Two-mean t-test (Hypothesis test)

Question: Sigma Apple Pi claims that their brothers get more daily Tinder matches than Alpha Blueberry Pi brothers. You decide to test this claim by randomly sampling brothers from each chapter and gathering their average daily Tinder match data. You compile the following results:

Sigma Apple PiAlpha Blueberry Pi
Sample size3540
Sample mean23.522.5
Sample standard deviation5.73.9

Conduct a hypothesis test with α = 0.01 to assess this rumor, with degrees of freedom equaling 60.

As with all hypothesis test problems, we're going to take the following steps to solve!

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

Step 1 - State the hypotheses

Let's start with the null hypothesis (H0), then we'll move onto the alternative hypothesis (Ha).

Defining your null hypothesis (H0)

Here's what we need to ask ourselves: what's the baseline claim that's being made about both the populations?

It's a little tricky when working with 2 means... unless one population claims their population mean is a specific number more/less than the other population's, then you can assume that the baseline claim is that both population means are equal to each other (a.k.a. there's no difference between the populations).

We can write that null hypothesis (H0) like so:

H0: µ1 = µ2

However, you'll typically see it written like so...

H0: µ1 - µ2 = 0

...because it's easier to assess with calculating test statistics and such.

Essentially, what we're saying here is that the true population mean for Sigma Apple Pi brothers daily Tinder matches (µ1)...

H0: µ1 - µ2 = 0

...minus the true population mean for Alpha Blueberry Pi brothers daily Tinder matches (µ2)...

H0: µ1 - µ2 = 0

...is zero.

H0: µ1 - µ2 = 0

In other words... they're equal to each other! There's no difference between the Sigma Apple Pi brothers and Alpha Blueberry Pi brothers average daily Tinder matches.

Defining your alternative hypothesis (Ha)

Here we need to ask ourselves: what's the claim that's being made that's different from the baseline claim about the population?

Sigma Apple Pi brothers are claiming that their average daily Tinder matches are greater than Alpha Blueberry Pi brothers!

Question: Sigma Apple Pi claims that their brothers get more daily Tinder matches than Alpha Blueberry Pi brothers. You decide to test this claim by randomly sampling brothers from each chapter and gathering their average daily Tinder match data. You compile the following results:

Sigma Apple PiAlpha Blueberry Pi
Sample size3540
Sample mean23.522.5
Sample standard deviation5.73.9

Conduct a hypothesis test with α = 0.01 to assess this rumor, with degrees of freedom equaling 60.

We'd write that alternative hypothesis (Ha) like so:

H0: µ1 - µ2 = 0
Ha: µ1 - µ2 > 0

What we're saying here is that the true population mean for Sigma Apple Pi brothers daily Tinder matches (µ1)...

H0: µ1 - µ2 = 0
Ha: µ1 - µ2 > 0

...minus the true population mean for Alpha Blueberry Pi brothers daily Tinder matches (µ2)...

H0: µ1 - µ2 = 0
Ha: µ1 - µ2 > 0

...is greater than zero.

H0: µ1 - µ2 = 0
Ha: µ1 - µ2 > 0

In other words... Sigma Apple Pi brothers have a higher average daily Tinder match value than Alpha Blueberry Pi brothers!

Step 2 - Calculate the test statistic

Before even beginning to calculate our test statistic, we have to check out assumptions for both samples!

Check your assumptions!

The assumptions for 2 means are very similar to 1 mean, with one additional assumption.

1. Both samples are randomly selected from the population
2. Both sample sizes (n1 and n2) are less than or equal to 10% of their respective population sizes
3. Both sample sizes (n1 and n2) are greater than or equal to 30, or the populations themselves are normally distributed
4. Both samples are independent of each other.

Concerning #1, we are randomly selecting our samples.

Question: Sigma Apple Pi claims that their brothers get more daily Tinder matches than Alpha Blueberry Pi brothers. You decide to test this claim by randomly sampling brothers from each chapter and gathering their average daily Tinder match data. You compile the following results:

Sigma Apple PiAlpha Blueberry Pi
Sample size3540
Sample mean23.522.5
Sample standard deviation5.73.9

Conduct a hypothesis test with α = 0.01 to assess this rumor, with degrees of freedom equaling 60.

Concerning #2, we're going to assume that both fraternities have 400+ members. Therefore, both samples are less than or equal to 10% of the population.

Question: Sigma Apple Pi claims that their brothers get more daily Tinder matches than Alpha Blueberry Pi brothers. You decide to test this claim by randomly sampling brothers from each chapter and gathering their average daily Tinder match data. You compile the following results:

Sigma Apple PiAlpha Blueberry Pi
Sample size3540
Sample mean23.522.5
Sample standard deviation5.73.9

Conduct a hypothesis test with α = 0.01 to assess this rumor, with degrees of freedom equaling 60.

Concerning #3, both samples are greater than 30, therefore the Central Limit Theorem applies!

Concerning #4, we're going to assume each chapter's average daily Tinder match data for each brother is independent of each other.

Therefore, all of our assumptions are passed! Now we can move onto calculating our test statistic!

Which should we use: z-score or t-score?

Based on this graphic in t-scores and sampling distributions...

...since we're given the sample standard deviations (s1 and s2) instead of the population standard deviations (σ)...

Sigma Apple Pi claims that their brothers get more daily Tinder matches than Alpha Blueberry Pi brothers. You decide to test this claim by randomly sampling brothers from each chapter and gathering their average daily Tinder match data. You compile the following results:

Sigma Apple PiAlpha Blueberry Pi
Sample size3540
Sample mean23.522.5
Sample standard deviation5.73.9

Conduct a hypothesis test with α = 0.01 to assess this rumor, with degrees of freedom equaling . The degrees of freedom is 60.

...we'll be calculating a t-score with the following formula:

What if we had both population standard deviations (σ1 and σ2), and therefore used z-score?

Then we'd utilize the following formula:

Notice how literally, the only difference here is that we're plugging in the population standard deviation (σ) instead of the sample standard deviation (s)!

Remember, even if we were given the population standard deviations (σ1 and σ2), we'd still have to have a sample size of at least 30 for both samples to utilize a z-score! (That way, the Central Limit Theorem would apply on our sampling distributions!)

Plugging in sample means (y-bar1 and y-bar2)

Based on the table within the prompt, the sample mean for Sigma Apple Pi brothers was 23.5...

Sigma Apple PiAlpha Blueberry Pi
Sample size3540
Sample mean23.522.5
Sample standard deviation5.73.9

...therefore, we'll plug in 23.5 for y-bar1.

The sample mean for Alpha Blueberry Pi brothers was 22.5...

Sigma Apple PiAlpha Blueberry Pi
Sample size3540
Sample mean23.522.5
Sample standard deviation5.73.9

therefore, we'll plug in 22.5 for y-bar2.

Plugging in sample standard deviations (s1 and s2)

The sample standard deviation for Sigma Apple Pi brothers was 5.7...

Sigma Apple PiAlpha Blueberry Pi
Sample size3540
Sample mean23.522.5
Sample standard deviation5.73.9

...therefore, we'll plug in 5.7 for s1.

The sample standard deviation for Alpha Blueberry Pi brothers was 3.9...

Sigma Apple PiAlpha Blueberry Pi
Sample size3540
Sample mean23.522.5
Sample standard deviation5.73.9

...therefore, we'll plug in 3.9 for s2.

Plugging in sample sizes (n1 and n2)

The sample size for Sigma Apple Pi brothers was 35...

Sigma Apple PiAlpha Blueberry Pi
Sample size3540
Sample mean23.522.5
Sample standard deviation5.73.9

...therefore, we'll plug in 35 for n1.

The sample size for Alpha Blueberry Pi brothers was 40...

Sigma Apple PiAlpha Blueberry Pi
Sample size3540
Sample mean23.522.5
Sample standard deviation5.73.9

...therefore, we'll plug in 40 for n2.

Plugging in the difference between the claimed population means (Δ0)

Now for Δ0... the first part of our numerator gives the difference between the sample means...

...therefore, Δ0 will give the difference between the claimed population means!

And based on our null hypothesis, we can see that the claimed difference is 0!

H0: µ1 - µ2 = 0
Ha: µ1 - µ2 > 0

So, let's go ahead and plug in 0 for Δ0!

Solving for t-score

When we solve this out, it results in a t-score of 1.144!

Step 3 - Find the p-value

First, let's determine if we're doing a one-tail or two-tail test, and then let's find our p-value in the z-table.

One-tail or two-tail?

In What is a hypothesis test?, we declared the following:

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.

Since our alternative hypothesis uses a ">", that means we're dealing with a one-tail test, specifically a right-tail test, since we're using the greater than symbol.

At an alpha level of 0.01...

Sigma Apple Pi claims that their brothers get more daily Tinder matches than Alpha Blueberry Pi brothers. You decide to test this claim by randomly sampling brothers from each chapter and gathering their average daily Tinder match data. You compile the following results:

Sigma Apple PiAlpha Blueberry Pi
Sample size3540
Sample mean23.522.5
Sample standard deviation5.73.9

Conduct a hypothesis test with α = 0.01 to assess this rumor, with degrees of freedom equaling . The degrees of freedom is 60.

...this means that we're looking to see if our p-value occurs in this area under our sampling distribution curve.

Finding our p-value range

We know our t-score is 1.144, so that means we'll be using the t-table to find a range of p-values.

To find that range, we first need to recognize in the problem, it states that we'll use 60 degrees of freedom (df).

Sigma Apple Pi claims that their brothers get more daily Tinder matches than Alpha Blueberry Pi brothers. You decide to test this claim by randomly sampling brothers from each chapter and gathering their average daily Tinder match data. You compile the following results:

Sigma Apple PiAlpha Blueberry Pi
Sample size3540
Sample mean23.522.5
Sample standard deviation5.73.9

Conduct a hypothesis test with α = 0.01 to assess this rumor, with degrees of freedom equaling . The degrees of freedom is 60.

What if when working with two means, I'm not given the degrees of freedom?

I would not fret too much about this. The vast majority of the time, you should be given it.

If you have to calculate it, you can utilize the below formula...

...but most of the time, you'll be given a calculator to solve this automatically or will be given it explicitly.

This means that within the row of the t-table corresponding to 60 degrees of freedom (df)...

...we need to identify that our t-score of 1.144 is less than the lowest value in this row of 1.296...

...meaning that our p-value some value greater than 0.10.

We can state that mathematically like so:

0.10 < p < 1.00

Why include the 1.00?

Because we know our p-value can't be greater than 1.00, that's the max p-value and represents the entire area under the sampling distribution curve!

Visualizing our p-value

The graphic at the top of the t-table shows the following...

...meaning that the p-values in the t-table correspond to the area to the right of each t-score. This is exactly what we are looking for, since we're working with the right-tail of our sampling distribution!

Based on the t-table, we found that our p-value falls somewhere in the range of 0.10 and 1.00.

And we know that all values between 0.10 and 1.00 are going to fall outside our alpha value (α) of 0.01!

Step 4 - Make your concluding statement

Your concluding statement is going to center around the alpha level declared in the problem. Each problem should explicitly state the alpha level. In our problem, it's 0.01.

Sigma Apple Pi claims that their brothers get more daily Tinder matches than Alpha Blueberry Pi brothers. You decide to test this claim by randomly sampling brothers from each chapter and gathering their average daily Tinder match data. You compile the following results:

Sigma Apple PiAlpha Blueberry Pi
Sample size3540
Sample mean23.522.5
Sample standard deviation5.73.9

Conduct a hypothesis test with α = 0.01 to assess this rumor, with degrees of freedom equaling . The degrees of freedom is 60.

As we stated in What is a hypothesis test?...

- 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.

In our case, our p-value range above the alpha level. Remember: all values between 0.10 and 1.00 are greater than 0.01!

0.10 < p < 1.00

This means that we fail to reject our null hypothesis!

Applying the t-score answer template

If you remember in What is a hypothesis test?, we gave the following answer template when working with t-scores:

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.

Applied to our question, this would give us the following answer to our original question!

Answer: Since our p-value range of 0.10 < p < 1.00 is greater than our alpha level of 0.01, we fail to reject the null hypothesis. We don't have enough evidence to support the alternative hypothesis, which states that the brothers of Sigma Apple Pi have an mean daily Tinder match value less than 25.

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