What is a hypothesis test?

Imagine you're a freshman at Crammer Nation University, and that Greek Life rush has just begun. You just called it quits on your high school relationship, so you’re looking to hop right into the college action. So, you download Tinder.

You hear through the grapevine that the brothers of Sigma Apple Pi are pushing some absolute numbers on Tinder… claiming to get an average of 25 matches per day. Before you completely ignore every other fraternity and solely rush Sigma Apple Pi for their Tinder clout, you want to gather some data to determine if this claim holds its weight.

You decide to ask 30 random Sigma Apple Pi brothers for their average Tinder matches per day. Keep in mind: you’re not checking to see if every brother has exactly 25 average Tinder matches per day… there’s obviously going to be some wiggle-room. However, if the brothers’ average Tinder matches are consistently below 25… there’s a chance Sigma Apple Pi doesn’t have as much Tinder game as they claim.

In this situation, you’ve essentially set up a hypothesis test!

Hypothesis testing is a way to test a claim about a given population. It enables you to determine whether or not an outcome of a given sample was due to random chance or was statistically significant.

What’s the population in this situation? The brothers of Sigma Apple Pi.

What’s the sample? The 30 brothers that you sampled.

What’s the claim that you’re testing? That the brothers of Sigma Apple Pi actually get on average 25 Tinder matches per day.

The outcome of every hypothesis test

Without digging into the math yet, there's two potential conclusions our hypothesis test will come to based on our sample.

  • We do have enough evidence to reject Sigma Apple Pi's claim of 25 average Tinder matches per day.
  • We don't have enough evidence to reject (a.k.a. "fail to reject") Sigma Apple Pi's claim of 25 average Tinder matches per day.

To arrive to either of the above two solutions, there's 4 crucial steps that we'll take:

  1. State the hypotheses
  2. Calculate the test statistic
  3. Find the p-value
  4. Make your concluding statement

Let's dig into the key elements of each step!

Step 1 - State the hypotheses

There will be two hypotheses that you need to state in any hypothesis test problem. (1) The null hypothesis and (2) the alternative hypothesis.

Each of these hypotheses will be making a claim about the population parameter (µ or p). We will not make claims about the sample parameter (x-bar or p-hat), because the whole point of us taking the sample is to figure out if we have enough evidence to support a claim made about the population!

Your hypotheses will involve the population parameter (µ or p), not the sample parameter (x-bar or p-hat)!

Understanding the null hypothesis

Your null hypothesis essentially restates the claim about the given population.

Your null hypothesis (H0) embodies the claim made about the population parameter.

In the case of the situation above, it's that Sigma Apple Pi brothers get an average of 25 Tinder matches per day. We'd write that null hypothesis (H0) like so:

H0: µ = 25

Essentially, what we're saying here is that the true population mean for the average Tinder matches per day of Sigma Apple Pi brothers...

H0: µ = 25

...is equal to 25.

H0: µ = 25

Something else important to note...

The null hypothesis will always have an equal sign.

Why? Because there will always be a claim made that the population parameter equals something.

Understanding the alternative hypothesis

Your alternative hypothesis in essence makes a claim that the population parameter is different than the null hypothesis says.

Your alternative hypothesis (Ha) makes a claim that the population parameter differs from what the H0 says.

In the case of the situation above, we're claiming that Sigma Apple Pi brothers actually get less than 25 average Tinder matches per day. We'd write that alternative hypothesis (Ha) like so:

H0: µ = 25
Ha: µ < 25

Essentially what we're saying here is that the true population mean for average Tinder matches per day of Sigma Apple Pi brothers...

H0: µ = 25
Ha: µ < 25

...is actually less than 25.

H0: µ = 25
Ha: µ < 25

A helpful tip when writing alternative hypotheses:

The alternative hypothesis can have the greater than (>), less than (<), or not equal to () sign.

In this situation, we're testing the claim that Sigma Apple Pi brothers average Tinder matches per day are actually less than 25 (<). However... we could've tested that they are greater than 25 (>) or that they do not equal 25 (≠).

Step 2 - Calculating your test statistic

In this step, you'll be calculating a z-score or t-score based on the given sample and population data.

We've covered this process in the following articles:

Step 3 - Find the p-value

While we've covered this process in the following articles...

...it's important to understand the difference between one-tail and two-tail tests.

A one-tail vs. two-tail test

Put simply...

If your alternative hypothesis has ">" or "<", that means it's a one-tail test.
If your alternative hypothesis has "", that means it's a two-tail test.

But... how does this work?

One-tail test visualized

An alternative hypothesis (Ha) with ">" might have a sampling distribution that would look like so...

...in which we are only considering the area to the right of our test statistic (this is referred to as the "right-tail" of the sampling distribution).

This occurs because we're using the greater than (>) symbol in our alternative hypothesis (Ha). In other words... we're assessing the probability of a sample occurring with a sample parameter greater than the one we found!

An alternative hypothesis (Ha) with "<" might have a sampling distribution that would look like so...

...in which we are only considering the area to the left of our test statistic (this is referred to as the "right-tail" of the sampling distribution).

This occurs because we're using the less than (>) symbol in our alternative hypothesis (Ha). In other words... we're assessing the probability of a sample occurring with a sample parameter less than the one we found!

Two-tail test visualized

An alternative hypothesis (Ha) with "≠" might have a sampling distribution that would look like so:

Notice how in this case, we've essentially reflected our test statistic onto both tails of the distribution and are considering the area in both tails? That's because if our alternative hypothesis is claiming that the true population parameter does not equal (≠) a certain value, it could potentially be greater than (>) or less than (<) said value!

The t-table only shows the p-values for one tail of the t-distribution...

...therefore...

Whenever working with a two-tail hypothesis test, make sure to double your p-value! This accounts for the fact that you have to reflect it on both tails of your sampling distribution, and the t-table only accounts for one of those tails!

Step 4 - Make your concluding statement

Your concluding statement is going to center around the alpha level declared in the problem. It is typically stated at the end of your problem, and will be either 0.01 or 0.05. Here's a quick example:

Sigma Apple Pi claims that their brothers get on average 25 Tinder matches per day. You want to test if they actually pull that many matches per day, or if they’re falsely stating their Tinder stats to boost recruitment numbers. You collect a random sample of 30 Sigma Apple Pi brothers average Tinder matches per day. You measure a mean of 24.2 average matches per day with a standard deviation of 3.1 average matches per day. Provide support for your claim using a hypothesis test with an alpha level of 0.05.

Put simply...

- If the p-value is below the alpha level, then we reject the null hypothesis
- If the p-value is above the alpha level, then we fail to reject the null hypothesis.

Before we get into what it means to reject vs. fail to reject the null hypothesis, let's breakdown what an alpha level truly is.

Alpha level (α) explained

In simple terms...

Your alpha level (α) is the threshold that determines whether your p-value is statistically significant. In other words, it determines whether or not you reject or fail to reject your null hypothesis in "Step 4 - Make your concluding statement".

If your p-value is below your alpha level (α), that means the probability of a sample of the same size occurring (the p-value) was so low (below the alpha level (α)) that it indicates the results of our sample hold statistical significance.

If your p-value is above your alpha level (α), that means the probability of a sample of the same size occurring (the p-value) was too high (above the alpha level (α)) and could've just been due to random chance.

What does it mean to "reject" the null hypothesis?

When we reject the null hypothesis, we're essentially saying that we have enough evidence to support the alternative hypothesis.

Why is this the case?

Because our p-value, or probability of our sample results occurring, is below our alpha level (α)!

If the null hypothesis was actually a truthful claim about the population, then the probability of our sample results occurring (the p-value) were extremely low (below the alpha level (α))... so low, that it might mean the null hypothesis isn't true!

Something else important to understand: we are NOT accepting the alternative hypothesis! That would mean that we are 100% certain that the alternative hypothesis is true... which is not the case. Rather, the outcome of our sample provided enough evidence to support the alternative hypothesis we proposed over the null hypothesis, and therefore we "reject" the null hypothesis.

What does it mean to "fail to reject" the null hypothesis?

When we fail to reject the null hypothesis, we're essentially saying that we don't have enough evidence to support the alternative hypothesis.

Why is this the case?

Because our p-value, or probability of our sample results occurring, is above our alpha level (α)!

Put in simpler terms, the probability of our sample results occurring (the p-value) was too high (above the alpha level (α)) to potentially disprove the null hypothesis. Therefore, we "fail to reject" the null hypothesis!

Something else important to understand: we are NOT accepting the null hypothesis! That would mean that we are 100% certain that the null hypothesis is true... which is not the case. Rather, the outcome of our sample did not provide enough evidence to support the alternative hypothesis we proposed, and therefore we "fail to reject" the null hypothesis.

Memorizing the concluding statement templates

Your concluding statements will vary slightly dependent on if you calculated a z-score or t-score in Step 2. That's because with z-scores, you're able to identify an exact p-value, and with t-scores you're only given a range that your p-value falls between.

Before digging into these concluding statement templates...

It is crucial that you copy these concluding statement templates word-for-word! You'll miss points if you incorrectly write your concluding statement!

Concluding statement with a z-score

When you calculate a z-score in Step 2, utilize the below template to write your concluding statement.

Since our p-value of p-value is less / greater than our alpha level of alpha level value, we reject / fail to reject the null hypothesis and do / don't have enough evidence to support the alternative hypothesis, implying that description of alternative hypothesis.

To visualize this in action, check out this example concluding statement:

Since our p-value of 0.8212 is greater than our alpha level of 0.05, we fail to reject the null hypothesis and do not have enough evidence to support the alternative hypothesis, implying that the proportion of Crammer Nation University freshmen who joined Greek Life this year does not equal 0.30.

Concluding statement with a t-score

When you calculate a t-score in Step 2, utilize the below template to write your concluding statement.

Since our p-value range of p-value range is less / greater than our alpha level of alpha level value, we reject / fail to reject the null hypothesis. We do / don't have enough evidence to support the alternative hypothesis, which states that description of the alternative hypothesis.

To visualize this in action, check out this example concluding statement:

Since our p-value range of 0.025 < p < 0.05 is less than our alpha level of 0.05, we reject the null hypothesis. We do have enough evidence to support the alternative hypothesis, which states that the brothers of Sigma Apple Pi have an average Tinder matches per day value less than 25.

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