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What is a Hypothesis?

Ever make bets with your family and friends about sports, singing or dancing competitions? You can think of these bets as a hypothesis. A hypothesis is defined as a proposed explanation for an event based on previous facts. A hypothesis is little more than an educated guess.

This is why, when you make a bet on a winning singer in a talent competition, you are making a hypothesis. You are using previous information in the form of past performances to propose an explanation for the future.

Take a look at the table below for an example of hypotheses in various fields.

Field Hypothesis
Psychology Twins have the same IQ
Biology Fertilizer increases average height growth for plants
Political Science Countries that received a loan are better off now than those who didn’t

 

What is Hypothesis Testing?

Now that you understand the definition of a hypothesis, you can start to delve into the basics of hypothesis testing. Hypothesis is exactly what it sounds like - you conduct a hypothesis test in order to test whether or not your hypothesis is probable.

Using the probabilities in your data, you can determine how likely your hypothesis is true most of the time or simply the result of chance. Hypotheses have a special notation, which are introduced below.

 

H_{o} Null hypothesis
H_{1}, H_{a} Alternative hypothesis
\mu Population mean
\sigma Population standard deviation
\bar{x} Sample mean
s Sample SD

 

Let’s focus on the first two lines - the rest will be explained further. The null and alternative hypotheses are the two types of hypotheses you can make about your data. The definitions of each are summarized in the table below.

 

H_{o} States that a parameter is equal to, less or greater than, or different from a hypothesized value
H_{1}, H_{a} States that a parameter is not equal to, less or greater, or different from a hypothesized value

 

Hypothesis tests can be conducted by finding the probability distribution of your data. The frequency distribution of your data set is simply the frequency at which each value in the data set occurs.

frequency_distribution

Probability distributions, on the other hand, use the mean and standard deviation of your data set to produce the likelihood of certain values occurring.

probability_distribution

 

Types of Hypothesis Testing

Building off of the previous section, you now know that we can make hypothesis tests based off of a data set’s probability distribution. In fact, most of the time you can make hypothesis tests without the entire data set and simply use the mean and standard deviation information.

Hypothesis tests can be made using probability distributions because you can see whether or not your hypothesized value is likely to occur. Recall, however, that there are many types of distributions. Take a look at the table below, which summarizes each hypothesis test, their required distributions and their test parameters.

 

Test Type Distribution Test Parameters
Z-test Normal Mean
T-test Student-t Mean
ANOVA F distribution Means
Chi-Square Chi-squared distribution Association between two categorical variables

 

As you can see, while the first three are used for the comparison of means, the last test is used to compare the association between two categorical variables.

Each hypothesis test uses these basic principles.

Element Example Description
Hypothesis with hypothesized value \mu > 45 The H_{o} here is that the population mean is greater than 45
Test value 45 This is used as a benchmark to test how likely a mean of 45 is given the population mean and SD
Confidence interval \alpha = 0.05 At the 95% confidence level (1-0.95 = 0.05), we can be certain that our test gets the true answer 95% of the time
Test statistic z \; score = 1.5 The test statistic gives you the standardized value of your test value on your test distribution
P-value p-value = 0.06 The p-value is the calculated probability of your value occuring

 

In hypothesis testing, the following rules are used to either reject or accept the hypothesis given a \alpha of 0.05. Keep in mind that if you were to have an \alpha of 0.1, you’re results would be given with 90% confidence and the example above, with a p-value of 0.06, would reject H_{o}.

 

P-value < 0.05 Region of rejection Reject H_{o}
P-value > 0.05 Region of acceptance Fail to reject H_{o}

 

Type 1 and Type 2 Error

To understand type 1 and type 2 error, let’s take a closer look at the confidence level. Take as an example a scenario in which you are conducting an experiment on average plant growth in one year as a result of a specific fertilizer. Recall that a population is the thing or idea that you want to study - here, it would be the plant we are studying.

population_sample

Because it is impossible for us to test our hypothesis on all of the plants in the world, we take samples in order to make estimates about the population. Let’s assume we’ve taken 100 different samples from this plant population.

A confidence level is defined as the percentage of all possible samples that we can expect to include the true population parameter. A 95% confidence level would therefore be the same as saying that we expect 95% of all 100 samples to include the true population parameter.

confidence_level

A confidence level can be set at a ton of different values. In fact, the DNA tests that people take nowadays to find out where their ancestors come from include varying confidence levels you can set. The table below summarizes common confidence levels with their corresponding \alpha, or alpha, value.

 

Confidence Level Alpha
99 \alpha = 0.01
95 \alpha = 0.05
90 \alpha = 0.1
85 \alpha = 0.15
75 \alpha = 0.25

 

Type 1 and 2 errors occur when we reject or accept our null hypothesis when, in reality, we shouldn’t have. This happens because, while statistics is powerful, there is a certain chance that you may be wrong. The table below summarizes these types of errors.

 

Accept H_{o} Reject H_{o}
In reality, H_{o} is actually true Correct: H_{o} is true and statistical test accepts H_{o} Incorrect: Type 1 error - H_{o} is true and statistical test rejects H_{o}
In reality, H_{o} is actually false Incorrect: Type 2 error - H_{o} is false and statistical test accepts H_{o} Correct: H_{o} is false and statistical test rejects H_{o}
 
Now that you’re familiar with hypothesis testing, let’s look at an example: 

The mean population IQ for adults is 100 with an SD of 15. You want to see whether those born prematurely have a lower IQ. To test this, you attain a sample of the IQ’s adults that were born prematurely with a sample mean of 95. Your hypothesis is that prematurely born people do not have lower IQs.

 

Because we know the population mean and standard deviation, as well as the distribution (IQ’s are generally normally distributed), we can use a z-test.

 

Null Hypothesis H_{o}: IQ of 95 or above is normal \mu > 95
Alternative Hypothesis H_{o}: IQ of 95 is not normal

First, we find the z-score

z-score

Next, we find the z-score on a z-table for negative values.

Z 0.00 0.01 0.02 0.03
0.2 0.42074 0.41683 0.41294 0.40905
0.3 0.38209 0.37828 0.37448 0.37070

With a p-value of 0.3707, we the fail to reject the null hypothesis.

hypothesis_test

 

Type 1 Error Example

In the example above, you can see that we have chosen our confidence level to be at 95%. This gives us an alpha of 0.05. As explained above, a type 1 error occurs when our statistical test rejects the null hypothesis when, in reality, the null hypothesis is true.

 

The main question is, how do we know when a type 1 error has occurred? The only way we could know for certain would be if we had all population values, which we don’t. Luckily, we can use the same logic as we do for the confidence level. If we are 95% certain of something occurring, this means that the probability that this thing really didn’t occur as the tail end of our rejection region. Therefore, the type 1 error is calculated simply as the 1 minus the probability that our hypothesis occurred, which is simply our p-value 0.3707.

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Danica

Located in Prague and studying to become a Statistician, I enjoy reading, writing, and exploring new places.