Steps in Hypothesis Testing: Study with Video Lessons, Practice Problems & Examples

In hypothesis testing, the process begins with formulating the null hypothesis (H₀) and the alternative hypothesis (H_a). The null hypothesis represents a claim about a population, while the alternative hypothesis challenges this claim. After establishing these hypotheses, a test statistic, such as a z-score or t-score, is calculated to assess how unusual the sample data is. The p-value, a probability indicating the likelihood of observing the sample data under the null hypothesis, is then compared to a significance level (α) to determine whether to reject or fail to reject H₀.

Step 1: Write Hypotheses

So as we talked about in our overview of hypothesis testing, every problem begins with this first step, which is we have to write our hypotheses. Now this is a really important step that determines how the rest of the problem is going to go. And unfortunately, there's a lot about this that isn't always intuitive, such as how and why we even write these two statements based on the problem text. But don't worry because I'm going to break all of this down for you. I'm going to show you exactly, using these problems, how to do this step.

So let's just jump right in and we'll get started here. Okay. So every hypothesis test begins with writing these two statements that are going to be mathematical with symbols, and basically, we're going to have to extract these from the problem text. So let's take a look at these two statements in more detail. The first one is always a claim that's being made about a population.

So for example, if I claimed that 30% of everybody in the United States likes vanilla ice cream, that's a claim that's being made about a population. We call that the null hypothesis. And, basically, there are a couple of symbols that we're going to use for this. The one that you're always going to see is \( H_0 \) or \( H_0 \) or \( H_0 \). And the way that this is always going to be written or usually going to be written is it's going to be a parameter, and it's going to be a parameter about a population.

So in other words, it's going to be \( \mu \) like a mean, \( p \) like a proportion, or \( \sigma \) like a standard deviation. And the symbol that we're going to use is always going to be an equal sign, and then it's just going to be some value from the problem. So for example, if I claim that 30% of people like vanilla ice cream, that's a proportion, so that's \( p = 0.30 \). So it's always a letter like \( \mu \), \( p \), or \( \sigma \) equals some number from the problem. Some words for this you may see are kind of like the default assumption or the status quo.

It's basically just giving you a number that you're going to assume to be true for the rest of the problem so that you can challenge it with the second statement. We'll talk about that in a second. Alright? So before we actually even get to the alternative hypothesis, let's just jump right into our problem and see if we can figure out what the null hypothesis is. Alright?

So we're going to write our two statements using these problems. Let's get started. So for the first one, you're a researcher investigating the average age of students at your university. The enrollment office claims that the mean age is 23. You're looking to test if current students are younger than this claimed average.

So again, we're going to write the null hypothesis. It's going to be a parameter equals some value. The first thing is what is the parameter that we're looking at? Again, there are 3 options here. Is it a \( \mu \), \( p \) or is it a \( \sigma \)?

In this case, we're looking at the mean age, so mean means we're looking at a \( \mu \). So the parameter we have here is we're looking at a \( \mu \). And what's the value? Well, the enrollment office claims that the population mean age is 23, so that's the value. So in other words, our null hypothesis, the thing that we're assuming to be true from now on, is that \( \mu = 23 \).

That's all there is to it. So that's the null hypothesis. Again, that's the status quo, the thing we're going to eventually try to find evidence against or trying to challenge. Alright? So that leads us to the second step or the second statement, which is basically the alternative hypothesis.

This is essentially just an opposing claim that you're trying to find evidence for. Alright? So the way this is usually written is like \( H_a \) instead of \( H_0 \), so you can distinguish the 2. And basically what's going to happen here is whatever parameter you're using for the null, you use in the alternative, whatever value you use in the null, you also use in the alternative. The only thing that's different is one of the 3 symbols that's going to go in the middle over here because it's going to be one of 3 possible things.

It's either going to be a less than, a greater than, or it's just going to be a not equals. So for example, in my vanilla ice cream problem, the \( p \) is still going to be the same. The 0.30 is going to be the same, but one alternative hypothesis is that the actual proportion of people who like vanilla ice cream is not 0.30 or 30%. So I would just use a not equal to. That's one of the three possibilities for those symbols.

Alright? So, basically, what's going to happen here is that in order to figure out which one of these symbols it is, you're going to have to interpret and look at the problem text. Alright? So, again, the enrollment office claims, just to go back to our problem, that the mean age is equal to 23. That's our null hypothesis.

You're looking to test, in other words, you're looking to find evidence that current students are actually younger than this claimed average. So these words like younger or older, taller, shorter, greater than, less than, more, fewer are always keywords that are going to help you figure out which one of these symbols it is. In this case, we're looking to find that the mean age is actually less than 23. So basically, what's the parameter? We're still going to use \( \mu \).

We're still going to use 23. All that happens here is that we're trying to figure out now we're trying to find evidence for the fact that \( \mu \) is actually less than 23. So that's our null hypothesis. That's our alternative hypothesis. Alright?

So basically, what happens is your null hypothesis is always going to have that equal sign in it, and your alternative hypothesis is always going to have something else. Alright? So that's really all there is to it. So basically, one thing you might be wondering here is why do you actually have to write these two statements? And really what's going to happen is what we're going to see is that this statement over here, the null hypothesis, gives us a value to test.

It gives us a value to assume is true, And this alternative hypothesis tells us how we're going to test against it, and we'll see that in just a second. Alright? So now problems won't always be this straightforward here, so let's take a look at our second example. So we're going to have a business journal that wants to estimate the percentage of companies with female CEOs within the United States. So they want to prove that greater than 20% of companies nationwide have a female CEO.

Alright? So these problems don't always explicitly tell you, hey. This thing is making this claim about the population, and then you're looking to find a sample against them. Sometimes it'll be a little bit more nuanced, and you'll have to look into the problem a bit more. So in this case, we have a business journal.

We're looking at a percentage. So in other words, what's the parameter? We're looking at not a mean or a standard deviation, but we're looking at a proportion. So this is going to be a parameter. This is going to be a \( p \).

Percentage, usually when we're dealing with percentages, that's a dead giveaway that it's a proportion. Right? What's the value? In this case, it is actually pretty obvious because obviously the value here is at 20%, or you would write this as a decimal 0.20. So what is the null hypothesis?

What's the default assumption that you're assuming is true? Well, basically, it's just that the proportion is equal to 0.20, and the claim that they're trying to find evidence for or prove is the fact that the proportion is actually greater than, 0.20. So, again, that's that word over here, this greater than 20%, is going to tell you basically what's the alternative hypothesis. Alright? So now that we have a good understanding of how to write the null and alternative hypotheses, let's go ahead and take a look at some practice.

Thanks for watching, and let's move on.

A popular theme park claims that their weekly attendance is around $100,000$. You believe that the weekly attendance is different than this claimed value, so you gather sample data. Write the null and alternative hypotheses.

• A. $H_0:$ $\mu =100,000$$H_{a}:$ $\mu >100,000$
• B. $H_0:$ $\mu =100,000$$H_{a}:$ $\mu \ne 100,000$
• C. $H_0:$ $\mu=100,000$$H_{a}:$ $\mu$ < $100,000$
• D. $H_0:$ $\mu\ne100,000$$H_{a}:$ $\mu=100,000$

A candy manufacturer seeking to minimize the variation in weights of their candies claims to produce candies with a standard deviation less than $0.300$ g. Write the null and alternative hypotheses.

• A. $H_0:$ $\sigma =0.300$ g$H_{a}:$ $\sigma >0.300$ g
• B. $H_0:$ $\sigma=0.300$ g$H_{a}:$ $\sigma \ne 0.300$ g
• C. $H_0:$ $\sigma=0.300$ g$H_{a}:$ $\sigma$ < $0.300$ g
• D. $H_0:$ $\sigma>0.300$ g$H_{a}:$ $\sigma$ < $0.300$ g

Step 1: Write Hypotheses Example 1

Let's take a look at this one here. So, in this problem, we actually have no context for what the problem is. There's no problem text. All we're told here is that the following statement, which is that \( \mu \) is equal to or is less than 120, represents some claim that's being made. You have no idea what it is.

We're going to go ahead and use it to write the null and alternative hypothesis. Alright? Let's go ahead and get started here. Remember, the null hypothesis is \( H_0 \), and it's always going to be a parameter equals some value. In this case, the claim that's being made is that \( \mu \) is less than 120.

That's a parameter, \( \mu \). That's the 120, that's the value, but the symbol that's there is a less than symbol. So this is clearly not the null hypothesis. So using this claim that's being made, again, we don't know what it is, can we actually write the null hypothesis? And remember, we can because it's just going to be the parameter, \( \mu \).

We have the value, which is at a 120, and our default assumption, the null hypothesis, is that \( \mu \) is equal to 120. Again, we have no idea what that is, but that's just going to be our default assumption for right now. Because what happens is that this is actually representative of the alternative hypothesis already, so it's already kind of given to you. What happens is this alternative claim, this opposing claim that you're trying to find evidence for is that \( \mu \) is actually less than 120. So the claim that was being made in this problem statement over here is actually already the null or the alternative hypothesis.

Alright? So this is your \( H_a \). Your \( H_0 \) is that \( \mu \) is equal to 120. And sometimes you may actually run across problems like this. Well, they won't give you any context, but you'll have to be able to write and extract what \( H_0 \) and are.

Alright? That’s how you solve these types of problems. Let me know if you have any questions.

Step 2: Calculate Test Statistic

In the last couple of videos, we spent a lot of time talking about how to write these two statements, H₀ and H_a. Remember, the goal of a hypothesis test is we want to use data from a small sample to test a claim that's being made about a population. That's H₀. How do we actually go about doing that? We're going to have to get a little bit more mathematical here.

We're going to, and in the second step, we're going to talk about how to calculate something called a test statistic. This can sound kind of scary at first, and there's lots of different equations flying around. But don't worry, because all I'm going to show you here is that these test statistics are really something like z's and t scores, things we've already seen how to do. There's nothing new here that you don't already know how to do. It's just now you apply it to a hypothesis test.

So let's just jump right into this problem here so I can show you how this works. There's nothing to be afraid of. Let's get started. Alright. So basically in this problem here, we're a researcher looking to see if students at our university are actually younger than students from last year who had a mean age of 23.

We've already seen how to write the null and alternative hypothesis. We said that μ is 23. And for the alternative, μ is less than 23. That's the thing we're trying to find evidence for. Now, the standard deviation of this population that we're given is σ equals 4 years.

Now just be careful here. Our null hypothesis was μ equals 23. That's our null, but they're just giving us some information about a different parameter, which is σ. So don't get those confused. Alright.

So we have a claim that we're trying to test against, which is that μ is actually less than 23. We can't go out and talk to everybody at the university. So what do we do? We have to gather a sample. So, you gather data from a smaller sample of n equals 35 students, and you find a sample mean of 𝑥 = 22 years old.

That's 𝑥 because it's from a sample, not μ. Alright? So we're going to determine which test statistic to use and calculate it. Alright? So basically, what's going on here is that we have a lot of information.

We have 𝑥 is equal to 22, we have n is equal to 35, and then we have σ is equal to 3. A lot of things to process here. So one of the things you might be wondering is we went out and grabbed a sample of 35 students. We found a sample mean of 22, and that's less than the thing we're trying to test against, which is that μ is equal to 23. So doesn't that mean we're done already and we don't have to go any further?

Unfortunately, what happens is we're taking data from a sample, right, only 35 students. We could have gotten a weird sample. We have to be a little bit more rigorous about this. So we're gonna have to use a test statistic. Alright?

Basically, what's going to happen here is we're going to calculate our sample statistics, and we're going to convert those things into scores. Now if that sounds kind of familiar, it should because these scores are really just z's and t scores, things that we've already seen before. When you do this in a hypothesis test, it just gets a fancy name, which is a test statistic. Alright? This is going to be kind of frustrating about this whole topic, which is you're going to use a lot of things you've already seen before.

It's just now they get a fancy name. In this case, it's a test statistic. Alright? So really what happens is there are these 4 equations for z's and t scores and chi-squared and things like that, depending on the parameter that you're trying to test against. Alright?

So in this problem, what happens is that we're looking at a mean age. So, in other words, we're not going to use the equation for proportion and we're not going to use the equation for variance. It's basically going to be one of these 2. It's going to be a z or a t score. Now remember, the difference between these two is that you use z when you know σ and t when you don't.

In this case, we actually know what σ is. It was given to us explicitly that σ equals 4 years. And by the way, this is usually how this is going to happen in problems. Alright? So what happens here is because we know σ, we're going to use this equation over here.

Alright? So this is going to be our right equation. We have that z is equal to, we have 𝑥 minus μ, that's the population, divided by σ, divided by the square root of n. Alright? So we actually already know a lot of these things.

We already have 𝑥. We know what σ is, and we have the square root of n. So let's just go ahead and plug in those numbers. We have 𝑥 is 22 minus, and then we're gonna have to subtract μ, which is the population mean. What do we actually plug in for this number over here?

Well, basically, what happens here is that we're going to use the value from your null hypothesis whenever you're doing your test statistic. So whenever you have to plug in a number for either this μ or this p or this σ squared, you're going to be using the value that you wrote inside of your H₀, your null hypothesis. And that's going to be the number that you use. That's why it's, again, it's really important to write those two statements. The H₀ tells you what value to use, and then the H_a tells you what kind of test to use.

Alright? So really, what we're going to do here is we're just going to do 22 minus 23 because that's our population mean. We're going to divide this by our standard deviation of 4 divided by the square root of our n, which is 35. Now if you go ahead and work all of this out over here, what you're gonna get is you're gonna get a z score of negative 1.479. Alright?

So again, we convert these things to z scores because this tells us a lot more information rather than just 𝑥 is 22. We don't exactly know how different that is from our population mean, which we're assuming is 23. So, we convert it to a test score, a z score, which is negative 1.479. Alright. So again, what's going to happen in these problems is you're going to calculate some z scores.

And remember, when we worked with z scores, we always drew a graph. So what I want you to do in these problems is always draw out a graph. It's going to help you to minimize mistakes and kind of help you visualize the problem a little bit better. Basically, what's going to happen here is if we're assuming that μ is actually equal to 23, then if we grab a bunch of samples, they should form a normal distribution around this number. So basically, at the peak of this graph, we're saying that μ is equal to 23.

And what happens is we went out and grabbed a sample of a bunch of students, and we found that their mean age was actually had a z score of negative 1.479. We're trying to figure out how different that is than the mean of the population. So basically, what we found here is that 𝑥 is equal to 22, which doesn't seem different at first, but clearly, we can see that the z score is almost negative 1.45, which may be significant. Alright? So here's what happens.

We basically get our z score, and we actually know where the sample lies relative to the normal distribution. Right? So basically, what we found here is that z is equal to negative 1.479. Alright? So later on, we're going to see how we can actually use the z score to run our hypothesis test.

Basically, all you should remember for now is that remember that your test statistics, your z's and t scores, represent essentially how different the sample data is from the claimed parameter. So we calculated the Z score, it was negative 1.479, which tells us how different it is from the population parameter that we're trying to test against. Alright? So let's go ahead and take a look at some practice problems now that we understand how to calculate test statistics. Thanks for watching.

A survey claimed that $30\%$ of adults prefer electric cars over traditional cars. A car manufacturer believes the true proportion is higher than $30\%$. To test this, they survey a random sample of $50$ adults and find that $19$ say they prefer electric cars. Determine which test statistic to use & calculate it.

• A. $z=1.23$
• B. $z=-1.23$
• C. $z=4.38$
• D. $z=-4.38$

Step 2: Calculate Test Statistic Example 2

Everyone, let's take a look at this example here. So a school claims that the average score on the math final is 75. Now, one of the teachers has reason to believe that the average score is actually lower than 75. So, in order to test this, they randomly select a sample of 15 students and find the scores that are shown down here in this list below. So again, we're going to determine which test statistic to use, and then we're going to go ahead and calculate it.

Alright. So again, what happens is you should always write your null hypothesis. We have that the \(H_0\) is that the average score on the math final is 75. That's the claim about the entire population. The population is all of the math final exam scores.

So in other words, \(H_0\) is that \(\mu\), that's a mean, and the average score is equal to 75. Now a teacher is trying to find evidence against that null hypothesis, that's \(H_a\), and they're actually trying to say that \(\mu\) is actually less than 75. Alright? So that's your null and alternative hypothesis. So clearly, we're dealing with a \(\mu\) over here.

Okay? So what else do we know? Basically, we have that we're going to select a random sample of 15 students. In other words, that's \(n\) over here. This is going to be your \(n\), and you're going to find the scores that are shown over here.

Sometimes what's going to happen is instead of already having what the sample data is, you may actually just be given a list of that actual sample data, and you're going to have to go compute some stuff yourself. So here's what's going to happen. Right? We're going to determine which test statistic to use and then calculate it. So remember, we're not dealing with a proportion or a variance.

It's going to be one of the mean equations, and it's either going to be \(\sigma\) known or \(\sigma\) unknown. Do we have any mention? Do we have anything at all involving \(\Sigma\) in this problem? Unfortunately, we don't. All we know is the claim, which is the 75.

We have a sample of 15 students, which is less than 30, and also, we're not clearly told what that \(\mu\) what that \(\Sigma\) is. So it's not going to be this, the z-score. It's actually going to be our t-score. We're going to have to use our t distribution in order to calculate this.

So, basically, we're going to use this equation over here. So let's go ahead and actually solve it. So we're going to have that \(t\) is equal to, so \(t\). t= ( x̅ - μ ) s n

Okay. So do you need any of those numbers? Well, we actually know what \(\mu\) is equal to because remember, this number always comes from our null hypothesis. We know that this is just going to equal 75. So we have that.

Do we have what \(x\)̅ is equal to? Remember, that's our sample mean. And unfortunately, we're not given that explicitly in the problem. But because we're given all the data in the problem, we actually could go figure that out. And then we also don't know what the sample standard deviation is, but we do know that the square root we know this \(n\) over here because it's 15.

In other words, what really happens in this problem is you can go ahead and figure out the \(x\)̅ and the \(s\) by actually using the sample data. And basically, what's going to happen over here is if you go ahead and calculate \(x\)̅ from this sample data so I'm just going to write this down over here. You can plug this into a calculator. You can plug and chug.

You can use whatever you want, whatever kind of calculator you want. You can find out that the sample mean of this is actually 72.4. Alright. So that's the sample mean. And you can also use a calculator.

In fact, you can use your TI 84 calculators to calculate the sample standard deviation, and you should find this is actually 3.89. Alright. So just to spare you some calculations. So basically, what happens here is now you can take these numbers and you can plug them back into this equation. Okay.

And what you're gonna find here is now that we have \(x\)̅ and \(s\), this equation becomes t= 72.4-75 3.89 15 or that's your \(\mu\). And then what happens is we're going to take the \(s\), which is 3.89, and then divide it by the square root of the \(n\), which is the sample size, which is 15. Alright? Again, be really careful when you plug this stuff in, but what you should get is a t-score of negative 2.59.

You may have gotten it to a negative 2.6 depending on how you rounded things, but it's going to be something around this number. Alright? So, again, just to draw it, you should always draw this. Essentially, what happens is this is not going to be a z distribution. This is going to be a slightly different t distribution, which basically kinda looks like a z distribution when \(n\) gets a little bit bigger.

It's going to be a little bit flatter. Right? Essentially, what you're going to do is you're going to assume that this \(\mu\) is equal to 75. That's your assumption. And what happens is if you take a bunch of samples, it should form a distribution around that 75, a t distribution.

What we found was that our \(t\) was equal to negative 2.59. So basically, what happens is we got something that was all the way over here. We got that, our \(x\)̅ was equal to 72.4, and therefore it's actually pretty far away from what we're assuming to be true about that mean. Alright. So in other words, this negative 2.59 gives us another way, a standard score of figuring out how different our sample is away from the population.

And that's your test statistic. All right? Thanks so much for watching. Let me know if you have any questions. Let's move on.

Step 3: Get P-Value

In the last couple of videos, we've seen how to write different hypotheses and how to calculate test statistics, which were z or t scores, which basically show how different our sample is from the population, our H₀. The problem is that that alone doesn't tell us whether to reject our null hypothesis or not. To do that, we're next going to have to take a look at how unusual our sample is. We've used language before like that, unusual and unlikely, when we talked about probabilities. And that's exactly what we’re going to do here.

In this next step, we're going to have to get something called a p-value. And it sounds really scary at first, but all I'm going to show you here is that a p-value is just a probability. And we've seen how to get probabilities from z's and t scores. We're just going to do it now on a hypothesis test. So we're going to jump right into this problem here so I can show you there's nothing to be afraid of.

It's pretty straightforward. Let's get started here. So, again, in this problem, you're a researcher looking to see how the mean age of students has changed from last year, who had a mean age of 23 years old. So in other words, that's your null hypothesis. μ is equal to 23.

We've seen that number before. So you're going to go out and get a bunch of samples and calculate test statistics. In each one of these following situations, we're going to write a slightly different alternative hypothesis. We'll talk about that in a second. And we're going to use our z scores from that second step to get the p-value.

So let's look all the way on the left here. Right? So let's say you were looking to see if students were now younger than they were last year. You would write your alternative hypothesis like μ is less than 23. If you had that they were now older, you would write it the opposite where they're greater than 23, so on and so forth.

Alright? So, again, what's going on here is you're going to assume for a second that the null hypothesis is true. If that μ is equal to 23, that should form the peak of your distribution. What happens if you go out and get a bunch of samples, then those sample means should form a normal distribution around this number. So you're going to do this and you find that for one of the samples, you get a z score of negative 2.0.

Alright? We're going to go over here. We're going to draw the little graph. Always should draw the graph. And basically, what this means here is that we got a sample in which the z score puts it right around here.

We actually don't know what that sample mean is, but it doesn't matter because all we found out was that this z score over here means that it's 2 standard deviations away from what we're assuming is the mean. Alright? Now, again, that doesn't tell us how unusual the sample is. So now what we have to do is get this z score and turn it into a p-value. Alright?

A p-value is really just a probability. It's the probability of getting the sample data that you actually got, assuming the null hypothesis is true. So what does that mean? Doesn't that mean that I could have gotten the sample data wrong or I didn't get the sample? No. It just means that assuming that the peak of your distribution is 23, what's the likelihood or probability that you actually got something that's all the way out here 2 standard deviations away from the mean? And when we talk about probabilities, we always link them back to areas, and that's exactly what's going on here in the graph. So when you think of probabilities and p-values, they're really just areas, and these are going to be areas that are outside of the test statistic.

Alright? Here's what's going on. If we have an alternative hypothesis where μ is less than 23, we got a z score of negative 2. All that's happening for this p-value is it's basically just the area to the outside of this test statistic. So it's basically just going to be all of this area to the left of the z score.

We know exactly how to get this. You can either use a z table or a calculator, however you want to do this. Basically, this p-value is just the area to the left of the Z score of -2.0.

p ( z < − 2.0 )

Again, you can use a calculator or a Z table and you'll find out that this number is really just, sorry, 0.0228. Alright? So this p-value that corresponds to this z score of negative 2 is basically just the area to the left of this number, and it's 0.0228. That's the p-value, and that's the area. And because that number is very low, it's very unusual.

Alright? So, by the way, because we're looking at an area to the left of some number, this is usually called a left-tailed test, and this is always going to happen whenever the symbol that you're using for your hypothesis is a less than symbol. Alright? Now let's go ahead and see what happens when you have a right-tailed test. And this is basically the same principle.

Everything is just sort of flipped. In this case, what happens is if you were looking at students are older instead of younger than they were last year, then the symbol you would have used is a greater than sign. This is always going to happen whenever you have a right-tailed test. So let's say you did a sample and you got a z score of 0.8. Usually, what's going to happen in these problems, this z score is going to be positive.

So again, you don't know exactly what that mean, that sample mean is. All you know is that it's 0.8 standard deviations away from what you're assuming to be true about the mean. So what happens? What's this p-value here? It's basically just going to be all of the area that's to the right of that number.

That's really all there is to it. So the p-value is always the outside area, and it's the area to the right of z. In this case, what happens is this ends up being pylint: disable-next-line=maybe-no-member p ( z > 0.8 ) And again, if you use a calculator, you're going to see that this is equal to 0.2119. Alright?

So that's this area over here. That's the p-value, and this is a right-tailed test. Okay? Really all there is to it. Let's take a look at the last one over here.

Let's say you were looking to find that students had a different mean age instead of younger or older. What you would do here is this symbol would not be less than or greater than. It would just be a not equal to symbol. It's always going to happen whenever you have a two-tailed test. Basically, what's going to happen here, the reason it's called a two-tailed test, is that instead of looking to the right or to the left, you're actually going to be looking at both of them.

What happens here is that let's say you get a z score of negative one. You're going to come over here to the graph, and you're going to draw your sample over here. This is going to be some \( \bar{x} \). We don't know what it is, but you have a z score of negative one. What happens is because you're not specifying to the left or to the right, what you're always going to have to do in these types of problems is you're always going to have to draw basically another sample that's to the outbound, that's to the mirror image, basically symmetrical to the other side of this, and your p-value is going to be the outside area.

In this case, what happens here is you're looking to find how different your sample is to the left or to the right. And so in other words, what happens is your p-value is going to be both this area and also this area. So basically, what happens here is your p-value is actually 2 times the area that's beyond z. So basically, what's going to happen is this is called a two-tailed test because you're looking at the two tails of the graph to the left and to the right. Alright?

So all that's happening here is this is going to be 2 * p ( z < − 1 ) , p ( z > 1 ) Either one works, right? Because they’re going to be the same. So basically, this is going to be easier to work with because it's a less than, and what you're going to find here is that this is equal to 0.3173. Alright?

So this is the area that corresponds to actually both of these things combined. That would be your p-value. Alright? That's really all there is to it when it comes to p-values. They're just probability from z scores.

Let me know if you have any questions. Thanks for watching. Let's get some practice.

Determine whether the hypothesis test is left-tailed, right-tailed, or two-tailed.

$H_0:$ $p=0.4$

$H_a:$ $p\ne 0.4$

• A. Left-tailed
• B. Right-tailed
• C. Two-tailed

In a certain hypothesis test, $H_0:p=0.4$, $H_a:p$ < $0.4$. You collect a sample and calculate a test statistic $z=-1.32$. Find the $P$-value.

• A. $P\left(z\right.$< $-1.32)=0.8132$
• B. $P\left(z\right.$< $-1.32)=0.9066$
• C. $P\left(z\right.$< $-1.32)=0.0934$
• D. $P(z$ < $-1.32)=0.1868$

Step 3: Get P-Value Example 3

Welcome back, folks. So in this example, we're going to match this p-value that's given to us at the end of this problem with a graph that represents the corresponding area. We're actually given a p-value, that is p=0.2420. Let's just mention here, this is not a proportion. This is actually a p-value.

Basically, what's given to us are three options, three graphs. We just have to figure out which one of these graphs represents a p-value of 0.2420. Is it this one, this one, or is it the one on the left? Alright. Let's go ahead and get started here because again, we know how the relationship between z-scores and probabilities.

We've seen how to do this before. In this case, what we have is a given z-score of negative 0.82, and we have this area that's over here to the left. Remember that the p-value is always the area that lies to the outside or beyond those z-scores. In this case, this is what this p-value would represent.

Now mathematically, what this would be is we just have capital P, that z in this case is less than this z-score z=−0.82. Alright. Now we've seen how to do this before. I'm going to assume that you know how to do this on a table or a z-table or a graphing calculator. But basically, you could just look this up.

You'll look up a z-score of negative 0.82 to find an area, and you should find over here that that area is 0.2061. Alright, so that's roughly what this area represents over here. Now that's not quite our answer of 0.2420, so let's go ahead and move on to the next one. I'm actually going to skip this one for just a second. I'm going to go over here because this one's a little bit easier to solve.

Basically, what's going on is whenever we have a right-tailed test, essentially, what this area represents over here is this is going to be p,z>1.54. That's going to be this area over here. Alright? Now again, if you are using a table, you're going to actually have to find the area associated with this z-score, and you're going to have to take the one minus that because remember that the table always assumes that you're looking at the areas to the left.

However, what's going to happen is if you go ahead and look this up or you use a graphing calculator, which I recommend that you do, then what you're going to see over here is that this area is 0.0618. Alright? So that's the area that corresponds to this over here. Clearly, we can see here that this is a pretty small area, so this number actually makes a lot of sense.

And again, this is not quite the area that we have. Alright? So clearly, what that means here is that the answer is B, but I want to go ahead and show you why. So let's go ahead and take a look at this.

This is a z-score of negative 1.17, and this is a two-tailed test. Remember, it's two-tailed because you're looking at both ends of the graph, both tails to the left and to the right. Alright? So this is a two-tailed test. And remember, a two-tailed test essentially means that you have to do two times the probability of whatever z-score that you have.

Right? In this case, it's the probability of finding a z that is to the left or less than that number that's given to us. If I had this number over here, which is 1.17, then I just would have done pz>1.17, but it's the same thing because it's symmetrical. Right? I always recommend that you use this one because it's easier to look up on a z-table.

So go ahead and do that, and you're going to find here that this is 2×0.1210. That's the number that you get from this. So if you multiply this by 2, what you're going to get here is, lo and behold, 0.2420, and that is the right answer. So again, what this represents over here is this is the combined area of both of these tails together.

So each one of these things is 0.1210. And if you combine them, then they become 0.2420. Okay? So this is the correct answer over here. This is the combined area.

Alright, folks. That's it for this one. Thanks for watching. Let's move on.

Step 4: State Conclusion

So in the first three steps of the hypothesis test, we wrote our hypothesis, calculated some test statistics like z's and t scores, and then found p values, which were basically how unusual our sample was. Now we're going to put all of these three things together to tackle the very last step in our hypothesis test and state the conclusion. Do we have enough evidence to reject the null hypothesis and say it's probably not true? Or do we not have enough evidence and say it's reasonable, maybe it actually is? To answer this question, we're going to look at one thing, which is the p value.

We're going to have to ask ourselves, how unusual is too unusual of a sample? So what I'm going to do here is I'm going to jump right into this problem, and I'm going to show you exactly how to construct the conclusion, because it turns out there's a very formulaic systematic way of doing this. So let's just jump right in, and we'll see how this works. Let's get started.

So you're facing this problem. You're a researcher looking to see how the mean age has currently changed at your university from last year, which had a mean age of 23 years old. We've seen this problem before. So our null hypothesis here is that mu is equal to 23. So how do you do this?

You collect a sample, calculate a test statistic, which is that z is equal to negative 1.43, and you get a p value, which is 0.076. All of the stuff we've seen how to do before. So here's what's new. We're going to state our conclusion for the hypothesis test, and we're going to be using the given levels of significance below, and I'll talk about that in just a second.

So here is what we need to consider next: p values, because remember that a p value is just the probability of getting the sample data that you actually got. So, in other words, you're going to get some sample data, and assuming the null hypothesis is actually true, what you do is you draw a little line on the diagram for your z score. In this case, what we got over here was that z is equal to negative 1.43.

And basically what you do is, using a table or calculator, you find this p value, which is just the probability or likelihood that a random sample is actually this far away from mu equals 23, which is the assumed actual population parameter. And what we saw here is that if you've got a really low number, then just like when we did normal distributions, a low p value or low area means that it's actually kind of unusual. But how unusual is too unusual? Well, here's where we'll talk about the significance level.

This is where we're going to talk about alpha, which is very similar to how we talked about it when we discussed confidence intervals. It's a threshold, commonly one of 3 values: 0.10, 0.05, or 0.01, with 0.05 being the most common.

It's a threshold for how unusual your sample can be before you reject the null hypothesis. The most important thing here to know is that it's a number that is always given to you in the problem; it's never something that you calculate. So in this example, we're going to state a conclusion for the hypothesis test, and we're actually given 2 levels of significance.

For part A, we have alpha equal to 0.10. For part B, we have alpha equals 0.05. So let's take a look at part A and see how this works. We already see our p value, which basically you just draw a little line like this, and all of the area that 0.076 represents how unlikely it is to get a sample that's this far away from the mean.

Right? So what this alpha is..., it's a threshold. Notice how this 0.10 over here is slightly bigger than 0.076. So I'm going to draw this line somewhere over here, and this alpha sets up a little area like this, which is the region where if any of your p values fall into this region, you're going to reject the null hypothesis. This is sometimes referred to as the critical region.

So for the first step of a conclusion, you're going to draw the graph. You always draw the graph, and all you have to do is just compare this p value to your alpha. Is it bigger or smaller? Always draw the graph so you can see it, but you can also just quickly look at the numbers to tell.

What we can see here is that our p value is smaller than the alpha. In other words, our p value for our sample lies entirely inside this critical or rejection region.

Now let's take a look over here at part B, because everything is the same. All the means are the same. The p value is the same. The only thing that's different is just the given level of significance. Remember, that's going to be given to you in the problem. In this case, it's alpha equals 0.05. What this means here is, basically, we're going to draw just like an area like we did for part A, but it's going to be a little bit different.

Because here we can tell that this alpha is a number that's smaller than the 0.076. So, in other words, when we draw the area, it should be smaller than the blue one. So this is going to be this region for alpha equals 0.05. There's another z score associated with that, z-alpha over here, and it's not the same as this z-alpha.

So, basically, what happens here is that, unlike part A, we actually have that p relative to alpha is greater than or equal to. Right? So we actually have a p value that's bigger than this number.

Alright. So that's the first part of a conclusion for a hypothesis test. So what does all of this stuff mean and why does it all matter? Because basically, once you have what the relationship between p and alpha is, now you know whether to reject or fail to reject the null hypothesis.

So that leads us to step 2 of a conclusion. What you're going to do here is once you've compared the p versus alpha, you already know which one of these two it's going to be. So what you're going to do here is you're always going to write this sentence: 'Because p is [less than or greater than] alpha, you [reject or fail to reject] the null hypothesis.'

In part A, we see that our p value is less than alpha. Therefore, you reject the null hypothesis. In part B, we see that the p value is greater than or equal to alpha. Thus, you fail to reject the null hypothesis.

Lastly, you should never just leave it here at step 2. You should always restate the alternative hypothesis in plain English and not using symbols here. So remember that the alternative hypothesis was that students at your university are younger than they were 10 years ago. What you're going to do is restate this using either enough evidence or not enough evidence to support this claim.

If there is a reject in your statements, then there is enough evidence to suggest that students at your university are younger than they were 10 years ago. If you have failed to reject, then there is not enough evidence because your sample over here didn't quite meet that threshold of 0.05.

So, just to recap, you're always going to draw the graph and compare the p value to alpha. If you have a less than symbol, you reject the null hypothesis, and there's going to be enough evidence for your alternative hypothesis. If p is greater than or equal to alpha, then you always fail to reject the null hypothesis, and there isn't enough evidence for your alternative hypothesis. These are the three important components to the conclusion of a hypothesis test, but now you know how to do all of these things. Let's go ahead and take a look and get some practice. Thanks for watching.

Step 4: State Conclusion Example 4

Welcome back, everyone. Let's do this example problem here. So we have a College organization. They claim that more than 10% of students read print newspapers. Now you do a hypothesis test, and it results in a p-value that is 0.1632.

Remember, you're going to do a sample, you're going to get a p-value from some kind of a z-score. We actually don't know what that is. Now we're going to go ahead and state our conclusion using a significance level of alpha equal to 0.05. So, remember, in a normal hypothesis test, if you were going through all of the steps, you would calculate a z-score and you would find a p-value; this is something that you would calculate on your own. The alpha, the significance level, is always just going to be something that's given to you right off the bat.

Alright? So remember what happens here is we're just going to go ahead and set up our null hypothesis and alternative hypothesis really quickly here. Now, this college organization is claiming that more than 10% of students read print newspapers, which means that the default assumption, because we're working with a proportion here, is that little p is equal to 0.10. Right? That's the default assumption, which means that they are trying to find evidence that p is actually greater than 0.10.

So remember, when you draw your normal distribution over here because this is going to be a z-score, then this is basically where your null hypothesis goes. We're going to assume for right now that this actually is equal to 0.10. So if you grab a bunch of samples and calculate a bunch of sample proportions, they should form a normal distribution around this number. So what happens here is, remember, because we're looking at a greater than sign, then what's usually going to happen over here is that you're going to get something that's to the right of the peak. So let's say you were to go out and grab a sample and calculate a z-score for this, it would be a positive number, and the p-value for this is 0.1632, which is going to be somewhere over here.

You could basically just use this big p-value over here, which is 0.1632. Actually, I'm sorry. That's what this equals over here. That's what this p-value equals. This would be if you calculated a sample proportion.

This area over here would be 16.1632 over here, which is going to roughly cover about 16% of the normal distribution. Okay. So again, how unusual is this sample that we got over here? Well, now we have to compare it to the significance level, which is alpha. So if you just look at these two numbers over here, this is going to be your big P-value.

Again, this can get very confusing sometimes. Remember, little p is your population proportion. P hat is your sample proportion, and this big P, which you may see sometimes, is just your P-value. They mean different things and don't get confused. Alright?

So your alpha, your significance level of 0.05, clearly is a smaller number than this. So therefore, what that represents is it's going to be a smaller piece of the graph over here, which means that's your threshold. It's going to be somewhere over here. If you remember from confidence intervals, the thing that results in 5% of the graph being to the right of this is going to be a critical z-score of about 1.65, and it's going to be somewhere over here in this vicinity. Alright. So clearly, we can see that our sample over here actually has a bigger area, bigger p-value. So therefore, because p is greater than or equal to alpha, we have p versus alpha, and as this is greater than or equal to, now we actually know the rest of the conclusion, and we're ready to start our statement of the conclusion.

Alright. So remember, it all starts out with a little statement that's kind of like this. I've just written it here just to save some time. Because our p-value, which in this case is equal to 0.1632, is relative to alpha, which is 0.05, which is, in our case, it's going to be greater than or equal to, that's the symbol that we use, then what does that mean? Do we reject or do we fail to reject the null hypothesis?

Well, hopefully, so you see that we fail to reject. So, in other words, our sample is actually not too unusual relative to our threshold over here of 0.05. Right? All you have to do is compare your p-value to your alpha, and in this case, it was larger. So, we fail to reject the null hypothesis.

Now, we don't stop there. The last thing we have to do is we have to basically restate, just in plain words, what the alternative hypothesis is, which is basically that there are more than 10% of students read print newspapers. Is there enough evidence to suggest that, or is there not enough evidence to suggest that? Again, whenever you have a fail to reject, this becomes not enough. There's not enough evidence to support the idea that more than 10% of students read print newspapers.

Alright. So this is how you construct a statement of a conclusion to a hypothesis test. Let me know if you have any questions. Thanks for watching.