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• 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.

Table of contents

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.

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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Formulating and Testing Hypotheses

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The term hypothesis has been mentioned several times in the preceding chapters. The definition that will be used here is that a hypothesis is a proposition set forth as explanation for the occurrence of a specified phenomenon. The basis of scientific investigation is the collection of information that is used either to formulate or to test hypotheses. One assesses the important variables and tries to build a model or hypothesis that explains the observed phenomenon. In general, a hypothesis is formulated by rephrasing the objective of a study as a statement, e.g., if the objective of an investigation is to determine if a pesticide is safe, the resulting hypothesis might be “ the pesticide is not safe ”, or alternatively that “ the pesticide is safe ”. A hypothesis is a statistical hypothesis only if it is stated in terms related to the distribution of populations. The general hypothesis above might be refined to: “ this pesticide, when used as directed, has no effect on the average number of robins in an area ”, which is a testable hypothesis. The hypothesis to be tested is called the null hypothesis (H 0 ). The alternative hypothesis (H 1 ) for the above example would be “ this pesticide, when used as directed, has an effect on the average number of robins in an area”. In testing a hypothesis, H 0 is considered to be true, unless the sample data indicate otherwise, (i.e., that the pesticide is innocent, unless proven guilty). Testing cannot prove H 0 to be true but the results can cause it to be rejected. In accepting or rejecting H 0 , two types of error may be made. If H 0 is rejected when, in fact, it is true a type 1 error has been committed. If Ho is not true and the test fails to reject it, a type 2 error has been made.

“ Research in the field, through study of disease as it manifests itself in nature, is an important and independent approach to solution of medical problems. Modern medical progress has been so thoroughly associated with research in the biological laboratory, and it has been so largely a development of the experimental method, that this other and older method has come in recent years to be overshadowed ” (Gordon, 1950)

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Wobeser, G.A. (1994). Formulating and Testing Hypotheses. In: Investigation and Management of Disease in Wild Animals. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-5609-8_6

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HYPOTHESIS FORMULATION

The third step in the research process after formulating problem statement and literature review is to formulate hypotheses. The hypothesis is a tentative solution of a problem. The research activities are planned to verify the hypothesis and not to find out the solution of the problem or to seek an answer of a question. It is very essential to a research worker to understand the meaning and nature of hypothesis. This paper will discuss about definition of hypothesis, nature, different of hypothesis and assumption and postulate, function and importance of hypothesis, kind of hypothesis, and formulating hypothesis.

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Hypothesis For Kids

Ai generator.

Crafting a hypothesis isn’t just for scientists in white lab coats; even young budding researchers can join in the fun! When kids learn to frame their curious wonders as hypothesis statements, they pave the way for exciting discoveries. Our guide breaks down the world of hypothesis writing into kid-friendly chunks, complete with relatable thesis statement examples and easy-to-follow tips. Dive in to spark a love for inquiry and nurture young scientific minds!

What is an example of a Hypothesis for Kids?

Question: Do plants grow taller when they are watered with coffee instead of water?

Hypothesis: If I water a plant with coffee instead of water, then the plant will not grow as tall because coffee might have substances that aren’t good for plants.

This hypothesis is based on a simple observation or question a child might have, and it predicts a specific outcome (the plant not growing as tall) due to a specific condition (being watered with coffee). It’s presented in simple language suitable for kids.

100 Kids Hypothesis Statement Examples

Size: 170 KB

Children’s innate curiosity lays the foundation for numerous questions about the world around them. Framing these questions as good hypothesis statements can transform them into exciting learning experiments. Presented below are relatable and straightforward examples crafted especially for young minds, offering them a structured way to articulate their wonders and predictions.

• Sunlight & Plant Growth : If a plant gets more sunlight, then it will grow taller.
• Sugary Drinks & Tooth Decay : Drinking sugary drinks daily will lead to faster tooth decay.
• Chocolates & Energy : Eating chocolate will make me feel more energetic.
• Moon Phases & Sleep : I’ll sleep more during a full moon night.
• Homework & Weekend Moods : If I finish my homework on Friday, I’ll be happier over the weekend.
• Pets & Happiness : Owning a pet will make a child happier.
• Rain & Worms : Worms come out more after it rains.
• Shadows & Time of Day : Shadows are longer in the evening than at noon.
• Snow & School Holidays : More snow means there’s a better chance of school being canceled.
• Ice Cream & Brain Freeze : Eating ice cream too fast will give me a brain freeze.
• Video Games & Dreams : Playing video games before bed might make my dreams more vivid.
• Green Vegetables & Strength : Eating more green vegetables will make me stronger.
• Bicycles & Balance : The more I practice, the better I’ll get at riding my bike without training wheels.
• Stars & Wishes : If I wish on the first star I see at night, my wish might come true.
• Cartoons & Laughing : Watching my favorite cartoon will always make me laugh.
• Soda & Bone Health : Drinking soda every day will make my bones weaker.
• Beach Visits & Sunburn : If I don’t wear sunscreen at the beach, I’ll get sunburned.
• Loud Noises & Pet Behavior : My cat hides when she hears loud noises.
• Bedtime & Morning Energy : Going to bed early will make me feel more energetic in the morning.
• Healthy Snacks & Hunger : Eating a healthy snack will keep me full for longer. …
• Toys & Sharing : The more toys I have, the more I want to share with my friends.
• Homemade Cookies & Taste : Homemade cookies always taste better than store-bought ones.
• Books & Imagination : The more books I read, the more adventures I can imagine.
• Jumping & Height : The more I practice, the higher I can jump.
• Singing & Mood : Singing my favorite song always makes me happy.
• Snowmen & Temperature : If the temperature rises, my snowman will melt faster.
• Costumes & Play : Wearing a costume will make playtime more fun.
• Gardening & Patience : Waiting for my plants to grow teaches me patience.
• Night Lights & Sleep : Having a night light makes it easier for me to sleep.
• Handwriting & Practice : The more I practice, the better my handwriting will become.
• Painting & Creativity : Using more colors in my painting lets me express my creativity better.
• Puzzles & Problem Solving : The more puzzles I solve, the better I become at problem-solving.
• Dancing & Coordination : The more I dance, the more coordinated I will become.
• Stargazing & Constellations : If I stargaze every night, I’ll recognize more constellations.
• Bird Watching & Species Knowledge : The more I watch birds, the more species I can identify.
• Cooking & Skill : If I help in the kitchen often, I’ll become a better cook.
• Swimming & Confidence : The more I swim, the more confident I become in the water.
• Trees & Birds’ Nests : The taller the tree, the more likely it is to have birds’ nests.
• Roller Skating & Balance : If I roller skate every weekend, I’ll improve my balance.
• Drawing & Observation : The more I draw, the better I become at observing details.
• Sandcastles & Water : If I use wet sand, I can build a stronger sandcastle.
• Hiking & Endurance : The more I hike, the farther I can walk without getting tired.
• Camping & Outdoor Skills : If I go camping often, I’ll learn more about surviving outdoors.
• Magic Tricks & Practice : The more I practice a magic trick, the better I’ll get at performing it.
• Stickers & Collection : If I collect stickers, my album will become more colorful.
• Board Games & Strategy : The more board games I play, the better strategist I’ll become.
• Pets & Responsibility : The more I take care of my pet, the more responsible I become.
• Music & Concentration : Listening to calm music while studying will help me concentrate better.
• Photographs & Memories : The more photos I take, the more memories I can preserve.
• Rainbows & Rain : If it rains while the sun is out, I might see a rainbow.
• Museums & Knowledge : Every time I visit a museum, I learn something new.
• Fruits & Health : Eating more fruits will keep me healthier.
• Stories & Vocabulary : The more stories I listen to, the more new words I learn.
• Trees & Fresh Air : The more trees there are in a park, the fresher the air will be.
• Diary & Feelings : Writing in my diary helps me understand my feelings better.
• Planets & Telescopes : If I look through a telescope, I’ll see more planets clearly.
• Crafting & Creativity : The more crafts I make, the more creative I become.
• Snowflakes & Patterns : Every snowflake has a unique pattern.
• Jokes & Laughter : The funnier the joke, the louder I’ll laugh.
• Riddles & Thinking : Solving riddles makes me think harder.
• Nature Walks & Observations : The quieter I am on a nature walk, the more animals I’ll spot.
• Building Blocks & Structures : The more blocks I use, the taller my tower will be.
• Kites & Wind : If there’s more wind, my kite will fly higher.
• Popcorn & Movie Nights : Watching a movie with popcorn makes it more enjoyable.
• Stars & Wishes : If I see a shooting star, I should make a wish.
• Diets & Energy : Eating a balanced diet gives me more energy for playtime.
• Clay & Sculptures : The more I play with clay, the better my sculptures will be.
• Insects & Magnifying Glass : Using a magnifying glass will let me see more details of tiny insects.
• Aquarium Visits & Marine Knowledge : Every time I visit the aquarium, I discover a new marine creature.
• Yoga & Flexibility : If I practice yoga daily, I’ll become more flexible.
• Toothpaste & Bubbles : The more toothpaste I use, the more bubbles I’ll get while brushing.
• Journals & Memories : Writing in my journal every day helps me remember special moments.
• Piggy Banks & Savings : The more coins I save, the heavier my piggy bank will get.
• Baking & Measurements : If I measure ingredients accurately, my cake will turn out better.
• Coloring Books & Art Skills : The more I color, the better I get at staying inside the lines.
• Picnics & Outdoor Fun : Having a picnic makes a sunny day even more enjoyable.
• Recycling & Environment : The more I recycle, the cleaner my environment will be.
• Treasure Hunts & Discoveries : Every treasure hunt has a new discovery waiting.
• Milk & Bone Health : Drinking milk daily will make my bones stronger.
• Puppet Shows & Stories : The more puppet shows I watch, the more stories I learn.
• Field Trips & Learning : Every field trip to a new place teaches me something different.
• Chores & Responsibility : The more chores I do, the more responsible I feel.
• Fishing & Patience : Fishing teaches me to be patient while waiting for a catch.
• Fairy Tales & Imagination : Listening to fairy tales expands my imagination.
• Homemade Pizza & Toppings : The more toppings I add, the tastier my homemade pizza will be.
• Gardens & Butterflies : If I plant more flowers, I’ll see more butterflies in my garden.
• Raincoats & Puddles : Wearing a raincoat lets me jump in puddles without getting wet.
• Gymnastics & Balance : The more I practice gymnastics, the better my balance will be.
• Origami & Craft Skills : The more origami I fold, the better my craft skills become.
• Basketball & Shooting Skills : The more I practice, the better I get at shooting baskets.
• Fireflies & Night Beauty : Catching fireflies makes summer nights magical.
• Books & Knowledge : The more books I read, the smarter I become.
• Pillows & Forts : With more pillows, I can build a bigger fort.
• Lemonade & Summers : Drinking lemonade makes hot summer days refreshing.
• Bicycles & Balance : The more I practice, the better I get at riding my bike without training wheels.
• Pencils & Drawings : If I have colored pencils, my drawings will be more colorful.
• Ice Cream & Happiness : Eating ice cream always makes me happy.
• Beach Visits & Shell Collections : Every time I visit the beach, I find new shells for my collection.
• Jump Ropes & Fitness : The more I jump rope, the fitter I become.
• Tea Parties & Imagination : Hosting tea parties lets my imagination run wild.

Simple Hypothesis Statement Examples for Kids

Simple hypothesis are straightforward predictions that can be tested easily. They help children understand the relationship between two variables. Here are some examples tailored just for kids.

• Plants & Sunlight : Plants placed near the window will grow taller than those in the dark.
• Chocolates & Happiness : Eating chocolates can make kids feel happier.
• Rain & Puddles : The more it rains, the bigger the puddles become.
• Homework & Learning : Doing homework helps kids understand lessons better.
• Toys & Sharing : Sharing toys with friends makes playtime more fun.
• Pets & Care : Taking care of a pet fish helps it live longer.
• Storytime & Sleep : Listening to a bedtime story helps kids sleep faster.
• Brushing & Cavity : Brushing teeth daily prevents cavities.
• Games & Skill : Playing a new game every day improves problem-solving skills.
• Baking & Patience : Waiting for cookies to bake teaches patience.

Hypothesis Statement Examples for Kids Psychology

Child psychology hypothesis delves into how kids think, behave, and process emotions. These hypotheses help understand the psychological aspects of children’s behaviors.

• Emotions & Colors : Kids might feel calm when surrounded by blue and energetic with red.
• Friendship & Self-esteem : Making friends can boost a child’s self-confidence.
• Learning Styles & Memory : Some kids remember better by seeing, while others by doing.
• Play & Development : Pretend play is crucial for cognitive development.
• Rewards & Motivation : Giving small rewards can motivate kids to finish tasks.
• Music & Mood : Listening to soft music can calm a child’s anxiety.
• Sibling Bonds & Sharing : Having siblings can influence a child’s willingness to share.
• Feedback & Performance : Positive feedback can improve a kid’s academic performance.
• Outdoor Play & Attention Span : Playing outside can help kids concentrate better in class.
• Dreams & Reality : Kids sometimes can’t differentiate between dreams and reality.

Hypothesis Examples in Kid Friendly Words

Phrasing hypothesis in simple words makes it relatable and easier for kids to grasp. Here are examples with kid-friendly language.

• Socks & Warmth : Wearing socks will keep my toes toasty.
• Jumping & Energy : The more I jump, the more energy I feel.
• Sandcastles & Water : A little water makes my sandcastle stand tall.
• Stickers & Smiles : Getting a sticker makes my day shine brighter.
• Rainbows & Rain : After the rain, I might see a rainbow.
• Slides & Speed : The taller the slide, the faster I go.
• Hugs & Love : Giving hugs makes me and my friends feel loved.
• Stars & Counting : The darker it is, the more stars I can count.
• Paint & Mess : The more paint I use, the messier it gets.
• Bubbles & Wind : If I blow my bubble wand, the wind will carry them high.

Hypothesis Statement Examples for Kids in Research

Even in a research setting, research hypothesis should be age-appropriate for kids. These examples focus on concepts children might encounter in structured studies.

• Reading & Vocabulary : Kids who read daily might have a richer vocabulary.
• Games & Math Skills : Playing number games can improve math skills.
• Experiments & Curiosity : Conducting science experiments can make kids more curious.
• Doodles & Creativity : Drawing daily might enhance a child’s creativity.
• Learning Methods & Retention : Kids who learn with visuals might remember lessons better.
• Discussions & Understanding : Talking about a topic can deepen understanding.
• Observation & Knowledge : Observing nature can increase a kid’s knowledge about the environment.
• Puzzles & Cognitive Skills : Solving puzzles regularly might enhance logical thinking.
• Music & Rhythmic Abilities : Kids who practice music might develop better rhythm skills.
• Teamwork & Social Skills : Group projects can boost a child’s social skills.

Hypothesis Statement Examples for Kids Science Fair

Science fairs are a chance for kids to delve into the world of experiments and observations. Here are hypotheses suitable for these events.

• Magnet & Metals : Certain metals will be attracted to a magnet.
• Plants & Colored Light : Plants might grow differently under blue and red lights.
• Eggs & Vinegar : An egg in vinegar might become bouncy.
• Solar Panels & Sunlight : Solar panels will generate more power on sunny days.
• Volcanoes & Eruptions : Mixing baking soda and vinegar will make a mini eruption.
• Mirrors & Reflection : Shiny surfaces can reflect light better than dull ones.
• Battery & Energy : Fresh batteries will make a toy run faster.
• Density & Floating : Objects with lower density will float in water.
• Shadows & Light Source : Moving the light source will change the shadow’s direction.
• Freezing & States : Water turns solid when kept in the freezer.

Hypothesis Statement Examples for Science Experiments

Experiments let kids test out their predictions in real-time. Here are hypotheses crafted for various scientific tests.

• Salt & Boiling Point : Adding salt will make water boil at a higher temperature.
• Plants & Music : Playing music might affect a plant’s growth rate.
• Rust & Moisture : Metals kept in a moist environment will rust faster.
• Candles & Oxygen : A candle will burn out faster in an enclosed jar.
• Fruits & Browning : Lemon juice can prevent cut fruits from browning.
• Yeast & Sugar : Adding sugar will make yeast activate more vigorously.
• Density & Layers : Different liquids will form layers based on their density.
• Acids & Bases : Red cabbage juice will change color in acids and bases.
• Soil Types & Water : Sandy soil will drain water faster than clay.
• Thermometers & Temperatures : Thermometers will show higher readings in the sun.

Hypothesis Statement Examples for Kids At Home

These hypotheses are crafted for experiments and observations kids can easily make at home, using everyday items.

• Chores & Time : Setting a timer will make me finish my chores faster.
• Pets & Behavior : My cat sleeps more during the day than at night.
• Recycling & Environment : Recycling more can reduce the trash in my home.
• Cooking & Tastes : Adding spices will change the taste of my food.
• Family Time & Bonding : Playing board games strengthens our family bond.
• Cleaning & Organization : Organizing my toys daily will keep my room tidier.
• Watering & Plant Health : Watering my plant regularly will keep its leaves green.
• Decor & Mood : Changing the room decor can influence my mood.
• Journals & Memories : Writing in my journal daily will help me remember fun events.
• Photos & Growth : Taking monthly photos will show how much I’ve grown.

How do you write a hypothesis for kids? – A Step by Step Guide

Step 1: Start with Curiosity Begin with a question that your child is curious about. This could be something simple, like “Why is the sky blue?” or “Do plants need sunlight to grow?”

Step 2: Observe and Research Before formulating the hypothesis, encourage your child to observe the world around them. If possible, read or watch videos about the topic to gather information. The idea is to get a general understanding of the subject.

Step 3: Keep it Simple For kids, it’s essential to keep the hypothesis straightforward and concise. Use language that is easy to understand and relatable to their age.

Step 4: Make a Predictable Statement Help your child frame their hypothesis as an “If… then…” statement. For example, “If I water a plant every day, then it will grow taller.”

Step 5: Ensure Testability Ensure that the hypothesis can be tested using simple experiments or observations. It should be something they can prove or disprove through hands-on activities.

Step 6: Avoid Certainty Teach kids that a hypothesis is not a definitive statement of fact but rather a best guess based on what they know. It’s okay if the hypothesis turns out to be wrong; the learning process is more important.

Step 7: Review and Refine After forming the initial hypothesis, review it with your child. Discuss if it can be made simpler or clearer. Refinement aids in better understanding and testing.

Step 8: Test the Hypothesis This is the fun part! Plan an experiment or set of observations to test the hypothesis. Whether the hypothesis is proven correct or not, the experience provides a learning opportunity.

Tips for Writing Hypothesis for Kids

• Encourage Curiosity : Always encourage your child to ask questions about the world around them. It’s the first step to formulating a hypothesis.
• Use Familiar Language : Use words that the child understands and can relate to. Avoid jargon or technical terms.
• Make it Fun : Turn the process of forming a hypothesis into a game or a storytelling session. This will keep kids engaged.
• Use Visual Aids : Kids often respond well to visuals. Drawing or using props can help in understanding and formulating the hypothesis.
• Stay Open-minded : It’s essential to teach kids that it’s okay if their hypothesis is wrong. The process of discovery and learning is what’s crucial.
• Practice Regularly : The more often kids practice forming hypotheses, the better they get at it. Use everyday situations as opportunities.
• Link to Real-life Scenarios : Relate the hypothesis to real-life situations or personal experiences. For instance, if discussing plants, you can relate it to a plant you have at home.
• Collaborate : Sometimes, two heads are better than one. Encourage group activities where kids can discuss and come up with hypotheses together.
• Encourage Documentation : Keeping a journal or notebook where they document their hypotheses and results can be a great learning tool.
• Celebrate Efforts : Regardless of whether the hypothesis was correct, celebrate the effort and the learning journey. This reinforces the idea that the process is more important than the outcome.

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‘Sensational’ Proof Delivers New Insights Into Prime Numbers

July 15, 2024

Nico Roper/ Quanta Magazine

Introduction

Sometimes mathematicians try to tackle a problem head on, and sometimes they come at it sideways. That’s especially true when the mathematical stakes are high, as with the Riemann hypothesis, whose solution comes with a $1 million reward from the Clay Mathematics Institute. Its proof would give mathematicians much deeper certainty about how prime numbers are distributed, while also implying a host of other consequences — making it arguably the most important open question in math.

Mathematicians have no idea how to prove the Riemann hypothesis. But they can still get useful results just by showing that the number of possible exceptions to it is limited. “In many cases, that can be as good as the Riemann hypothesis itself,” said James Maynard of the University of Oxford. “We can get similar results about prime numbers from this.”

In a breakthrough result posted online in May, Maynard and Larry Guth of the Massachusetts Institute of Technology established a new cap on the number of exceptions of a particular type, finally beating a record that had been set more than 80 years earlier. “It’s a sensational result,” said Henryk Iwaniec of Rutgers University. “It’s very, very, very hard. But it’s a gem.”

The new proof automatically leads to better approximations of how many primes exist in short intervals on the number line, and stands to offer many other insights into how primes behave.

A Careful Sidestep

The Riemann hypothesis is a statement about a central formula in number theory called the Riemann zeta function. The zeta ($latex \zeta$) function is a generalization of a straightforward sum:

$latex 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \cdots $.

This series will become arbitrarily large as more and more terms are added to it — mathematicians say that it diverges. But if instead you were to sum up

$latex 1 + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \frac{1}{5^2} + \cdots = 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \frac{1}{25} + \cdots $

you would get $latex \frac{\pi^2}{6}$, or about 1.64. Riemann’s surprisingly powerful idea was to turn a series like this into a function, like so:

$latex \zeta (s) = 1 + \frac{1}{2^s} + \frac{1}{3^s} + \frac{1}{4^s} + \frac{1}{5^s} + \cdots$.

So $latex \zeta (1)$ is infinite, but $latex \zeta (2) = \frac{\pi^2}{6}$.

Things get really interesting when you let s be a complex number, which has two parts: a “real” part, which is an everyday number, and an “imaginary” part, which is an everyday number multiplied by the square root of −1 (or i , as mathematicians write it). Complex numbers can be plotted on a plane, with the real part on the x -axis and the imaginary part on the y -axis. Here, for example, is 3 + 4 i .

The zeta function takes points on the complex plane as inputs, and it produces other complex numbers as outputs. It turns out that for some complex numbers, the zeta function is equal to zero. Figuring out where those zeros are located on the complex plane is one of the most interesting questions in mathematics.

In 1859, Bernhard Riemann conjectured that all the zeros are concentrated on two lines. If you extend the zeta function so you can compute it for negative inputs, you’ll find that it equals zero for all negative even numbers: −2, −4, −6 and so on. This is relatively easy to show, so these are called trivial zeros. Riemann conjectured that all the other zeros of the function, called nontrivial zeros, have a real part of 1/2, and so are located on this vertical line.

This is the Riemann hypothesis, and proving it has been prohibitively difficult. Mathematicians know that every nontrivial zero must have a real part between zero and 1, but they can’t rule out that some zeros might have a real part of, say, 0.499.

What they can do is show that such zeros must be incredibly rare. In 1940, an English mathematician named Albert Ingham established an upper bound on the number of zeros whose real part is not equal to 1/2 that mathematicians continue to use as a point of reference today.

A few decades later, in the 1960s and ’70s, other mathematicians figured out how to translate Ingham’s result into statements about how clumped or spread out prime numbers are as you move further along the number line, and about other patterns they might form. Around the same time, mathematicians also introduced new techniques that improved Ingham’s bounds for zeros with a real part greater than 3/4.

But it turned out that the most important zeros to cap were those with a real part of exactly 3/4. “Lots of headline results about prime numbers were limited by our understanding of zeros with real part 3/4,” Maynard said.

About a decade ago, Maynard started thinking about how to improve Ingham’s estimate for those particular zeros. “It’s been one of my favorite problems in analytic number theory,” he said. “It always felt tempting that you just have to work a bit harder, and you’ll be able to get an improvement.” But year after year, no matter how many times he came back to it, he kept getting stuck. “It almost sucked you in, and it looked much more innocent than I think it was.”

Then, in early 2020, during a plane trip to a conference in Colorado, an idea came to him. Perhaps, Maynard thought, tools from another area of math called harmonic analysis might be useful.

Larry Guth, an expert in harmonic analysis who was at the same conference, just happened to already be thinking along similar lines. “But I didn’t know the analytic number theory at all well,” he said. Maynard explained the number theory side of the story to him over lunch and gave him a test case to work with. Guth studied it on and off for a few years, only to realize that his techniques from harmonic analysis wouldn’t work.

But he didn’t stop thinking about the problem, and he experimented with new approaches. He got back in touch with Maynard in February. The two started collaborating in earnest, combining their different perspectives. A few months later, they had their result.

A Mathematical Gambit

Guth and Maynard started out by converting the problem they wanted to solve into another one. If you have a zero that doesn’t have a real part of 1/2, then a related function, called a Dirichlet polynomial, must produce a very large output. As a result, proving that there are few exceptions to the Riemann hypothesis is equivalent to showing that the Dirichlet polynomial cannot get large too often.

Video : Alex Kontorovich, professor of mathematics at Rutgers University, breaks down the notoriously difficult Riemann hypothesis in this comprehensive explainer.

Emily Buder/Quanta Magazine; Guan-Huei Wu and Clay Shonkwiler for Quanta Magazine

The mathematicians then performed another act of translation. First, they used the Dirichlet polynomial to build a matrix, or a table of numbers. “Mathematicians love to see matrices, because matrices are one of the things that we understand really well,” Guth said. “You learn to keep your ears open and be ready to see that there are matrices all over the place.”

Matrices can “act on” a mathematical arrow called a vector, which is defined by a length and direction, to produce another vector. They generally change both the length and direction of the vector when they do so. Sometimes there are special vectors that, when run through a matrix, change only in length but not direction. These are called eigenvectors. Mathematicians measure the size of those changes using numbers called eigenvalues.

Guth and Maynard rewrote their problem so that it was now about the largest eigenvalue of their matrix. If they could show that the largest eigenvalue could not get too big, they’d be done. To do that, they used a formula that gave them a complicated sum, and searched for ways to make the positive and negative values in that sum cancel each other out as much as possible. “You have to rearrange the sequence, or look at it from the right angle, in order to see some symmetry that gives some cancellation,” Guth said.

That process involved several surprising steps, including “what’s in my mind the most important idea, which still seems a bit magical to me,” Maynard said. At one point, there was a seemingly obvious step they should have taken to simplify their sum. Instead, they left it in its longer and more complicated form. “We do something that at first sight looks completely stupid. We just refuse to do the standard simplification,” Maynard said. “And this gives up a lot. It means that now we can’t get any easy bound for this sum.”

But in the long run, this turned out to be an advantageous move. “In chess you call it a gambit, where you sacrifice a piece to get a better position on the board,” Maynard said. Guth likened it to playing with a Rubik’s Cube; sometimes you have to undo previous moves and make everything look worse before finding a way to get more colors in the right place.

Larry Guth’s expertise in harmonic analysis gave him a fresh perspective on a number theory problem that had resisted proof for decades.

Bryce Vickmark

“You have to be really brave to throw away an obvious improvement and hope that you can recover it later,” said Roger Heath-Brown , a mathematician at Oxford and Maynard’s former adviser. “That goes against everything that I thought you should be doing.”

In fact, he added of his own experiences working on this problem, “now that I think about it, that’s where I got stuck.”

Maynard said that Guth’s expertise as a harmonic analyst rather than a number theorist made this gambit possible. “He doesn’t inherently have these rules drilled into him, so he was more happy to consider things that go against the grain.”

Ultimately, they were able to get a good enough bound on the largest eigenvalue, which in turn translated to a better bound on the number of potential counterexamples to the Riemann hypothesis. Although their work began with the ideas from harmonic analysis that had inspired Guth, the mathematicians were ultimately able to cut those more complicated techniques out of the picture. “Now it looks exactly like the sort of thing I might have tried to do 40 years ago,” Heath-Brown said.

By giving a better bound on the number of zeros with a real part of 3/4, Guth and Maynard automatically proved results about how prime numbers are distributed. For example, estimates of how many primes are found in a given interval get less accurate for shorter intervals. The new work has allowed mathematicians to shorten the intervals in which they can get good estimates.

Mathematicians suspect that the proof will yield improvements to other statements about primes as well. There seems to be room to push Guth and Maynard’s techniques further. But “I feel that these aren’t the right techniques to solve the Riemann hypothesis itself,” Maynard said. “It’s going to need some big idea from somewhere else.”

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Lab on a Chip

Optimized microfluidic formulation and organic excipients for improved lipid nanoparticle mediated genome editing.

* Corresponding authors

a Department of Bioengineering, University of Pennsylvania, Philadelphia, PA, USA E-mail: [email protected]

b Center for Fetal Research, Children's Hospital of Philadelphia, Philadelphia, PA, USA E-mail: [email protected]

c Abramson Cancer Center, Perelman School of Medicine, University of Pennsylvania, Philadelphia, PA 19104, USA

d Center for Cellular Immunotherapies, Perelman School of Medicine, University of Pennsylvania, Philadelphia, PA 19104, USA

e Penn Institute for RNA Innovation, Perelman School of Medicine, University of Pennsylvania, Philadelphia, PA 19104, USA

f Institute for Immunology, Perelman School of Medicine, University of Pennsylvania, Philadelphia, PA 19104, USA

g Cardiovascular Institute, Perelman School of Medicine, University of Pennsylvania, Philadelphia, PA 19104, USA

h Institute for Regenerative Medicine, Perelman School of Medicine, University of Pennsylvania, Philadelphia, PA 19104, USA

mRNA-based gene editing platforms have tremendous promise in the treatment of genetic diseases. However, for this potential to be realized in vivo , these nucleic acid cargos must be delivered safely and effectively to cells of interest. Ionizable lipid nanoparticles (LNPs), the most clinically advanced non-viral RNA delivery system, have been well-studied for the delivery of mRNA but have not been systematically optimized for the delivery of mRNA-based CRISPR-Cas9 platforms. In this study, we investigated the effect of microfluidic and lipid excipient parameters on LNP gene editing efficacy. Through in vitro screening in liver cells, we discovered distinct trends in delivery based on phospholipid, cholesterol, and lipid-PEG structure in LNP formulations. Combination of top-performing lipid excipients produced an LNP formulation that resulted in 3-fold greater gene editing in vitro and facilitated 3-fold greater reduction of a therapeutically-relevant protein in vivo relative to the unoptimized LNP formulation. Thus, systematic optimization of LNP formulation parameters revealed a novel LNP formulation that has strong potential for delivery of gene editors to the liver to treat metabolic disease.

• This article is part of the themed collection: Lab on a Chip Emerging Investigators

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R. Palanki, E. L. Han, A. M. Murray, R. Maganti, S. Tang, K. L. Swingle, D. Kim, H. Yamagata, H. C. Safford, K. Mrksich, W. H. Peranteau and M. J. Mitchell, Lab Chip , 2024, Advance Article , DOI: 10.1039/D4LC00283K

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How to Write a Hypothesis? [Tips with Examples]

Click here if you have ever found yourself in the position of having to wrestle with the development of a hypothesis for your research paper. As an expert writer, I have seen that this is where most students begin to sweat. It is a potpourri of theory and practice, hence rather intimidating. But not to worry because I have got your back. This guide is a pool of tips and tricks for writing a hypothesis to set the stage for compelling research.

What is a Hypothesis?

A hypothesis is a tentative statement, usually in the form of an educated guess, that provides a probable explanation for something either a phenomenon or a relationship between variables. This will, therefore, form a basis for conducting experiments and research studies, hence laying down the course of your investigation and mainly laying the ground for your conclusion.

A good hypothesis should be:

Specific and clear

Testable and falsifiable

Based upon existing knowledge

Logically consistent

Types of Hypothesis

There are different kinds of hypotheses used in research, all of which serve different purposes depending on the nature of the study. Here are eight common types:

1. The null hypothesis (H0):  asserts that there is no effect or relationship between variables. This forms a baseline for comparison. Example: "There is no difference in test scores for students who study music and for those who do not."

2. Alternative Hypothesis (H1): The hypothesis that postulates some effect or relationship between variables; it is, therefore, the opposite of the null hypothesis. For instance, "Students who study with music have different test scores than those who study in silence."

3. Simple Hypothesis: The hypothesis that states a relationship between two variables: one independent and one dependent. For example, "More sunlight increases plant growth."

4. Complex Hypothesis: This hypothesis involves the relationship of more than one variable. For example, "More sunlight and water increase plant growth."

5. Directional Hypothesis: The hypothesis which specifies the direction of the effect between variables. For instance, "Students who study with music will have higher test scores than students who study in silence."

6. Non-Directional Hypothesis: This is a hypothesis used where the relationship is indicated, but the direction is not specified. For example, "There is a difference in test scores between students who study with music and those who study in silence."

7. Associative Hypothesis: This hypothesis merely states that the change in one variable is associated with a change in another. It does not indicate cause and effect. For example: "There is a relationship between study habits and academic performance."

8. Causal Hypothesis: This hypothesis states that one variable causes a change in another. For example: "Increased study time results in higher test scores."

Understanding such types of hypotheses will help in the selection of the correct hypothesis for your research and in making your analysis clear and effective.

5 Steps to Write a Good Hypothesis [With Examples]

An excellent hypothesis provides a backbone to any scientific research. Leave some help behind in writing one? Follow this easy guide:

Step 1: Ask a Question

First, you must understand what your research question is. Suppose you want to carry out an experiment on plant growth. Your question can be, "How does sunlight affect plant growth?"

Use WPS AI to help when you get stuck. Feed it a topic, and it will come up with related questions to ask.

Step 2: Do Preliminary Research

Do some research to see what's already known about your topic. That way, you can build upon existing knowledge.

Research information in journals, books and credible websites. Then summarize what you read. This will help you formulate your hypothesis.

Step 3: Define Variables

Identify your variables:

Independent Variable: What you manipulate. For example, the amount of sun.

Dependent Variable: What you measure. For example, plant growth rate.

Clearly defining these makes your hypothesis specific and testable.

Step 4: State Your Hypothesis

State your question in the form of a hypothesis. Here are some examples:

If  then: "If plants receive more sunlight, then they will grow faster."

Comparative statements: "Plants receiving more sunlight grow faster than plants receiving less."

Correlation statements: "There is positive correlation between sunlight and plant growth." This kind of pattern makes your hypothesis easy to test.

Step 5: Refine Your Hypothesis

Revise your hypothesis to be clear and specific, and elicit feedback to improve it.

You will also need a null hypothesis, which says that there is no effect or relationship between variables. An example would be, "Sunlight has no effect on the growth of plants."

With these steps, you are now bound to come up with a testable hypothesis. WPS AI can help you in this process more efficiently.

Characteristics of a Good Hypothesis

A good hypothesis is seen as the backbone of doing effective research. Following are some key characteristics that define a good hypothesis:

A good hypothesis has to be testable either by experimentation or observation. The hypothesis should clearly predict what can be measured or observed. For example, "If it receives more sunlight, the plant will grow taller" is a testable hypothesis since it states what can be measured.

Falsifiable

A hypothesis has to be falsifiable: it should be able to prove it wrong. This feature is important because it accommodates testing in science. For example, the statement "All swans are white" is falsifiable since it just takes one black swan to disprove the claim.

A good hypothesis should be grounded in current knowledge and should be properly reasoned. It should be broad or reasonable within existing knowledge. For example, "Increasing the amount of sunlight will boost plant growth" makes sense, in that it tallies with generally known facts about photosynthesis.

Specific and Clear

What is needed is clarity and specificity. A hypothesis has to be brief, yet free from ambiguity. For instance, "Increased sunlight leads to taller plants" is clear and specific whereas "Sunlight affects plants" is too vague.

Built upon Prior Knowledge

A good hypothesis is informed by prior research and existing theories. The available knowledge enlightens it to build on what is known to find new relationships or effects. For example, "Given photosynthesis requires sunlight, increasing sunlight will enhance plant growth" is informed by available scientific understanding.

Ethical Considerations

Finally, a good hypothesis needs to consider the ethics involved. The research should not bring damage to participants or the environment. For instance, "How the new drug will affect a human when tested without testing it on animals" may present an ethical concern.

Checklist for Reviewing Your Hypothesis

To be certain that your hypothesis has the following characteristics, use this checklist to review your hypothesis:

1. Is the hypothesis testable through experimentation or observation?

2. Can the hypothesis be proven false?

3. Is the hypothesis logically deduced from known facts?

4. Is your hypothesis clear and specific?

5. Does your hypothesis relate to previous research or theories?

6. Will there be any ethical issues with the proposed research?

7. Are your independent and dependent variables well defined?

8. Is your hypothesis concise and ambiguity free?

9. Did you get feedback to help in refining your hypothesis?

10. Does your hypothesis contain a null hypothesis for comparison?

By making sure that your hypothesis has these qualities, you are much more likely to set yourself on the course of higher-quality research and larger impacts. WPS AI can help fine-tune a hypothesis to ensure it is well-structured and clear.

Using WPS to Perfect your Hypothesis

Drafting a good hypothesis is the real inception of any research project. WPS AI, with its advanced language functions, can very strongly improve this stage of your study. Here's how WPS AI can help you perfect your hypothesis:

Check Grammar and Syntax

Grammar and punctuation errors can make your hypothesis weak. WPS AI checks and corrects this with the assurance that your hypothesis is as clear as possible and professional in its presentation. For example, when your hypothesis is written, "If the temperature increases then plant growth will increases", WPS AI can correct it to "If the temperature increases, then plant growth will increase."

Rewrite Your Hypothesis for Clarity

There needs to be a clear hypothesis. WPS AI can suggest ways to reword your hypothesis so that it makes sense. If your original hypothesis is, "More sunlight will result in more significant plant growth due to photosynthesis," WPS AI can suggest, "Increased sunlight will lead to greater plant growth through enhanced photosynthesis."

Automatic Content Expansion

Sometimes, your hypothesis or the related paragraphs may require more detail. WPS AI's [Continue Writing] feature can help enlarge the content. For example, after having written, "This study will examine the effects of sunlight on plant growth", using [Continue Writing] it can enlarge it to, "This research paper is going to study how sunlight affects the growth of plants by measuring their height and their health under different amounts of sunlight over a period of six weeks."

WPS AI is a great tool that can help you in drafting a good hypothesis for your research. It will help you check grammar, syntax, clarity, and completeness. Using WPS AI , you will be assured that the results of your hypothesis will be well-written and clear to understand.

What is the difference between a hypothesis and a theory?

The hypothesis is one single testable prediction regarding some phenomenon. The theory is an explanation for some part of the natural world which is well-substantiated by a body of evidence, together with multiple hypotheses.

What do I do if my hypothesis isn't supported by my data?

If your results turn out not to support your hypothesis, analyze the data again to see why your result rejects your hypothesis. Do not manipulate the observations or experiment so that it leads to your hypothesis.

Can there be more than one hypothesis in a research study?

Yes, there may be more than one hypothesis, especially when one research study is examining several interrelated phenomena or variables. Each hypothesis has to be separately and clearly stated and tested.

Correct formulation of a strong, testable hypothesis is one of the most critical steps in the application of the scientific method and within academic research. The steps provided in this article will help you write a hypothesis that is clear, specific, and based on available knowledge. Give the tools and tips a try to elevate your academic writing and kick your research up a notch.

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