Z-scores for means vs. proportions | Midterm Exam

In our IQ example from What is a Z-score?, we were simply given a data point and asked to compute its Z-score based on the known population mean and population standard deviation.

This differs from problems that you'll face in class, which typically will deal with computing a Z-score for the outcome of a given sample.

These samples will fall into one of two categories: means or proportions.

For each of these categories, you'll calculate Z-scores! Therefore, it's important to know the differences.

Finding Z-score with sample mean

Here's an example problem that deals with calculating a Z-score for a given a sample mean:

Question: Red Ryder BB Guns has measured that their rifles shoot at an average velocity of 350 feet per second (fps) and a standard deviation of 15 fps. Ralphie from A Christmas Story is curious if their guns actually shoot that fast. He collects a random sample of 35 Red Ryder BB guns and measures their fps values. What's 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 from a sample.

Question: Red Ryder BB Guns has measured that their rifles shoot at an average velocity of 350 feet per second (fps) and a standard deviation of 15 fps. Ralphie from A Christmas Story is curious if their guns actually shoot that fast. He collects a random sample of 35 Red Ryder BB guns and measures their fps values. What's the probability that the mean of the sample will be below 345 fps?

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 has measured that their rifles shoot at an average velocity of 350 feet per second (fps) and a standard deviation of 15 fps. Ralphie from A Christmas Story is curious if their guns actually shoot that fast. He collects a random sample of 35 Red Ryder BB guns and measures their fps values. What's the probability that the mean of the sample will be below 345 fps?

Recognizing why we'll use Z-score

In What is a t-score? we'll learn how to distinguish between whether or not to use a t-score or Z-score. In essence, you'll utilize a Z-score whenever you have enough relevant information about a given population.

The graphic below gives a good depiction of the relevant information necessary:

In relation to our problem, we're given the population standard deviation...

Question: Red Ryder BB Guns has measured that their rifles shoot at an average velocity of 350 feet per second (fps) and a standard deviation of 15 fps. Ralphie from A Christmas Story is curious if their guns actually shoot that fast. He collects a random sample of 35 Red Ryder BB guns and measures their fps values. What's the probability that the mean of the sample will be below 345 fps?

...and our sample size is greater than 30.

Question: Red Ryder BB Guns has measured that their rifles shoot at an average velocity of 350 feet per second (fps) and a standard deviation of 15 fps. Ralphie from A Christmas Story is curious if their guns actually shoot that fast. He collects a random sample of 35 Red Ryder BB guns and measures their fps values. What's the probability that the mean of the sample will be below 345 fps?

Therefore, we're good to use Z-score!

Understanding the formula

Here is the formula for calculating Z-score with a sample mean:

What do each of these variables represent?

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

Calculating the Z-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 has measured that their rifles shoot at an average velocity of 350 feet per second (fps) and a standard deviation of 15 fps. Ralphie from A Christmas Story is curious if their guns actually shoot that fast. He collects a random sample of 35 Red Ryder BB guns and measures their fps values. What's 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 has measured that their rifles shoot at an average velocity of 350 feet per second (fps) and a standard deviation of 15 fps. Ralphie from A Christmas Story is curious if their guns actually shoot that fast. He collects a random sample of 35 Red Ryder BB guns and measures their fps values. What's the probability that the mean of the sample will be below 345 fps?

...so let's plug in 350 for µ.

What's σ? Red Ryder states their BB Guns have a standard deviation of 15 fps...

Question: Red Ryder BB Guns has measured that their rifles shoot at an average velocity of 350 feet per second (fps) and a standard deviation of 15 fps. Ralphie from A Christmas Story is curious if their guns actually shoot that fast. He collects a random sample of 35 Red Ryder BB guns and measures their fps values. What's the probability that the mean of the sample will be below 345 fps?

...therefore, we'll plug in 15 for σ.

Lastly, what's n? Ralphie took a sample of 35 BB Guns...

Question: Red Ryder BB Guns has measured that their rifles shoot at an average velocity of 350 feet per second (fps) and a standard deviation of 15 fps. Ralphie from A Christmas Story is curious if their guns actually shoot that fast. He collects a random sample of 35 Red Ryder BB guns and measures their fps values. What's the probability that the mean of the sample will be below 345 fps?

...so let's plug in 35 for n.

This results in a Z-score of -1.97!

Locating p-value

Utilizing the below Z-score table...

...let's identify the value -1.97 in the column on the left...

...and then the value -1.97 in the row on the top...

...which results in a p-value of 0.0244.

Graphically speaking, the total area under the normal distribution curve is 1.000. Our p-value of 0.0244 can be visualized in the red area under the curve to the left of our Z-score below:

In other words, the probability of the mean of our sample of 35 Red Ryder BB guns being below 345 is 2.44%!

Finding Z-score with sample proportion

Here's an example problem that deals with calculating a Z-score for a given a sample proportion:

Question: Crammer Nation University released news that 30% of the most recent freshmen class joined Greek Life. You conduct a sample of 50 randomly selected freshmen and find that 18 of them joined a Greek Life chapter. What is the probability of running another sample with a proportion greater than the one found in your first sample?

This is a sample proportion problem because there's two potential outcomes for each respondent: either they joined a Greek Life chapter, or they didn't.

A problem is working with a sample proportion if it's dealing with a ratio derived from a set of Yes/No values.

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

In this case, it's 0.36. How? Because 18 students in our sample joined a Greek Life chapter...

Question: Crammer Nation University released news that 30% of the most recent freshmen class joined Greek Life. You conduct a sample of 50 randomly selected freshmen and find that 18 of them joined a Greek Life chapter. What is the probability of running another sample with a proportion greater than the one found in your first sample?

...out of 50 total students in the sample.

Question: Crammer Nation University released news that 30% of the most recent freshmen class joined Greek Life. You conduct a sample of 50 randomly selected freshmen and find that 18 of them joined a Greek Life chapter. What is the probability of running another sample with a proportion greater than the one found in your first sample?

Then you can do the simple math equation to arrive at 0.36.

18 / 50 = 0.36

Recognizing why we'll use Z-score

We'll learn more about this in the What is a t-score? article. However, for the sake of this article, all you need to understand is that you'll always use Z-scores with sample proportions!

Understanding the formula

Here is the formula for calculating Z-score with a sample proportion:

What do each of these variables represent?

p-hat represents the sample proportion.
p represents the population proportion.
q represents the population proportion of failure.
n represents the sample size.

Calculating the Z-score

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

Based on the problem, it's 0.36, because 18 students said they joined Greek Life...

Question: Crammer Nation University released news that 30% of the most recent freshmen class joined Greek Life. You conduct a sample of 50 randomly selected freshmen and find that 18 of them joined a Greek Life chapter. What is the probability of running another sample with a proportion greater than the one found in your first sample?

...from the sample of 50 students.

Question: Crammer Nation University released news that 30% of the most recent freshmen class joined Greek Life. You conduct a sample of 50 randomly selected freshmen and find that 18 of them joined a Greek Life chapter. What is the probability of running another sample with a proportion greater than the one found in your first sample?

And when you do the math, this proportion equates to 0.36.

18 / 50 = 0.36

Therefore, let's plug in 0.36 for p-hat!

What's p going to equal? Based on the problem...

Question: Crammer Nation University released news that 30% of the most recent freshmen class joined Greek Life. You conduct a sample of 50 randomly selected freshmen and find that 18 of them joined a Greek Life chapter. What is the probability of running another sample with a proportion greater than the one found in your first sample?

...it's 30%, or 0.30!

What's q going to be? It's basically the opposite of p, so all we need to do is solve the following equation...

q = 1 - p

...to get our q value of 0.70!

q = 1 - 0.30 = 0.70

To find q, just subtract p from 1.00!

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

Lastly, our sample size was 50 students...

Question: Crammer Nation University released news that 30% of the most recent freshmen class joined Greek Life. You conduct a sample of 50 randomly selected freshmen and find that 18 of them joined a Greek Life chapter. What is the probability of running another sample with a proportion greater than the one found in your first sample?

...therefore we'll plug in 50 for n.

When we solve this out, we get a Z-score of 0.92!

Locating p-value

Utilizing the below Z-score table...

...let's identify the value 0.92 in the column on the left...

...and then the value 0.92 in the row on the top...

...which results in a p-value of 0.0244.

Graphically speaking, the total area under the normal distribution curve is 1.000. Our p-value of 0.8212 can be visualized in the red area under the curve to the left of our Z-score below:

However... in the problem we're asking for the probably of obtaining another sample with a higher proportion than 0.36.

Question: Crammer Nation University released news that 30% of the most recent freshmen class joined Greek Life. You conduct a sample of 50 randomly selected freshmen and find that 18 of them joined a Greek Life chapter. What is the probability of running another sample with a proportion greater than the one found in your first sample?

Therefore, we need to figure out the area to the right of our Z-score...

...which is equal to 0.1788.

1.000 - 0.8212 = 0.1788

In other words, the probability of the proportion of another sample of the same size being greater than 0.36 is 17.88%!

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