples
Introduction to logic. Necessary and sufficient.
a) In a valid argument, if the hypothesis is true, then the conclusion a) must be true. This in fact is all that logic can tell us about truth. b) In a valid argument, if the hypothesis is false, then the conclusion b) may be true or false. If at least one statement of the hypothesis is b) false, then we consider the hypothesis to be false.
Identify hypotheses and conclusions (Geometry practice)
Identify hypotheses and conclusions (Geometry practice)
Understanding Logical Statements
A logical statement A statement that allows drawing a conclusion or result based on a hypothesis or premise. is a statement that, when true, allows us to take a known set of facts and infer (or assume) a new fact from them. Logical statements have two parts: The hypothesis The part of a logical statement that provides the premise on which the conclusion is based.
Hypothesis -- from Wolfram MathWorld
A hypothesis is a proposition that is consistent with known data, but has been neither verified nor shown to be false. In statistics, a hypothesis (sometimes called a statistical hypothesis) refers to a statement on which hypothesis testing will be based. Particularly important statistical hypotheses include the null hypothesis and alternative hypothesis. In symbolic logic, a hypothesis is the ...
2.3: Implications
Example 2.3.4 2.3. 4. To show that "if x = 2 x = 2, then x2 = 4 x 2 = 4 " is true, we need not worry about those x x -values that are not equal to 2, because the implication is immediately true if x ≠ 2 x ≠ 2. It suffices to assume that x = 2 x = 2, and try to prove that we will get x2 = 4 x 2 = 4.
9.1: Introduction to Hypothesis Testing
In hypothesis testing, the goal is to see if there is sufficient statistical evidence to reject a presumed null hypothesis in favor of a conjectured alternative hypothesis.The null hypothesis is usually denoted $H_0$ while the alternative hypothesis is usually denoted $H_1$. An hypothesis test is a statistical decision; the conclusion will either be to reject the null hypothesis in favor ...
8.1: The Elements of Hypothesis Testing
A standardized test statistic for a hypothesis test is the statistic that is formed by subtracting from the statistic of interest its mean and dividing by its standard deviation. For example, reviewing Example 8.1.3 8.1. 3, if instead of working with the sample mean X¯¯¯¯ X ¯ we instead work with the test statistic.
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Actually, you swapped it. The hypothesis is what you already know what is true. The conclusion is what you want to prove. Then: hypothesis: The n n items are the first n n positive integers. conclusion: Their sum is n(n + 1)/2 n (n + 1) / 2. Share.
5.2
5.2 - Writing Hypotheses. The first step in conducting a hypothesis test is to write the hypothesis statements that are going to be tested. For each test you will have a null hypothesis (H 0) and an alternative hypothesis (H a). Null Hypothesis. The statement that there is not a difference in the population (s), denoted as H 0.
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1 Answer. anor277. Dec 30, 2017. An hypothesis is what is proposed.... Explanation: ...and a conclusion is whether that proposal was justified. From a dictionary of philosophy (that I happened to pick up last week for £2-00 at a second hand bookstore), an hypothesis is a proposition made as a basis for reasoning, without any assumption of its ...
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When writing the conclusion of a hypothesis test, we typically include: Whether we reject or fail to reject the null hypothesis. The significance level. A short explanation in the context of the hypothesis test. For example, we would write: We reject the null hypothesis at the 5% significance level.
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10.2: Null and Alternative Hypotheses
They are called the null hypothesis and the alternative hypothesis. These hypotheses contain opposing viewpoints. The null hypothesis (\ (H_ {0}\)) 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.
discrete mathematics
See also Denying the antecedent. It is a fallacy exactly because from the two premisse (or : assumptions, or hypothesis) : p → q p → q and ¬p ¬ p. it is not possible to validly conclude with : ¬q ¬ q. The truth-functional properties of the conditional : → → state that when p p is FALSE F A L S E and q q is TRUE T R U E, the ...

Research Question	Is the population mean different from \( \mu_{0} \)?	Is the population mean greater than \(\mu_{0}\)?	Is the population mean less than \(\mu_{0}\)?
Null Hypothesis, \(H_{0}\)	\(\mu=\mu_{0} \)	\(\mu=\mu_{0} \)	\(\mu=\mu_{0} \)
Alternative Hypothesis, \(H_{a}\)	\(\mu\neq \mu_{0} \)	\(\mu> \mu_{0} \)	\(\mu<\mu_{0} \)
Type of Hypothesis Test	Two-tailed, non-directional	Right-tailed, directional	Left-tailed, directional

Research Question	Is the population proportion different from \(p_0\)?	Is the population proportion greater than \(p_0\)?	Is the population proportion less than \(p_0\)?
Null Hypothesis, \(H_{0}\)	\(p=p_0\)	\(p= p_0\)	\(p= p_0\)
Alternative Hypothesis, \(H_{a}\)	\(p\neq p_0\)	\(p> p_0\)	\(p< p_0\)
Type of Hypothesis Test	Two-tailed, non-directional	Right-tailed, directional	Left-tailed, directional