One-mean t-test (Confidence interval)

Question: You're going through fraternity rush at Crammer Nation University and hear that Sigma Apple Pi brothers get a lot of Tinder matches. You take a random sample of 35 brothers and find a sample mean of 23.2 daily Tinder matches with a standard deviation of 3.2 matches. Based on this, find a 95% confidence interval for the true mean daily Tinder matches of Sigma Apple Pi brothers.

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, 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 deviation (s) in the prompt and not the population standard deviation (σ)...

Question: You're going through fraternity rush at Crammer Nation University and hear that Sigma Apple Pi brothers get a lot of Tinder matches. You take a random sample of 35 brothers and find a sample mean of 23.2 daily Tinder matches with a standard deviation of 3.2 matches. Based on this, find a 95% confidence interval for the true mean daily Tinder matches of Sigma Apple Pi brothers.

...we'll 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 the population standard deviation (σ)?

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

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

Let's go through each of the variables and plug them in!

Plugging in sample mean (x-bar)

As stated in the prompt, our sample mean is 23.2 daily Tinder matches...

You're going through fraternity rush at Crammer Nation University and hear that Sigma Apple Pi brothers get a lot of Tinder matches. You take a random sample of 35 brothers and find a sample mean of 23.2 daily Tinder matches with a standard deviation of 3.2 matches. Based on this, find a 95% confidence interval for the true mean daily Tinder matches of Sigma Apple Pi brothers.

...therefore, we'll plug in 23.2 for x-bar!

Recognizing Margin of Error (MoE)

Now that we've plugged in our sample mean (x-bar), it's easier to visualize that our confidence interval will have upper and lower bounds from our sample mean, 23.2.

That Margin of Error (MoE) will occur here...

...and will create the upper and lower bounds with the (±) here:

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.

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!

Calculating our degrees of freedom (df)

We can calculate our degrees of freedom (df) easily with the below formula:

df = n - 1

In our case, we've got a sample size of 35...

df = 35 - 1

...therefore our degrees of freedom are 34.

df = 34

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 34, let's go to the row in the t-table corresponding to 34 degrees of freedom...

...and in the column corresponding to an p-value of 0.025 (which is our alpha level (α))...

...find our critical t-score value of 2.032!

Let's go ahead and plug in that t-score of 2.032 for t*n-1!

Plugging in sample standard deviation (s)

Based on the prompt, our sample standard deviation is 2.3 matches...

You're going through fraternity rush at Crammer Nation University and hear that Sigma Apple Pi brothers get a lot of Tinder matches. You take a random sample of 35 brothers and find a sample mean of 23.2 daily Tinder matches with a standard deviation of 3.2 matches. Based on this, find a 95% confidence interval for the true mean daily Tinder matches of Sigma Apple Pi brothers.

...therefore, let's plug it in for s!

Plugging in sample size (n)

Our sample size is 35 based on the prompt...

You're going through fraternity rush at Crammer Nation University and hear that Sigma Apple Pi brothers get a lot of Tinder matches. You take a random sample of 35 brothers and find a sample mean of 23.2 daily Tinder matches with a standard deviation of 3.2 matches. Based on this, find a 95% confidence interval for the true mean daily Tinder matches of Sigma Apple Pi brothers.

...therefore, let's plug in 35 for n!

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: You're going through fraternity rush at Crammer Nation University and hear a rumor that Sigma Apple Pi brothers get an average of 25 daily Tinder matches. You take a random sample of 35 brothers and find a sample mean of 23.2 daily Tinder matches with a standard deviation of 3.2 matches. Based on this, find a 95% confidence interval for the true mean daily Tinder matches of Sigma Apple Pi brothers.

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

We are confidence level % confidence that the true 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% confidence that the true population mean daily Tinder matches for Sigma Apple Pi brothers is between 22.1 and 24.3.

Don't get tricked by this question!

Sometimes, you'll be a given a follow-up question to this interpretation like so:

Follow-up Question: Can you be sure that the true mean daily Tinder matches for Sigma Apple Pi brothers is not 25?

This is a trick question! But... how so?

Considering that 25 is outside our confidence interval of (22.1, 24.3), it's possible that it's the true population mean is not 25... however, it's not guaranteed. Remember, we're just 95% confident it's in the range of 22.1 and 24.3... not 100% confident!!

For this reason, we can not be sure that the true mean daily Tinder matches for Sigma Apple Pi brothers is not 25.

Just because a value falls outside the confidence interval doesn't mean the true population parameter is definitely not that value!

Don't get tricked by that on your exam!

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