Associating p-value to t-score on sampling distribution

To get some practice finding p-values with t-scores, we'll utilize the following scenario:

At Crammer Nation University, the brothers of Sigma Apple Pi fraternity get an average of 25.2 Tinder matches per day. You take a sample of 35 brothers, resulting in a standard deviation of 2.7 matches. What's the probability that the mean of the sample will be above 25.9?

First, we'll calculate t-score, and the we'll find the corresponding p-value range in the t-table!

But first...

Why should we use t-score?

In relation to this graphic...

...we are not given the population standard deviation of all Sigma Apple Pi brothers' average Tinder matches per day. We're given the sample standard deviation of 2.7 matches.

Therefore, we must use t-score!

Calculating the t-score

To calculate t-score, we'll utilize the below formula:

What do each of these variables mean?

x-bar is the sample mean that you want to obtain the z-score of.
μ is the mean value of the given population.
s is the standard deviation of the given sample.
n is the sample size.

What if I'm given the population standard deviation (σ) instead of sample standard deviation (s)?

Then use population standard deviation instead of sample standard deviation!

Remember, just because you're using the population standard deviation, it doesn't mean you're calculating a t-score! If your sample size is not above 30 (according to the graphic above)... that still means you'll be calculating t-score!

Identifying Standard Error (SE)

As stated in Associating p-value to z-score on sampling distribution, the standard error occurs in the denominator of your t-score function.

But wait... I thought in Associating p-value to z-score on sampling distribution, we said the formula for Standard Error (SE) was this?

Well... it is, but when we're not given the population standard deviation (σ), we have to make-due with the sample standard deviation (s)! It's still Standard Error (SE), but now it's with s instead of σ!

When calculating Standard Error (SE), if you're not given the population standard deviation (σ), you can make-due with the sample standard deviation (s) instead!

Plugging in population mean (µ)

In the prompt, it states that the population mean for the average Tinder matches of all Sigma Apple Pi brothers is 25.2...

At Crammer Nation University, the brothers of Sigma Apple Pi fraternity get an average of 25.2 Tinder matches per day. You take a sample of 35 brothers, resulting in a standard deviation of 2.7 matches. What's the probability that the mean of the sample will be above 25.9?

...therefore we'll plug in 25.2 for µ.

Plugging in sample mean (x-bar)

In this case, we're assessing the chances that our sample has a mean above 25.9.

At Crammer Nation University, the brothers of Sigma Apple Pi fraternity get an average of 25.2 Tinder matches per day. You take a sample of 35 brothers, resulting in a standard deviation of 2.7 matches. What's the probability that the mean of the sample will be above 25.9?

We'll get to the whole "above" aspect of this when working with the t-table. For now, all we need to do is plug in 25.9 for x-bar.

Plugging in sample standard deviation (s)

The sample that we took of 35 brothers had a standard deviation of 2.7 Tinder matches...

At Crammer Nation University, the brothers of Sigma Apple Pi fraternity get an average of 25.2 Tinder matches per day. You take a sample of 35 brothers, resulting in a standard deviation of 2.7 matches. What's the probability that the mean of the sample will be above 25.9?

...therefore let's plug in 2.7 for s.

Plugging in sample size (n)

Our sample was of 35 brothers...

At Crammer Nation University, the brothers of Sigma Apple Pi fraternity get an average of 25.2 Tinder matches per day. You take a sample of 35 brothers, resulting in a standard deviation of 2.7 matches. What's the probability that the mean of the sample will be above 25.9?

...so we'll plug in 35 for n.

Solving for t-score

When we solve this equation...

...we get a t-score of 1.535!

How to associate t-scores with p-values

This works a little different than using the z-table... here's what the z-table looked like:

Here's what the t-table looks like:

With Z-scores, we calculated our value and then found the corresponding p-value in the table. For example, if we had a Z-score of -1.97, we could identify its associated p-value of 0.0244 like so:

With t-scores, it'll be a little different. We'll calculate our t-score and our degrees of freedom. Afterwards, we'll identify the range of values that our t-score falls between... and in turn, the range of values that our p-value falls between.

Determining degrees of freedom

In t-scores and t-distributions, we stated the following about degrees of freedom:

When working with t-scores, to calculate your degrees of freedom (df), just subtract one from your sample size!

When we were calculating the t-score, we determined that the sample size (n) was 35... therefore, our degrees of freedom (df) equal 34!

Locating p-value range on t-table

Now that we know that our degrees of freedom (df) equals 34, we must identify the row corresponding to the "df" value of 34...

...then, find the range of values to the right that contain our t-score of 1.535. In our case, our t-score lies between 1.307 and 1.691.

Based on this range of values, we can identify our range of p-values in the header row above!

This means that our p-value falls between 0.10 and 0.05! Written mathematically, txhat'd look like so:

0.05 < p < 0.10

Understanding your p-value visually

This process is pretty similar to with z-scores, with a couple small tweaks.

Notice how at the top of the t-table, the area to the right of the "t" value is filled in?

That means that the area under the curve to the right of our t-score value of 1.535...

...is somewhere in our p-value range of "0.05 < p < 0.10".

In relation to our prompt...

At Crammer Nation University, the brothers of Sigma Apple Pi fraternity get an average of 25.2 Tinder matches per day. You take a sample of 35 brothers, resulting in a standard deviation of 2.7 matches. What's the probability that the mean of the sample will be above 25.9?

...this means that the probability of our sample mean being above 25.9 is somewhere between 5% and 10%.

...Why don't we need to calculate an exact p-value?

You'll mainly be utilizing t-scores for confidence intervals and hypothesis tests. In both of those situations, it's not necessary to know the exact p-value for your given t-score.

It's not necessary for confidence intervals because...

For confidence intervals, you'll often be calculating 95% confidence intervals with a certain degrees of freedom value.

In the case of one-tail tests (> or <), a 95% confidence interval means you'll be utilizing the column associated with a p-value of 0.05.

In the case of two tail tests (≠), you'll be utilizing the column associated with a p-value of 0.025.

Don't get caught up in the numbers or "one/two-tail tests" here. What you need to understand is...

In relation to confidence intervals, the t-score table already contains all the necessary t-scores for relevant p-values.

It's not necessary for hypothesis tests because...

For hypothesis tests, you'll often be determining whether your t-score results in a p-value below 0.05 or 0.01. If it falls below those p-values, that means that your sample holds statistical significance.

In these situations, the only thing that matters is whether or not our t-score's p-value falls above or below the declared threshold. We don't need an exact value!

Take, for example, a hypothesis test in relation to a sample with 25 degrees of freedom.

In this hypothesis test, if our p-value is less than or equal to 0.05, that means that our sample holds statistical significance. If it's above 0.05, that means that it does not hold statistical significance.

Let's say our calculated t-score value turned out to be 1.800. In the row corresponding to 25 degrees of freedom, we can identify that a t-score of 1.800 falls between 1.708 and 2.060.

This corresponds to a p-value range of 0.05 and 0.025.

Any values between 0.05 and 0.025 are less than or equal to our threshold of 0.05... therefore this hypothesis test would result in statistical significance!

Now let's change things up: say our calculated t-score value turned out to be 1.500. In the row corresponding to 25 degrees of freedom, we can identify that a t-score of 1.500 falls between 1.316 and 1.708.

This corresponds to a p-value range of 0.10 and 0.05.

Any values between 0.10 and 0.05 are greater than our threshold of 0.05... therefore this hypothesis test would not result in statistical significance!

What you need to understand is...

In relation to hypothesis tests, we only need to determine if our t-score results in a p-value above or below a declared threshold. This can be accomplished through the ranges of t-score values contain in the t-score table.

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