Hypothesis Testing - Writing, Examples and Steps
An empirical study begins with writing a hypothesis. If there is no hypothesis, we will not be able to test any cause and effect relationship. Therefore, it’s important to write a hypothesis that can be tested and can offer some great insights into a situation.
We’ve been using the word “hypothesis” quite frequently in previous econometrics articles. In fact, we have represented a hypothesis statistically, developed econometrics models and calculated the extent to which an independent variable affects a dependent variable. However, we haven’t formally defined it. So, here we go:
In the simplest words, a hypothesis:
- is an assumption or a supposition,
- which can be tested and analyzed
- on the basis of one’s observation
- of a set of random variables
And a statistical hypothesis is an assumption about a situation or a population that can be represented and tested via any or a combination of statistical methods.
Therefore, the main elements of a hypothesis include:
- a guess, a theory or a statement
- observation (study of something)
- population (data to be gathered by observing population)
However, a meticulously thought and refined hypothesis is not a guess.
How to Write a Hypothesis ?
You know what a hypothesis is; what purpose it serves; how it is to be tested. The entire study or experience revolves around a hypothesis. So, a slight mistake in writing a hypothesis could result in wastage of time, money and effort.
While testing a hypothesis is a complex procedure, writing a hypothesis is the trickiest part. Needless to say, you need to be extremely careful when writing a hypothesis that you’re going to test. It is thinking about the right question – a question that can be tested and results obtained from it can enhance your understanding or meet your objectives.
Remember that there is no single tried and tested method of writing a hypothesis. You can see a generic relationship between two variables and then can refine it. Here is an example:
“Males and females differently handle employee issues”.
In this statement, we wrote a generic hypothesis. It is not measurable.
“Females handle employee issues better than males”.
The second statement provides a direction, as in who does better. When you compare two things, it means a situation is measurable.
“If females are assigned the task of handling employee issues, then they will do a better job than males because females have higher emotional quotient”.
The third statement, as you can see, offers specific details. The difference in the level of emotional quotient of males and females sets the scene. It is measurable and quantifiable.
Therefore, a well written hypothesis should be:
- Measurable
- Quantifiable
- Testable
Hypothesis Testing
Hypothesis testing refers to a formal process of investigating a supposition or statement to accept or reject it. The econometricians examine a random sample from the population. If it is consistent with the hypothesis, it is accepted. Otherwise it is rejected.
Types of Hypothesis
There are two types of hypothesis – Null and Alternative.
- Null Hypothesis: It is denoted by H_{0}. A null hypothesis is the one in which sample observations result purely from chance. This means that the observations are not influenced by some non-random cause.
- Alternative Hypothesis: It is denoted by H_{a} or H_{1}. An alternative hypothesis is the one in which sample observations are influenced by some non-random cause.
A hypothesis test concludes whether to reject the null hypothesis and accept the alternative hypothesis or to fail to reject the null hypothesis. The decision is based on the value of X and R.
Points to be noted:
- Rejection implies that the null hypothesis is not true and alternative hypothesis can be accepted.
- Fail to reject implies that there are no sufficient reasons to completely reject the null hypothesis or prefer alternative hypothesis over null hypothesis.
Decision Errors in Hypothesis Testing
Before we jump onto the process of hypothesis testing, let’s learn about the errors that can result from it. The errors are divided into two categories:
- Type I Error: Type I error occurs when null hypothesis is rejected even when it’s true. This error leads to a conclusion that a relationship or a supposed effect exists when in reality it doesn’t.
- The probability of committing Type I error is called alpha and is denoted by α.
- This probability is called the significance level.
- Also known as the error of first kind.
- It’s a focus of skepticism
Examples:
- A fair and balanced coin should result in equal heads and tails when flipped. Let’s say a coin is flipped 20 times, it will result in 10 heads and 10 tails while in fact it may not.
- An experiment showing that a medical treatment is curing a disease when actually it is not.
- Type II Error: Type II error occurs when a null hypothesis is not rejected even when it’s false. This is actually a case of failure to reject. There are not sufficient evidences or reasons for preferring alternative hypothesis over null hypothesis.
- Type II error is when a researcher fails to detect an effect that is present.
- The probability of committing Type II error is called Beta and is denoted by β.
- And the probability of not committing a Type II error is called Power Test (1 – β).
- Also known as the error of second kind.
Examples:
- Fire breaks out but fire alarm doesn’t ring.
- A survey showing that stress doesn’t hamper one’s life when it actually does.
When Null Hypothesis is:
True False Reject Type I Error Correct Decision Do Not Reject Correct Decision Type II Error When Alternative Hypothesis is:
True False Reject Type II Error Correct Decision Do Not Reject Correct Decision Type I Error
Steps in Hypothesis Testing
Econometricians follow a formal process to test a hypothesis and determine whether it is to be rejected. The steps include:
- Stating the Hypotheses
The first step involves positioning the null and alternative hypotheses. Remember, that these are mutually exclusive. If one hypothesis states a fact, the other must reject it.
- Making Statistical Assumptions
Consider statistical assumptions – such as independence of observations from each other, normality of observations, random errors and probability distribution of random errors, randomization during sampling, etc.
- Formulating an Analysis Plan
This includes deciding the test which is to be carried out to test the hypothesis. At the same time, we need to decide how sample data will be used to test the null hypothesis.
- Investigating Sample Data
At this stage, sample data is examined. It’s when we find scores – mean values, normal distribution, t distribution, z score, etc.
- Interpreting Results
This stage involves making decision to either reject the null hypothesis in favor of alternative hypothesis or not to reject the null hypothesis.
Accepting or Rejecting Null Hypothesis
This is an extension of the last step - interpreting results in the process of hypothesis testing. A null hypothesis is accepted or rejected basis P value and the region of acceptance.
P value – it is a function of the observed sample results. A threshold value is chosen before the test is conducted and is called the significance level, which is represented as α. If the calculated value of P ≤ α, it suggests the inconsistency between the observed data and the assumption that the null hypothesis is true. This suggests that the null hypothesis must be rejected. However, this doesn’t mean that alternative hypothesis can be accepted as true. This is when Type I error occurs.
Example: You roll a pair of dice once and assume that these are fair and hence the result shown by rolling the dice would be fair.
The null hypothesis is – the dice are fair. You’ve assumed a significance level (α) of 0.04.
Now you roll the dice and observe that both show 6. The p value will be 1/36 or 1/ (6*6) assuming that the test static is uniformly distributed. The p value comes out to be 0.028 which is less than the assumed value of α. On this basis the null hypothesis is rejected. It suggests that the assumption suggesting that dice are fair is not correct.
Region of Acceptance – It is the range of values that leads you to accept the null hypothesis. When you collect and observe sample data, you compute a test static. If its value falls within the specific range, the null hypothesis is accepted.
Example: You might hypothesize that the average weight of the students in a school is 30 kgs. To test this hypothesis, you collect a random sample and compute the mean score. If the sample mean falls close to the hypothesized mean, say between 29 and 31, you accept the null hypothesis. The region of acceptance, therefore, is 29 and 31. The values falling outside this region will fall in the region of rejection.
Hypothesis Tests – One-Tailed and Two-Tailed
The region of acceptance or rejection can be directional or non-directional. Basis this, we decide whether to perform one-tailed or two-tailed test to accept or reject the hypothesis.
One-Tailed Test
When the region of acceptance falls entirely on one side of the tail of distribution, one-tailed test is conducted. This means in a test of a statistical hypothesis when values fall outside the specific region only on one side of the sampling distribution, it is one-tailed test.
Example: A null hypothesis says that the marriageable age of a person is greater than or equal to 24. Then, the alternative hypothesis would be that the marriageable age is less than 24. The region of rejection, in this case, would be on the left hand side of the sampling distribution, which is the set of numbers less than 24.
Two-Tailed Test
When the region of rejection falls on the both sides of sampling distribution, it’s a two-tailed test.
Example: The null hypothesis says that the marriageable age of a person is equal to 24. Then, the alternative hypothesis would be that the marriageable age is less than or greater than 24. The region of rejection, in this case, would be on both sides of the sampling distribution, which are two sets of numbers – one greater than 24 and the other less than 24.
Sample Problem
Election commission supposes that at least 80% of the 1,000,000 voters will turn up to vote in upcoming elections. A survey of 100 randomly sampled voters finds that only 71 percent will turn up. How to find the region of acceptance, assuming a significance level of 0.05 or 5%?
Solution:
Formulate hypotheses
Null Hypothesis: At least 80% of the voters will turn up to vote.
H_{0} suggest that P ≥ 80
Alternative Hypothesis: Less than 80% will turn up to vote.
H_{a} suggests P ≤ 80
Data Sampling
The sample of the population is taken randomly.
Formulating an Analysis Plan
The proportion of sample voters who say that they will turn up to vote is 71% or .71. This is also the test of statistic.
Investigating the Data
Let’s assume that the mean of sample data is .80, which is hypothesized proportion of sample which will turn to vote.
Standard deviation (σ) = √ [ {P*(1-P)/n} * {(N-n)/(N-1)} ]
P = test value specified in null hypothesis
n = sample size
N = population size
σ = √ [ {(0.80 * 0.20)/100} * {(1,000,000 – 100)/(1,000,000 – 1)} ]
σ = √ [0.0016 * 0.9999] = √ 0.0015998 = √0.0016 = 0.04
Finding the lower and upper limits of region of acceptance
The upper limit will be equal to 100% or 1 since this is the highest proportion of the population.
The lower limit (LL) = P(X’ ≤LL) = α = 0.05
If we put the values in a statistical normal distribution calculator, LL comes out to be 0.734.
This means that the region of acceptance lies between 0.734 and 1.
Accepting or Rejecting the Hypothesis
The survey on sample proportion suggested that 71% voters will turn up to vote. But the region of acceptance is between 0.734 and 1. It means that .71 falls out of the region of acceptance and falls in the region of rejection on the left hand side. Therefore, we reject the null hypothesis that 80% of the voters will turn out to vote in upcoming elections.
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