experiment in math meaning

In probability and statistics, an experiment typically refers to a study in which the experimenter is trying to determine whether there is a relationship between two or more variables. In an experiment, the subjects are randomly assigned to either a treatment group or a control group (there can be more than one of either group).

Generally, the control group in an experiment receives a placebo (substance that has no effect) or no treatment at all. The treatment group receives the experimental treatment. The goal of the experiment is to determine whether or not the treatment has the desired/any effect that differs from the control group to a degree that the difference can be attributed to the treatment rather than to random chance or variability. Well-designed experiments can yield informative and unambiguous conclusions about cause and effect relationships.

As an example, if a scientist wants to test whether a new medication they developed has any effect, they would select subjects from a common population and randomly assign them to either a treatment group or a control group. They would then administer the treatment to the treatment group, and either a placebo or no treatment to the control group, and study the effects of each using statistical measures to determine whether the medication had any effect beyond chance or variability.

Note that an experiment does not necessarily need to have a physical treatment. The term "treatment" is used fairly loosely. Another experiment could look at the effects of getting advice from a college counselor on admission rates compared to not getting advice from a college counselor. In this case, the "treatment" would be getting advice from a college counselor. The control group would get no advice from a college counselor.

Importance of experimental design

Like survey methodology , experimental design is essential to the validity of the results of the experiment. A poorly designed experiment can result in false or incorrect conclusions. Proper statistical experiment design generally involves the following:

• Identification of the explanatory variable, also referred to as the independent variable . The explanatory variable is the "treatment," or the thing that causes the change, and can be anything that causes a change in the response variable.
• Identification of the response variable, also referred to as the dependent variable . It is the variable that may be affected by the explanatory/independent variable.
• Defining the population of interest and taking a random sample from the population. Generally the larger the random sample, the less potential for sample error, since the larger sample will likely be more representative of the population.
• Random assignment of the subjects in the sample to either the treatment group or the control group.
• Administration of the treatment to the treatment group, and placebo (or nothing) to the control group), possibly using a blind experiment (the subject doesn't know whether they are receiving the treatment or the placebo) or double blind experiment (neither experimenter nor subject knows which treatment they are getting).
• Measurement of the response over a chosen period of time.
• Statistical analysis of the supposed response to determine whether there is an actual response, or the response can be attributed to chance, to determine whether there is a causal relationship between the treatment and the response.
• Replication of the experiment by peers, assuming there is a causal relationship between the treatment and the response.

Experiments vs surveys

Experiments and surveys are both techniques used as part of inferential statistics . A survey involves the use of a random sample of the population, rather than the whole, with the goal that all subjects in the population have an equal chance of being selected. The random sample of the population is then used to draw conclusions or make inferences about the population as a whole.

In contrast, an experiment typically involves the use of random assignment such that all subjects have an equal chance of being assigned to the groups (treatment and control) in the study, which minimizes potential biases, as well as allows the experimenters to evaluate the role of variability in the experiment. This in turn allows them to determine whether any observed differences between the groups merit further study or not based on whether or not the differences can be attributed to variability or chance.

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Basics of an Experiment 

Controlled variables, independent variable, dependent variable, types of experiments, experiment|definition & meaning.

An experiment is a series of procedures and results that are carried out to answer a specific issue or problem or to confirm or disprove a theory or body of knowledge about a phenomenon.

Figure 1: Illustration of an Experiment

The scientific method, a methodical approach to learning about the world around you, is founded on the concept of the experiment . Even while some experiments are conducted in labs, you can conduct an experiment at every time and everywhere.

• The key stages of the scientific process are as follows:
• Keenly Observe things.
• Develop a hypothesis .
• Create and carry out an experiment to verify Your hypothesis.
• Analyze the findings of your experiment.
• Based on the analysis of your results, approve or refute your hypothesis.
• Create a new hypothesis , if required, and evaluate it.

Types of Variables in an Experiment

A variable is, to put it simply, everything that can be altered or managed throughout an experiment. Humidity, the length of the study, the structure of an element, the intensity of sunlight, etc. are typical examples of variables. In any study, there are 3 major types of variables :

• Controlled variables (c.v)
• Independent variables (i.v)
• Dependent variables (d.v).

Figure 2: Illustration of Types of Variables

Variables that are maintained constant or unchangeable are known as controlled variables, sometimes known as constant variables. For instance, if you were evaluating the amount of fizz emitted by various sodas, you may regulate the bottle size to ensure that all soda manufacturers were in 12- ounce bottles. If you were conducting an experiment on the effects of spraying plants with various chemicals, you will attempt to keep a similar pressure and perhaps a similar amount when spraying the plants.

The only variable that  you can modify  is the independent variable. It is one factor as you typically try to adjust one element at a time in experiments. As a result, measuring and interpretation of data are made quite simple. For instance, if you’re attempting to establish whether raising the temperature makes it possible to solvate more amount of sugar in the water, the water temperature is the independent variable. This is the factor that you are consciously in control of.

The variable that is  monitored  to determine whether or not your independent variable has an impact is known as the dependent variable. For instance, in the case where you raise the water temperature to observe if it has an impact on the solubility of sugar in it, the weight or volume of sugar (depending on which one you want to calculate) will be the dependent variable.

There are three main types of experiments. Each has its own pros and cons and is carried out according to the nature of the given scenario and desired outcomes. Following are the names of these three types.

Quasi Experiment

Controlled experiment, field experiment.

Figure 3: Illustration of Types of Experiments

Each of these experiments is discussed below along with their strengths and weaknesses.

These are often conducted in a natural environment and involve measuring the impact of one object on another to determine its impact (D.V.). In Quasi-experiments, the research is simply assessing the impact of an event that is already occurring because there is no intentional modification of the variable in this instance; rather, it is changing naturally.

Owing to the unavailability of the researcher, variables occur   naturally , allowing for easy generalization of results to other (real-life) situations, which leads to greater ecological validity.

Absence of control – Quasi-experiments possess poor internal validity since the experimenter cannot always precisely analyze the impact of the independent variable because there is no influence over the environment or other supplementary variables.

Non-repeatable – Because the researcher has no control over the research process, the validity of the findings cannot be verified.

Controlled experiments are also known as lab experiments . Controlled experiments are carried out under carefully monitored conditions, with the researcher purposefully altering one variable (Independent Variable) to determine how it affects another (dependent Variable).

Control – lab studies have a higher level of environmental and other extrinsic variable control, which allows the scientist to precisely examine the impact of the Independent Variable, increasing internal validity.

Replicable – because of the researcher’s greater degree of control, research techniques may be replicated so that the accuracy of the findings can be verified.

Absence of ecological validity — results are difficult to generalize to other (real-life) situations because of the researcher’s participation in modifying and regulating variables, which leads to poor external validity.

A field experiment could be a controlled or a Quasi-experiment. Instead of taking place in a laboratory, it occurs in the actual world. An illustration of a field experiment could be one that involved an organism in its natural environment.

Validity : Because field experiments are carried out in a natural setting and with a certain level of control, they are considered to possess adequate internal and external validity.

Internal validity is believed to be poorer because there is less control than in lab trials, making it more probable that uncontrollable factors would skew results.

An Example of Identifying the Variables in an Experiment

A farmer wants to determine the effect of different amounts of fertilizer on his crop yield. The farmer does not change the amount of water given to the crop for different amounts of fertilizer applied to the field. Determine which of the variables is the controlled variable, independent variable, and independent variable. Also, mention the reasons behind it.

Figure 4: Illustration of the Example

Controlled variable : The amount of water given to the crop is a controlled variable since it is not changed when different amounts of fertilizer are applied.

Independent variable : The amount of fertilizer added to crops is the independent variable. This is because it is the variable which is being manipulated to determine its impact on crop yield.

Dependent variable : Crop yield is the dependent variable in this example. This is because it is the variable on which the impact of the independent variable (Amount of fertilizer) is being monitored .

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Experimental Probability

Experimental probability: introduction, experimental probability: definition, experimental probability formula, solved examples, practice problems, frequently asked questions.

In mathematics, probability refers to the chance of occurrence of a specific event. Probability can be measured on a scale from 0 to 1. The probability is 0 for an impossible event. The probability is 1 if the occurrence of the event is certain.

There are two approaches to study probability: experimental and theoretical. 

Suppose you and your friend toss a coin to decide who gets the first turn to ride a new bicycle. You choose “heads” and your friend chooses “tails.” 

Can you guess who will win? No! You have $\frac{1}{2}$ a chance of winning and so does your friend. This is theoretical since you are predicting the outcome based on what is expected to happen and not on the basis of outcomes of an experiment.

So, what is the experimental probability? Experimental probability is calculated by repeating an experiment and observing the outcomes. Let’s understand this a little better.

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Experimental probability, or empirical probability, is the probability calculated by performing actual experiments and gathering or recording the necessary information. How would you define an experiment? The math definition of an experiment is “a process or procedure that can be repeated and that has a set of well-defined possible results or outcomes.”

Consider the same example. Suppose you flip the coin 50 times to see whether you get heads or tails, and you record the outcomes. Suppose you get heads 20 times and tails 30 times. Then the probability calculated using these outcomes is experimental probability. Here, t he experimental meaning is connected with such experiments used to determine the probability of an event.

Now that you know the meaning of experimental probability, let’s understand its formula.

Experimental Probability for an Event A can be calculated as follows:

P(E) $= \frac{Number of occurance of the event A}{Total number of trials}$

Let’s understand this with the help of the last example. 

A coin is flipped a total of 50 times. Heads appeared 20 times. Now, what is the experimental probability of getting heads?

E xperimental probability of getting heads $= \frac{Number of occurrences}{Total number of trials}$

P (Heads) $= \frac{20}{50} = \frac{2}{5}$

P (Tails) $= \frac{30}{50} = \frac{3}{5}$

Experimental Probability vs. Theoretical Probability

Theoretical probability expresses what is expected. On the other hand, experimental probability explains how frequently an event occurred in an experiment.

If you roll a die, the theoretical probability of getting any particular number, say 3, is $\frac{1}{6}$. 

However, if you roll the die 100 times and record how many times 3 appears on top, say 65 times, then the experimental probability of getting 3 is $\frac{65}{100}$.

Theoretical probability for Event A can be calculated as follows:

P(A) $= \frac{Number of outcomes favorable to Event A}{Number of possible outcomes}$

In the example of flipping a coin, the theoretical probability of the occurrence of heads (or tails) on tossing a coin is

P(H) $= \frac{1}{2}$ and  P(T) $= \frac{1}{2}$ (since possible outcomes are $2 -$ head or tail)

Experimental Probability: Examples

Let’s take a look at some of the examples of experimental probability .

Example 1: Ben tried to toss a ping-pong ball in a cup using 10 trials, out of which he succeeded 4 times. 

P(win) $= \frac{Number of success}{Number of trials}$

             $= \frac{4}{10}$

             $= \frac{2}{5}$

Example 2: Two students are playing a game of die. They want to know how many times they land on 2 on the dice if the die is rolled 20 times in a row. 

The experimental probability of rolling a 2 

$= \frac{Number of times 2 appeared}{Number of trials}$

$= \frac{5}{20}$

$= \frac{1}{4}$

1. Probability of an event always lies between 0 and 1.

2. You can also express the probability as a decimal and a percentage.

Experimental probability is a probability that is determined by the results of a series of experiments. Learn more such interesting concepts at SplashLearn .

1. Leo tosses a coin 25 times and observes that the “head” appears 10 times. What is the experimental probability of getting a head?

 P(Head) $= \frac{Number of times heads appeared}{Total number of trials}$

               $= \frac{10}{25}$

               $= \frac{2}{5}$

               $= 0.4$

2. The number of cakes a baker makes per day in a week is given as 7, 8, 6, 10, 2, 8, 3. What is the probability that the baker makes less than 6 cakes the next day?

Solution: 

Number of cakes baked each day in a week $= 7, 8, 6, 10, 2, 8, 3$

Out of 7 days, there were 2 days (highlighted in bold) on which the baker made less than 6 cookies.

P$(< 6 $cookies$) = \frac{2}{7}$

3. The chart below shows the number of times a number was shown on the face of a tossed die. What was the probability of getting a 3 in this experiment?

Number of times 3 showed $= 7$

Number of tosses $= 30$

P(3) $= \frac{7}{30}$

4. John kicked a ball 20 times. He kicked 16 field goals and missed 4 times . What is the experimental probability that John will kick a field goal during the game?

Solution:  

John succeeded in kicking 16 field goals. He attempted to kick a field goal 20 times. 

So, the number of trials $= 20$

John’s experimental probability of kicking a field goal $= \frac{Successful outcomes} {Trials attempted} = \frac{16}{20}$ 

$= \frac{4}{5}$

$= 0.8$ or $80%$

5. James recorded the color of bikes crossing his street. Of the 500 bikes, 10 were custom colors, 100 were white, 50 were red, 120 were black, 100 were silver, 60 were blue, and 60 were gray. What is the probability that the car crossing his street is white?

Number of white bikes $= 100$ 

Total number of bikes $= 500$

P(white bike) $=  \frac{100}{500} = \frac{1}{5}$

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In a class, a student is chosen randomly in five trials to participate in 5 different events. Out of chosen students, 3 were girls and 2 were boys. What is the experimental probability of choosing a boy in the next event?

A manufacturer makes 1000 tablets every month. after inspecting 100 tablets, the manufacturer found that 30 tablets were defective. what is the probability that you will buy a defective tablet, the 3 coins are tossed 1000 times simultaneously and we get three tails $= 160$, two tails $= 260$, one tail $= 320$, no tails $= 260$. what is the probability of occurrence of two tails, the table below shows the colors of shirts sold in a clothing store on a particular day and their respective frequencies. use the table to answer the questions that follow. what is the probability of selling a blue shirt.

Jason leaves for work at the same time each day. Over a period of 327 working days, on his way to work, he had to wait for a train at the railway crossing for 68 days. What is the experimental probability that Jason has to wait for a train on his way to work?

What is the importance of experimental probability?

Experimental probability is widely used in research and experiments in various fields, such as medicine, social sciences, investing, and weather forecasting.

Is experimental probability always accurate?

Predictions based on experimental probability are less reliable than those based on theoretical probability.

Can experimental probability change every time the experiment is performed?

Since the experimental probability is based on the actual results of an experiment, it can change when the results of an experiment change.

What is theoretical probability?

The theoretical probability is calculated by finding the ratio of the number of favorable outcomes to the total number of probable outcomes.

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Probability

How likely something is to happen.

Many events can't be predicted with total certainty. The best we can say is how likely they are to happen, using the idea of probability.

Tossing a Coin

When a coin is tossed, there are two possible outcomes:

Heads (H) or Tails (T)

• the probability of the coin landing H is ½
• the probability of the coin landing T is ½

Throwing Dice

When a single die is thrown, there are six possible outcomes: 1, 2, 3, 4, 5, 6 .

The probability of any one of them is 1 6

In general:

Probability of an event happening = Number of ways it can happen Total number of outcomes

Example: the chances of rolling a "4" with a die

Number of ways it can happen: 1 (there is only 1 face with a "4" on it)

Total number of outcomes: 6 (there are 6 faces altogether)

So the probability = 1 6

Example: there are 5 marbles in a bag: 4 are blue, and 1 is red. What is the probability that a blue marble gets picked?

Number of ways it can happen: 4 (there are 4 blues)

Total number of outcomes: 5 (there are 5 marbles in total)

So the probability = 4 5 = 0.8

Probability Line

We can show probability on a Probability Line :

Probability is always between 0 and 1

Probability is Just a Guide

Probability does not tell us exactly what will happen, it is just a guide

Example: toss a coin 100 times, how many Heads will come up?

Probability says that heads have a ½ chance, so we can expect 50 Heads .

But when we actually try it we might get 48 heads, or 55 heads ... or anything really, but in most cases it will be a number near 50.

Learn more at Probability Index .

Some words have special meaning in Probability:

Experiment : a repeatable procedure with a set of possible results.

Example: Throwing dice

We can throw the dice again and again, so it is repeatable.

The set of possible results from any single throw is {1, 2, 3, 4, 5, 6}

Outcome: A possible result.

Example: "6" is one of the outcomes of a throw of a die.

Trial: A single performance of an experiment.

Example: I conducted a coin toss experiment. After 4 trials I got these results:

Three trials had the outcome "Head", and one trial had the outcome "Tail"

Sample Space: all the possible outcomes of an experiment.

Example: choosing a card from a deck

There are 52 cards in a deck (not including Jokers)

So the Sample Space is all 52 possible cards : {Ace of Hearts, 2 of Hearts, etc... }

The Sample Space is made up of Sample Points:

Sample Point: just one of the possible outcomes

Example: Deck of Cards

• the 5 of Clubs is a sample point
• the King of Hearts is a sample point

"King" is not a sample point. There are 4 Kings, so that is 4 different sample points.

There are 6 different sample points in that sample space.

Event: one or more outcomes of an experiment

Example Events:

An event can be just one outcome:

• Getting a Tail when tossing a coin
• Rolling a "5"

An event can include more than one outcome:

• Choosing a "King" from a deck of cards (any of the 4 Kings)
• Rolling an "even number" (2, 4 or 6)

Hey, let's use those words, so you get used to them:

Example: Alex wants to see how many times a "double" comes up when throwing 2 dice.

The Sample Space is all possible Outcomes (36 Sample Points):

{1,1} {1,2} {1,3} {1,4} ... ... ... {6,3} {6,4} {6,5} {6,6}

The Event Alex is looking for is a "double", where both dice have the same number. It is made up of these 6 Sample Points :

{1,1} {2,2} {3,3} {4,4} {5,5} and {6,6}

These are Alex's Results:

 After 100 Trials , Alex has 19 "double" Events ... is that close to what you would expect?

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An experiment is a procedure carried out to support, refute, or validate a hypothesis within the framework of probability. It involves observing outcomes that can be analyzed to draw conclusions.

congrats on reading the definition of experiment . now let's actually learn it.

5 Must Know Facts For Your Next Test

• Experiments are fundamental in determining probabilities by providing empirical data.
• The outcome of an experiment is the result observed after performing the procedure.
• In probability, experiments can involve random events such as rolling dice or drawing cards.
• Tree diagrams and tables are tools used to visualize all possible outcomes of an experiment.
• The sample space of an experiment includes all possible outcomes.

Review Questions

• What role do experiments play in determining probabilities?
• How can tree diagrams and tables help in understanding the outcomes of an experiment?
• What is meant by the sample space of an experiment?

Related terms

Tree Diagram : A graphical representation used to display all possible outcomes of an experiment systematically.

The set of all possible outcomes that can occur in an experiment.

A specific result obtained from conducting an experiment.

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Experiment: Definitions and Examples

Experiment: Definitions, Formulas, & Examples

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Introduction

Experiments are a fundamental part of mathematics, used to test hypotheses and establish relationships between variables. They are used in many fields, including physics, chemistry, biology, psychology, and economics. In this article, we will explore the basics of experiments in math, including definitions, examples, and a quiz to test your understanding.

Definitions

Before we dive into examples, let’s define some key terms related to experiments in math.

• Experiment: A process used to test a hypothesis or investigate a phenomenon. The process involves manipulating one or more variables and measuring the effect on one or more outcomes.
• Hypothesis: A statement or assumption about a phenomenon that is being tested in an experiment.
• Independent variable: The variable that is being manipulated in an experiment. It is also called the predictor variable or the input variable.
• Dependent variable: The variable that is being measured in an experiment. It is also called the response variable or the output variable.
• Control group: A group in an experiment that does not receive the treatment being tested. It is used as a baseline for comparison with the experimental group.
• Experimental group: A group in an experiment that receives the treatment being tested.
• Randomization: The process of randomly assigning subjects to groups in an experiment. This is done to minimize the effect of confounding variables.

Now that we have defined some key terms, let’s explore some examples of experiments in math.

• A scientist wants to test the effect of caffeine on reaction time. She recruits 100 subjects and randomly assigns them to two groups: one group receives caffeine, and the other group receives a placebo. She then measures their reaction time using a computer-based test.
• A researcher wants to test the effect of a new drug on blood pressure. He recruits 200 subjects and randomly assigns them to two groups: one group receives the new drug, and the other group receives a placebo. He then measures their blood pressure at various time points.
• A teacher wants to test the effect of a new teaching method on student performance. She randomly assigns 50 students to two groups: one group receives the new teaching method, and the other group receives the traditional teaching method. She then measures their performance on a standardized test.
• A psychologist wants to test the effect of music on mood. He recruits 30 subjects and randomly assigns them to two groups: one group listens to classical music, and the other group listens to no music. He then measures their mood using a standardized questionnaire.
• An economist wants to test the effect of a tax cut on consumer spending. He collects data on 100 households and measures their spending before and after the tax cut.
• A physicist wants to test the effect of temperature on the viscosity of a liquid. She heats the liquid to different temperatures and measures its viscosity using a viscometer.
• A biologist wants to test the effect of light on plant growth. She grows plants under different light conditions and measures their height and weight after a specified time.
• A mathematician wants to test the effect of an online tutorial on student understanding of a concept. She randomly assigns 50 students to two groups: one group watches the tutorial, and the other group does not. She then measures their understanding using a standardized test.
• A sociologist wants to test the effect of social media on self-esteem. She recruits 100 subjects and randomly assigns them to two groups: one group uses social media for an hour each day, and the other group does not. She then measures their self-esteem using a standardized questionnaire.
• An engineer wants to test the effect of a new manufacturing process on product quality. He randomly assigns 50 products to two groups: one group is manufactured using the new process, and the other group is manufactured using the traditional process. He then measures their quality using a standardized metric.
• What is the purpose of an experiment in math?

The purpose of an experiment in math is to test a hypothesis or investigate a phenomenon by manipulating one or more variables and measuring the effect on one or more outcomes.

• What is the difference between an independent variable and a dependent variable?

The independent variable is the variable that is being manipulated in an experiment, while the dependent variable is the variable that is being measured in the experiment.

• Why is randomization important in experiments?

Randomization is important in experiments because it minimizes the effect of confounding variables and ensures that the groups being compared are as similar as possible, except for the variable being tested.

• What is a control group?

A control group is a group in an experiment that does not receive the treatment being tested. It is used as a baseline for comparison with the experimental group.

• What is a hypothesis?

A hypothesis is a statement or assumption about a phenomenon that is being tested in an experiment.

• What is the purpose of an experiment in math? A) To test a hypothesis or investigate a phenomenon B) To prove a theory C) To collect data randomly D) None of the above
• What is the difference between an independent variable and a dependent variable? A) The independent variable is the variable being measured, and the dependent variable is the variable being manipulated. B) The independent variable is the variable being manipulated, and the dependent variable is the variable being measured. C) There is no difference between the two. D) Both variables are manipulated.
• Why is randomization important in experiments? A) It ensures that the groups being compared are as similar as possible, except for the variable being tested. B) It ensures that the groups being compared are different in every possible way. C) It has no effect on the outcome of the experiment. D) Both A and B.
• What is a control group? A) A group in an experiment that receives the treatment being tested. B) A group in an experiment that does not receive the treatment being tested. C) A group in an experiment that is randomly assigned to a treatment or no-treatment condition. D) Both A and C.
• What is a hypothesis? A) A statement or assumption about a phenomenon that is being tested in an experiment. B) A group in an experiment that receives the treatment being tested. C) A variable being manipulated in an experiment. D) A variable being measured in an experiment.
• A scientist wants to test the effect of exercise on heart rate. She recruits 50 subjects and randomly assigns them to two groups: one group exercises for 30 minutes, and the other group does not. She then measures their heart rate. What is the independent variable? A) Heart rate B) Group assignment (exercise or no exercise) C) Time D) None of the above
• What is the dependent variable in the experiment described in question 6? A) Heart rate B) Group assignment (exercise or no exercise) C) Time D) None of the above
• What is the purpose of a control group? A) To provide a baseline for comparison with the experimental group. B) To manipulate the independent variable. C) To measure the dependent variable. D) Both B and C.
• A researcher wants to test the effect of a new drug on blood sugar levels. He recruits 100 subjects and randomly assigns them to two groups: one group receives the new drug, and the other group receives a placebo. He then measures their blood sugar levels. What is the experimental group in this experiment?
• A researcher wants to test the effect of a new drug on blood sugar levels. He recruits 100 subjects and randomly assigns them to two groups: one group receives the new drug, and the other group receives a placebo. He then measures their blood sugar levels. What is the experimental group in this experiment? A) The group that receives the new drug B) The group that receives the placebo C) Both groups D) Neither group

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Cite this as:.

Weisstein, Eric W. "Experiment." From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/Experiment.html

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