t-scores for means | Midterm Exam

Here's an example problem that deals with calculating a t-score for a given sample mean:

Question: Red Ryder BB Guns claims that their rifles shoot at an average velocity of 350 feet per second (fps). Ralphie from A Christmas Story is curious if their guns actually shoot that fast. He collects a random sample of 25 Red Ryder BB Guns to test their claim and measures their fps. His measurements result in a sample standard deviation of 17.5 fps. What is the probability that the mean of the sample will be below 345 fps?

This is a sample mean problem because we're measuring the mean of a set of values. In this case, those values are the fps measurements of Red Ryder BB Guns.

A problem is working with sample means if it's dealing with the average of a set of values from a sample.

You might be wondering, "What's the sample mean for this question?"

In this case, it's 345 fps, because that's the "sample mean" value that we're testing for in regards to Ralphie's random sample.

Question: Red Ryder BB Guns claims that their rifles shoot at an average velocity of 350 feet per second (fps). Ralphie from A Christmas Story is curious if their guns actually shoot that fast. He collects a random sample of 25 Red Ryder BB Guns to test their claim and measures their fps. His measurements result in a sample standard deviation of 17.5 fps. What is the probability that the mean of the sample will be below 345 fps?

Recognizing why we'll use t-score

Remember the visual from What is a t-score?

In this example problem, we're not given the population standard deviation of all Red Ryder BB Guns. Instead, we're just given the sample standard deviation.

Question: Red Ryder BB Guns claims that their rifles shoot at an average velocity of 350 feet per second (fps). Ralphie from A Christmas Story is curious if their guns actually shoot that fast. He collects a random sample of 25 Red Ryder BB Guns to test their claim and measures their fps. His measurements result in a sample standard deviation of 17.5 fps. What is the probability that the mean of the sample will be below 345 fps?

Even if we were given a population standard deviation, our sample is not greater than 30.

Question: Red Ryder BB Guns claims that their rifles shoot at an average velocity of 350 feet per second (fps). Ralphie from A Christmas Story is curious if their guns actually shoot that fast. He collects a random sample of 25 Red Ryder BB Guns to test their claim and measures their fps. His measurements result in a sample standard deviation of 17.5 fps. What is the probability that the mean of the sample will be below 345 fps?

Therefore, we'll use t-score!

Understanding the formula

Here's the formula for calculating t-score with a sample mean:

What do each of these variables represent?

x-bar represents the sample mean.
µ represents the population mean.
s represents the sample standard deviation.
n represents the sample size.

In the case that you're given the population standard deviation but your sample is not greater than 30, you'll plug in σ (population standard deviation) instead of s (sample standard deviation).

Calculating the t-score

Let's go ahead and plug each of the values in one-by-one, starting with x-bar. What is the value of x-bar?

As stated in the problem, the sample mean that we're assessing is 345 fps...

Question: Red Ryder BB Guns claims that their rifles shoot at an average velocity of 350 feet per second (fps). Ralphie from A Christmas Story is curious if their guns actually shoot that fast. He collects a random sample of 25 Red Ryder BB Guns to test their claim and measures their fps. His measurements result in a sample standard deviation of 17.5 fps. What is the probability that the mean of the sample will be below 345 fps?

...therefore, let's plug in 345 for x-bar!

What's µ? In the problem, it states that Red Ryder says their BB Guns shoot at an average velocity of 350 fps...

Question: Red Ryder BB Guns claims that their rifles shoot at an average velocity of 350 feet per second (fps). Ralphie from A Christmas Story is curious if their guns actually shoot that fast. He collects a random sample of 25 Red Ryder BB Guns to test their claim and measures their fps. His measurements result in a sample standard deviation of 17.5 fps. What is the probability that the mean of the sample will be below 345 fps?

...so let's plug in 350 for µ.

What about s? Our sample had a standard deviation of 17.5 fps...

Question: Red Ryder BB Guns claims that their rifles shoot at an average velocity of 350 feet per second (fps). Ralphie from A Christmas Story is curious if their guns actually shoot that fast. He collects a random sample of 25 Red Ryder BB Guns to test their claim and measures their fps. His measurements result in a sample standard deviation of 17.5 fps. What is the probability that the mean of the sample will be below 345 fps?

...therefore, we'll plug in 17.5 for s.

Lastly, what's n? Ralphie took a sample of 25 BB guns...

Question: Red Ryder BB Guns claims that their rifles shoot at an average velocity of 350 feet per second (fps). Ralphie from A Christmas Story is curious if their guns actually shoot that fast. He collects a random sample of 25 Red Ryder BB Guns to test their claim and measures their fps. His measurements result in a sample standard deviation of 17.5 fps. What is the probability that the mean of the sample will be below 345 fps?

...so let's plug in 25 for n.

This results in a t-score of -1.429!

Determining degrees of freedom

Here's the formula for degrees of freedom (df) for sample means:

df = n - 1

All we need to do is plug in our n value of 25 (since that's our sample size)...

df = 25 - 1

...and get a df value of 24!

df = 24

Locating p-value range

Utilizing the below t-score table...

...locate the row corresponding to our 24 degrees of freedom.

Then, recognize that our t-score of -1.429 falls between 1.318 and 1.711 (ignore the fact that our t-score is negative, I'll explain why in a second)...

...which results in our p-value falling in the range of 0.10 and 0.05.

In other words, the probability of the mean of our sample of 25 Red Ryder BB guns falling below 345 fps is between 10% and 5%!

Why are we ignoring that our t-score is negative?

What to do if you have a negative t-score?

As is, the t-score table does not have any negative t-score values.

That's because the table is only focused on right-tail tests (finding the area under the curve to the right of a given t-score), which only deal with positive t-score values.

Luckily, left-tail tests (finding the area under the t-distribution curve to the left of a given t-score)...

...have the exact same t-score values, except they're negative!

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