In this section we're going to talk about confidence rules for means. However, just as a recap, in the last section we're able to answer questions of the type, what's the true proportion of females in college? So proportion means like a percent of females in college. Okay, now that dealt with quantitative variables and now in this section we're going to be dealing with average of quantity variables so that we can answer questions like what's the true mean?
or average age of students in college. So we'll be building confidence intervals for mu, or remember that is population mean. Okay, just like before we'll be using sample distribution, but this time we'll be using the sampling distribution for mu.
But we have a problem. If we don't know the population, how will we know the standard deviation of your sampling distribution? So Here's the solution to that problem.
We're going to use the student t distribution. You kind of have seen the student t distribution just very briefly in the last section. I'll bring you back to it in a second.
Here's a graph or several graphs of the student t distribution. So you have several graphs. How do you know which graph is for what?
Now the sampling distribution depends on, sorry, the curves depend on something called the df. What does the df stand for? DF stands for the degrees of freedom. So if the degree of freedom is 1, 5, 10, and 30, they are all different curves. Now, if you look at when the degree of freedom is 1, you have this blue curve.
When the degree of freedom is 5, you have this brownish-maroon curve. And when the degree of freedom is 10, it looks like this neon lime green is kind of hidden back there. When the degree of freedom is 30, it's also like... It's a purple curve. It's definitely in there somewhere, like slightly hidden.
But the one I want to talk about is this one, the standard normal curve. Why do I have the standard normal curve all of a sudden here? And the standard normal curve is just like this, this pretty blue color above.
Okay, so in the last video, we saw this table, right? This is the student t distribution table. Now, I didn't explain what this meant before, but now you do know it. This is called degree of freedom. The degree of freedom is defined as n minus 1, where n represents the sample size.
So for example, if n equals 10, then your degree of freedom would be 1 less than, which would be 9, because it's just 1. n minus 1, so 10 minus 1, which gives you 10. One more thing, this top row always describes the area in the right tail. So in this symmetric bell-shaped curve, whatever the area is in the right town, that's what these values represent. Now, if I keep going down, going down, looking down, as my degree of freedom increases, let's look at what happened.
As my degree of freedom increases, look, as my degree of freedom increases, also in this last row, I have z, because it's basically what this curve, this graph shows us before, as your... As your degree of freedom increases, your curves try to become like the standard normal curve. It's basically kind of getting really close to the standard normal curve. Okay, so let's talk about some more characteristics that actually write these things down.
So the certainty distribution curves are bell-shaped but has different tails. Once again, Why do you have different tails? Because it depends on the degree of freedom. The standard t distribution has a mean of mu and the standard deviation of s over square root n, which is almost like something in the central limit down.
Okay, it's characterized by the degree of freedom, which once again we're going to denote by tf. It's always n minus 1, or little n. sample size. Each little n corresponds to a slightly different shape.
Okay, we already talked about that. The larger n is, the closer the shape is to the normal distribution curve. So let's work on finding actual t-scores. So no z-scores anymore, but t-scores. Let's find where the degree of intersection intersects with the area in the one tail where we define as alpha over 2 on the table.
If we can't find the degree of freedom in our table, we round to the nearest one in the table itself. You'll see in a second. So let's find the two values for the given scenarios.
Your degree of freedom is 7 and the area to the right is 0.005. Okay, so I'm just going to sketch this out really quickly. So we have a symmetric bell-shaped curve.
The tail to the right is 0.005. I am going to label this as t as well. Okay, and the degree of freedom is 7. Okay, so I'm going to go back to this table. Alright, so area to the right was 0.005, so this is the column we're working on. Okay.
And then the degree of freedom was seven, I believe. Let me just confirm. Yeah, degree of freedom is seven. Okay. So we're here and I go to the seventh row.
Seventh row. So it looks like our T value. So let me do that again.
That's the column. This is the degree of freedom. So it looks like the value we're interested in is 3.449.
Okay, so the t-score is 3.499 for this problem. In example B, our n-value is 15. Okay, so that means my degree freedom is one less than that, which is 14, and the area to the left is 0.10. All right, okay. So, the area to the left of my t-score that we're interested in is 0.10.
Okay, so it seems a little strange. Like, how am I going to find this t-score if I have given the area to the left? If you look at these curves up here, what's the center?
The center is 0. And also, remember, this thing is bell-shaped. So... if, or symmetric as well. So if I want to find a value on the left side, it's just the mirror image of what's on the right side.
And instead of it being a positive value, I'm gonna make it a negative value. So what I'm saying is I'm going to actually think about the right side of the tail. And the right side area will be 0.10.
If the right side is 0.10, let's just see what we get. So the.10 is right here. The degree of freedom this time is 14. So 1.345.
is the value on the right. Remember we want the t-score for the left so that would be negative 1.34. Okay in example C now we have n equals 20 which means the degree of freedom is 19. Now we're trying to find the t-score that corresponds to 98% confidence level. Okay, we know confidence levels represent the middle. So the middle represents 0.98.
So that means each tail represents 0.01. Okay, so we need to find these t-scores. The area to the right is 0.01. Degree of freedom is 19. So let's see where would that be? 0.01.
Degree of freedom is 19. So it looks like we have 2.539. 2.539 for the right side. What would the left side be then? Negative 2.539.
Okay, great. Let's try example D. Atn equals 54. That means your degree of freedom in this example will be 53. It's always one less than. We're trying to find the T-score that corresponds to 90 degree confidence level. So if the middle is now 90%, that's the same thing as same point, 90. Each tail would then represent 0.05.
So then we go to our table and look for... the right tail being 0.05 and the degree of 53. Point zero five looks like it's right. I'm using different colors as well. Looks like right here.
We're gonna go all the way down to 53. Go all the way down to the 53rd row. But now we see, hey, there is no 53rd row. So what we do is we choose the row that's closest to 53. So right now the degree of freedom is 53. Which row then is closest to 53? It would be... 50. So I'm going to highlight, let's highlight the 50th row instead.
So the t-score we're considering is 1.676. So that would be 1.576. The left value then would be negative 1.576.