Two-mean t-test (Confidence interval)

Question: Sigma Apple Pi and Alpha Blueberry Pi are known across Crammer Nation University as the two fraternities that pull the most Tinder game. Considering that you're about to go through fraternity rush, you decide you want to determine the difference between each fraternity's average daily Tinder matches. You randomly sample brothers from each chapter and gather the following data:

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

Find the 95% confidence interval for the true difference in mean daily Tinder matches between Sigma Apple Pi and Alpha Blueberry Pi. The degrees of freedom is 60.

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

  • Step 1 - Calculate the confidence interval
  • Step 2 - Interpret the confidence interval

Step 1 - Calculate the confidence interval

Whenever you're working with a sample means problem (in this case, we have two sample means), you first need to determine whether to use a t-score or z-score!

t-score vs. z-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)...

Question: Sigma Apple Pi and Alpha Blueberry Pi are known across Crammer Nation University as the two fraternities that pull the most Tinder game. Considering that you're about to go through fraternity rush, you decide you want to determine the difference between each fraternity's average daily Tinder matches. You randomly sample brothers from each chapter and gather the following data:

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

Find the 95% confidence interval for the true difference in mean daily Tinder matches between Sigma Apple Pi and Alpha Blueberry Pi. The degrees of freedom is 60.

...we'll be utilize a t-score as our critical value! It'll be represented by t*n-1, and it means we'll utilize the following formula to solve for our confidence interval!

What if we had both population standard deviations? (σ1 and σ2)

Then we'd plug in the population standard deviations...

...as well as a z-score critical value (represented by Z*) instead of a t-score critical value (t*n-1), as long as both our sample sizes are above 30 (due to Central Limit Theorem).

Let's go through each variable and plug them in!

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.

Determine the critical value (t*n-1)

Remember, t*n-1 represent s a t-score value that we must find in the t-table.

To determine where our t*n-1 value is within this table, we need to calculate our alpha level (α) and degrees of freedom (df).

Calculating our alpha level (α)

When working with confidence intervals, we can utilize the below formula to identify our alpha level:

alpha = (1 - Confidence Level) / 2

In our case, our Confidence Level is 95% (a.k.a. 0.95)...

alpha = (1 - 0.95) / 2

...so our alpha value is 0.025.

alpha = (1 - 0.95) / 2 = 0.05 / 2 = 0.025

This alpha level essentially means that on our sampling distribution, we're looking for the critical t-score value that enables us to have an area of 0.025 in the right-tail of our t-distribution (and in turn, the left-tail as well, due to the ± in our confidence interval formula).

You'll notice this corresponds to the graphic at the top of the t-table, with the area to the right of the "tα" (the t-score) shaded, resulting in a p-value of "α"!

Therefore, on the t-table, by looking for p-values corresponding to our alpha level (α) of 0.025, we'll be able to find the critical t-score value (t*n-1) that we need to compose our confidence interval!

What about the left-tail of our sampling distribution?

A t-distribution curve is symmetrical. Therefore, the t-score that we find for the right-side will be the same as the left-side, just negative.

This is accounted for by the fact that our confidence interval equation has the plus/minus (±) sign!

Identifying our degrees of freedom (df)

Our degrees of freedom was already given to us in the problem!

Question: Sigma Apple Pi and Alpha Blueberry Pi are known across Crammer Nation University as the two fraternities that pull the most Tinder game. Considering that you're about to go through fraternity rush, you decide you want to determine the difference between each fraternity's average daily Tinder matches. You randomly sample brothers from each chapter and gather the following data:

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

Find the 95% confidence interval for the true difference in mean daily Tinder matches between Sigma Apple Pi and Alpha Blueberry Pi. The degrees of freedom is 60.

This is typical for two-mean problems. They typically won't make you calculate the degrees of freedom on your own, because the formula is complex.

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.

Locating our t*n-1 value

Now that we've got our alpha level (α) of 0.025 and our degrees of freedom (df) of 60, let's go to the row in the t-table corresponding to 60 degrees of freedom...

...and within the "t.025" column (representing our 0.025 alpha level (α))...

...identify our critical t*n-1 value of 2.000!

Let's go ahead and plug that into our confidence interval formula!

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.

Solve for the confidence interval

When we solve this out, we get the following:

When we separate the "±" to create our lower bound (-) and upper bound (+)...

...it results in the following confidence interval!

Step 2 - Interpret the confidence interval

Before interpreting a confidence interval, I always recommend that you re-read the question to remind yourself what exactly you're interpreting. It's easy to get caught up in the math and forget.

Question: Sigma Apple Pi and Alpha Blueberry Pi are known across Crammer Nation University as the two fraternities that pull the most Tinder game. Considering that you're about to go through fraternity rush, you decide you want to determine the difference between each fraternity's average daily Tinder matches. You randomly sample brothers from each chapter and gather the following data:

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

Find the 95% confidence interval for the true difference in mean daily Tinder matches between Sigma Apple Pi and Alpha Blueberry Pi. The degrees of freedom is 60.

Now, to create our interpretation, let's refer back to the answer template referenced in What is a confidence interval?

We are confidence level % confident that the true population mean vs. proportion description of what problem is finding is between lower CI bound and upper CI bound .

We're going to tweak this slightly for two mean hypothesis tests...

We are confidence level % confident that the true difference in population mean vs. proportion description of what problem is finding is between lower CI bound and upper CI bound .

In our case, our answer would look like so:

Answer: We are 95% confident that the true difference in population mean daily Tinder matches between Sigma Apple Pi and Alpha Blueberry Pi is between -1.287 and 3.287.

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