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0:02
John Cristobal: All right. Welcome, everybody. Today we're going to be working on hypothesis testing both for one populations and we'll get a little bit into introducing two populations as well since the process is so similar. John Cristobal: and using our same tools, we can get these results. But in hypothesis testing similar to our work with confidence intervals.
0:23
John Cristobal: Where we're using sample data to try to estimate values or possible range of values that the population parameter would be between. John Cristobal: Now, with hypothesis testing, we're using sample data to kind of test whether those population values, those population parameters are accurate, or maybe they've shifted in some way testing these claims that we have about a population, these kind of assumed values, these values that we kind of are using as true for the moment.
0:51
John Cristobal: and hoping, or possibly checking to see, with sample data, if those values have changed. So we'll go into some notes, as well as some practice problems. John Cristobal: involving hypothesis testing.
1:02
John Cristobal: Alright. So going into our hypothesis hypothesis testing, as I mentioned, right? What we're gonna kind of do is what John Cristobal: we're going to use sample data.
1:18
John Cristobal: The test, right? Hence the testing, right?
1:22
John Cristobal: if Population.
1:28
John Cristobal: parameters, right? Those values that we compute. John Cristobal: using the entire population's worth of data, right? So that's going to be the mean, proportion… John Cristobal: Later, we can get into the standard deviation, sigma, or even variance, sigma squared, so on, right? Testing to see if these population parameters, right, are…
1:53
John Cristobal: I'm gonna say valid right? John Cristobal: are still true, or possibly there is some shift There's gonna be some claims. John Cristobal: by researchers that maybe the value for our parameter has changed, and we're using sample data to test, okay? And so. John Cristobal: We're going through hypothesis testing. We have John Cristobal: Typically, there's kind of 5 step. John Cristobal: these 5 main points. John Cristobal: to go through for this process.
2:28
John Cristobal: for hypothesis testing. John Cristobal: Right? And so the first step… John Cristobal: Is that we want to identify. John Cristobal: our hypotheses. John Cristobal: Right? So these are the statements.
2:40
John Cristobal: those ideas. John Cristobal: that we're going to be checking that we assume to be true 1st off. and then we're going to be checking as we go on right. And so for hypotheses, the 1st hypothesis is. John Cristobal: H sub 0, right? This is our Null hypothesis.
3:04
John Cristobal: Okay? And with our null hypothesis, I'm gonna put here in green. There's notes here with our null hypothesis. This is always
3:15
John Cristobal: gonna have that equal sign. John Cristobal: Right. John Cristobal: And so, with our null hypothesis, this is where here. John Cristobal: This is going to be our assumed, our kind of agreed upon, or the assumption that we're working on. John Cristobal: Value.
3:37
John Cristobal: for the parameter. John Cristobal: Right? So this is that kind of
3:43
John Cristobal: general value that so far everyone agrees to. Maybe it was done with a previous practice a previous John Cristobal: research project that they've gone through and identified this value for a parameter. So with our null hypothesis that age sub 0.
4:01
John Cristobal: That's gonna be where we have mu equals. John Cristobal: This given value right for the mean, or we have a population. John Cristobal: Proportion P is equal to that, right? This percent of citizens of this kind of belief. John Cristobal: right? This proportion of businesses
4:24
John Cristobal: satisfy this characteristic, so on. And we're going to present that as a decimal after the percent right, or likewise, even if we're talking about standard deviation variance, so on. John Cristobal: Sigma. John Cristobal: And this is gonna be again our assumed value. Right? This is gonna be And the value…
4:46
John Cristobal: for the population.
4:53
John Cristobal: and comments. We're gonna check to see if this value has possibly changed. John Cristobal: has shifted at all. John Cristobal: Right? So there's our null hypothesis. One of those 3 statements depending which parameter we're dealing with.
5:06
John Cristobal: Then, whether it be H sub one, or sometimes in different texts, we see H sub a. This is our alternative hypothesis.
5:20
John Cristobal: And for our alternative, right, as it kind of says there, this is that alternate idea, that possibility of change. Now, in this case, this is where we're going to use
5:30
John Cristobal: an inequality symbol.
5:34
John Cristobal: this is where we start to see where possibly, maybe we think that value has
5:39
John Cristobal: gone small. That is less than that quantity. John Cristobal: That is greater than it's gone up. It's increased. John Cristobal: Or just that it's different, and maybe it's changed, we're not sure.
5:50
John Cristobal: if it's gotten big or small, but definitely don't think it's the assumed value. John Cristobal: that we have started with, right? And so with these again, we're going to going to address John Cristobal: that assumed value for our alternative hypothesis. I'll use a sub one for my notation here.
6:06
John Cristobal: for that alternative hypothesis. John Cristobal: We're gonna see right that mu John Cristobal: is less than. And now, in this value here. John Cristobal: You're going to go back to this is our value from.
6:21
John Cristobal: or null hypothesis.
6:26
John Cristobal: Right? Because we're always comparing to what we assume for the population, the sample data that we're going to see, the sample values that we have. John Cristobal: are going to come up later in our process. For now, we're kind of assuming this is what's being held true for the population.
6:43
John Cristobal: and our ideas, our possibilities, our alternate results that possibly could occur, right? And I'll go with the other value, right, for proportion. John Cristobal: Maybe it's greater than, same deal. John Cristobal: Same action. John Cristobal: Take that value from the hypothesis. Maybe we think the percentage is greater, has gone up from 50%, 55%, and so on. Okay? Or has decreased, has gone down. Or…
7:10
John Cristobal: somebody's not equal to. And again, I'm using my different symbols. We're not always gonna have greater than with the proport John Cristobal: Mainly here, I'm focusing on the inequality symbols we're using the parameter will identify with our problem. John Cristobal: statement of which symbol we need to make sure. Even it's always going to be the population information with our hypotheses. John Cristobal: Alright, so this is the first step.
7:33
John Cristobal: That first action in our hypothesis testing is identifying our hypothesis. John Cristobal: Okay. John Cristobal: using those symbols primarily. John Cristobal: All right. John Cristobal: So the next… The next step here. Now, this is where our sample data comes in. John Cristobal: Step 2. John Cristobal: Now we're going to look at our test. John Cristobal: statistic. John Cristobal: And with that test statistic. John Cristobal: Okay, this is gonna be our value.
8:06
John Cristobal: computed.
8:09
John Cristobal: Right, based on…
8:14
John Cristobal: our sample data.
8:21
John Cristobal: And really here this is where we're going to see either a Z score.
8:28
John Cristobal: that T distribution T score. John Cristobal: Right? Or… When it comes into standard deviation, we're going to get this chi-square score.
8:42
John Cristobal: Okay. John Cristobal: And so… Often, when we're looking at these. John Cristobal: these different values, we're running to see a Z score, When we're doing proportions.
8:56
John Cristobal: Or likewise, if we're doing the mean, and we actually know the population's derivation, which is pretty… John Cristobal: Odd is not very common. John Cristobal: to know the population's derivation, not know the population mean, so that's why we don't really necessarily use that too often. The t-score, that t-distribution, very similar to our confidence interval work, we're going to see John Cristobal: The T score is being used primarily when we have the mean.
9:21
John Cristobal: and that chi-square score another distribution. We have the normal, that Z distribution. John Cristobal: We had our T distribution, that student T distribution. John Cristobal: And now, when it comes to standard deviation or variance. John Cristobal: Another…
9:40
John Cristobal: Distribution shows up this chi square. This is the Greek letter chi. John Cristobal: CHI, how you spell it out in English, but the chi-square score. John Cristobal: Here's what we're doing Standardization or variance. John Cristobal: Okay, very. John Cristobal: Make sure that my spelling right there we go. Okay. John Cristobal: Sigma. John Cristobal: See my squared fifth annotation right?
10:05
John Cristobal: and I guess I suppose for the mean we're doing mu for the null hypothesis right proportion. John Cristobal: So, okay, and so then we're going to base these values, these test statistics, kind of looking at those distributions. John Cristobal: And as we had, right when it comes to proportion. John Cristobal: If we're doing a test for proportions. John Cristobal: Our test statistic is a Z score.
10:36
John Cristobal: And when we calculate that.
10:39
John Cristobal: Z score. We're gonna take our sample proportion John Cristobal: Right? So we did a sample of 50 people. John Cristobal: of those 50, right? This percentage, right? This many successes, X over N, so on. John Cristobal: that sample proportion, P hat. John Cristobal: minus this value. I'm going to call P sub 0. This is
11:03
John Cristobal: our population proportion from the null hypothesis.
11:14
John Cristobal: that same value.
11:16
John Cristobal: from step one, that null hypothesis value we're going to use here. And then we're going to divide 5. John Cristobal: this square root.
11:28
John Cristobal: and the square root we're going to have here.
11:32
John Cristobal: that same. John Cristobal: Population portion P sub 0 for the null hypothesis times its complement, 1 minus P sub 0.
11:41
John Cristobal: right? Sometimes in subtext you might see that called queue. John Cristobal: Right? The complement of your population portion. John Cristobal: And then we're gonna divide that by… and your sample size. John Cristobal: And so in calculation, we'll do that subtraction first in the numerator. John Cristobal: We can do this square root work.
12:02
John Cristobal: from the denominator separately, and then bringing those together, type it all in at once, just being careful with order of operation, so on. John Cristobal: Okay.
12:12
John Cristobal: But here's our formula, we can calculate, but once it will go forward, we'll see StatCrunch. John Cristobal: a lot of these programs will calculate it directly for us, and we don't necessarily need the formulas. I bring in the formulas just so we can identify John Cristobal: how those results show up. Okay, so there's our test statistic, right? There's value
12:35
John Cristobal: for our sample data that we're going to use to compare. John Cristobal: for population value. John Cristobal: Then for the mean.
12:44
John Cristobal: we could do if we know the population standardization truthfully right going back to our work in earlier chapters. John Cristobal: We can calculate the Z score.
12:55
John Cristobal: Where X… is our sample mean. John Cristobal: Right? Then we have our population mean that we're trying to identify that from the null hypothesis divide by John Cristobal: sigma, the population over the square root of N. We get that Z score. John Cristobal: But most often, when it comes to the mean, we're not going to have the population standardization. John Cristobal: We only have sample standard deviations, so we're going to see a t-score, that t-dist
13:20
John Cristobal: And so we talked about right X bar or sample mean minus mu again. John Cristobal: that population mean that's from I'll use my symbols this time. A sub 0, the null hypothesis John Cristobal: Same deal there.
13:36
John Cristobal: And then we're going to divide by S, our sample standard deviation. John Cristobal: divided by the square root of N. The square of your sample size. John Cristobal: Okay. John Cristobal: And then, lastly, We can also deal with a chi-square if we're going to do a hypoth 1st innovation. John Cristobal: or variance, so on, that chi-square score.
14:03
John Cristobal: Get that chi square. John Cristobal: is going to be equal to.
14:08
John Cristobal: N minus one. Got a sample size. John Cristobal: minus one.
14:15
John Cristobal: times your sample standard deviation squared, or your sample variance And we're dividing by.
14:23
John Cristobal: your population standard deviation squared, or your population variance. John Cristobal: In denominator. John Cristobal: Okay, so remember that notation that we have here.
14:34
John Cristobal: S is your sample standard deviation.
14:41
John Cristobal: So there's our… sample data coming in, or depending, make sure, be careful with your John Cristobal: reading of the problem. If it's a sample variance. John Cristobal: that they provided, That means they've already done the squaring. John Cristobal: Right, and then similar. John Cristobal: Sigma population standard deviation.
15:05
John Cristobal: That value that we're assuming to be true. John Cristobal: in our null hypothesis. John Cristobal: Or again, if they've already done the squaring, they might provide you the population variance.
15:19
John Cristobal: So in this case, this is the value based on our sample data. John Cristobal: Primarily, we're going to talk about proportion and mean. Later we'll get into standard deviation, right? But it's a value that utilizes our sample information.
15:36
John Cristobal: Okay. John Cristobal: to help identify. Okay, let's focus. John Cristobal: So then, once we get that test statistic.
15:46
John Cristobal: Now we're going to start talking about this kind of comparison. Right? And so we have 2 options. We have option one.
15:55
John Cristobal: using critical values.
16:02
John Cristobal: Okay.
16:05
John Cristobal: or option 2.
16:08
John Cristobal: P. Value. John Cristobal: And with… more… Software that we use. John Cristobal: The easier it is to use these kind of special programs on our computer, even in calculators now. John Cristobal: We started to see p-values getting used a little bit more often. John Cristobal: And certainly with 2 populations later, we'll do that. John Cristobal: But… Critical values, a nice visual, kind of bringing us back to our visualization with that normal distribution.
16:37
John Cristobal: the T distribution, so on. John Cristobal: in this comparison. Okay? John Cristobal: And so, when we start to think about this p-value, or critical value. John Cristobal: in order to make our decision, and so on. In the next step we're going to need to have a value for Alpha.
16:55
John Cristobal: which is our significance level.
17:02
John Cristobal: Most commonly, our values for alpha are going to be like 1% or as a decimal point 0 1. John Cristobal: or… 5%. John Cristobal: 0 point 0 5, but similar to our work with confidence intervals. John Cristobal: Right? How we can use any confidence. John Cristobal: level. We wanted 95%, 99%. John Cristobal: 92%, so on.
17:30
John Cristobal: Significance levels can be any percent. John Cristobal: Most likely, though, 1%, 5% show up. We can have 10%. Okay? And you'll see that similarity, right? We have that alpha with our confidence levels, one minus the confidence level. So this, like having a 99% confidence. John Cristobal: 5, that 0.05, that 5% significance, like a 95% confidence interval, so on, kind of seeing that similarity. John Cristobal: And… a key statement here. John Cristobal: Is this alpha?
18:01
John Cristobal: right? This is our Hi. John Cristobal: one error. John Cristobal: probability.
18:15
John Cristobal: Right? So that's the probability chance that we make an error. We make the wrong decision.
18:22
John Cristobal: the wrong conclusion based on our work right? Because, based on our sample, we might have one rare sample where our value is much higher than what's normal in the population. John Cristobal: throwing at this error because we're choosing because we're making the wrong decision because we had some unusual, some rare event. John Cristobal: in our sample. Okay, so there is a possibility of making the wrong decision. John Cristobal: But we're hoping, right.
18:49
John Cristobal: that we're not to make that error that our significance of making such areas small. So okay, I'm making this comparison. John Cristobal: So with that said, with our critical values. Okay, doing this 1st time.
19:05
John Cristobal: Our critical value method.
19:10
John Cristobal: depending on whether we have the normal distribution, the Z John Cristobal: or the student T distribution. Or later, we can talk about the critical values using the chi-square distribution. John Cristobal: But with our critical value. John Cristobal: We have our distribution curve. John Cristobal: right? So that's at Z. John Cristobal: that T distribution, so on.
19:32
John Cristobal: And when we look at those critical values, we're kind of identifying that. Okay, if I have John Cristobal: that amount of area alpha in that tail.
19:46
John Cristobal: That cutoff value for it is your critical value. John Cristobal: Okay? And so, looking at this distribution, We have these. John Cristobal: possibilities.
20:00
John Cristobal: do my best to redraw that normal distribution or that T distribution right where maybe we have area in the left tail. John Cristobal: the area alpha in the right tail. And again, that cutoff value that critical values.
20:16
John Cristobal: marks there.
20:21
John Cristobal: or similar to our confidence interval work.
20:26
John Cristobal: we might split separate that amount of area. John Cristobal: evenly between. John Cristobal: Our team of sales. John Cristobal: Do not.
20:38
John Cristobal: Let's see for the left critical value for the left critical value on the right. Okay? And so these 2 are our one tail tests. John Cristobal: And we kind of identify that if we have in our Alternate. John Cristobal: hypothesis. John Cristobal: that we have that our parameter. I'm gonna use mu for my example. John Cristobal: is less than this value. This means that we're going to have a left pill test.
21:07
John Cristobal: Alright, we're putting all that significance. John Cristobal: All that area on the left side. John Cristobal: You can probably guess what's gonna happen here if our
21:17
John Cristobal: inequality symbol in the alternative hypothesis, A sub A sub 1, is a greater. That's going to be a right-tailed test, that inequality symbol.
21:27
John Cristobal: Pointing us in the direction that we're gonna put that area. John Cristobal: On that right side.
21:34
John Cristobal: Or if we're going to split it, this is where we have John Cristobal: The 2 tail test. We have that area split between the 2, then not equal to. I kind of think of it. Okay, we're separating. We're cutting that in half. John Cristobal: To put that amount of area. John Cristobal: evenly between the 2. Okay? And here's our 2 gel test.
21:55
John Cristobal: All right. So with that 2 tail test. John Cristobal: One tail tests with left side, right side. John Cristobal: We're gonna critical value. Okay? And so, based on having that much area kind of like we found those cut off values. John Cristobal: back with our normal distribution work before having that amount of area in the tail. We can go to our particular distribution, whether it be in the normal Z distribution.
22:19
John Cristobal: that student T distribution using those calculators so on to find the cutoff value that Z-score that T-score. John Cristobal: to identify that amount of area in the tail. Okay? John Cristobal: And so then… once we have those critical values identified. John Cristobal: what we started to see. And again, kind of using
22:43
John Cristobal: that left tail test is my 1st example. Once we go through and we have
22:51
John Cristobal: that alpha, that amount of area in our tail. We see our critical value.
22:59
John Cristobal: Now, when it comes to making that decision, as we go through. John Cristobal: If I see that my test statistic.
23:10
John Cristobal: that value computed based on our sample.
23:15
John Cristobal: Is in that shaded region. John Cristobal: Right?
23:20
John Cristobal: or…
23:28
John Cristobal: I'll use that left tail test as a example. It can be the same when it comes to our right tail test, or even the 2 tail test. If our test statistic is outside.
23:44
John Cristobal: that shaded region beyond the critical value. John Cristobal: Right. John Cristobal: It's outside that critical region. Sometimes it's called okay. John Cristobal: When it comes to making our set, our next statement, step 4, our decision.
24:01
John Cristobal: Okay? Our decisions either… to reject. John Cristobal: The null hypothesis.
24:13
John Cristobal: or… We do not. Oh, you fail to reject.
24:20
John Cristobal: or null hypothesis.
24:25
John Cristobal: And we're always comparing to the null hypothesis, right? What we're assum John Cristobal: for the population, because we were using sample data this test could check if John Cristobal: the population value is still valid, or if it's shifted, so on, okay? And so, looking at these statements, looking at these results, that when our test statistic John Cristobal: is in that critical zone critical region when it's in the shaded space. Right? That's gonna be
24:56
John Cristobal: Where we reject.
24:59
John Cristobal: The null hypothesis, whether it be on the left tail, or it could be in a right tail test right in my test statistics here. John Cristobal: We're gonna reject. John Cristobal: Either side with a two-tail test, okay?
25:11
John Cristobal: But when… your test statistic. John Cristobal: is outside the shade region. John Cristobal: That's when we do not, or you fail to reject.
25:26
John Cristobal: the null hypothesis right? And so what this kind of idea is, and we'll get into with our summary. John Cristobal: is that when we're rejecting the null hypothesis, rejecting that assumed John Cristobal: that population value that we've been working from, that everybody assumes to be true at the moment, so we're rejecting that. Our sample data is different enough, it's significantly different, that we say, okay, that population value must be
25:48
John Cristobal: Invalid. It's not true anymore. John Cristobal: Right? We can go further. John Cristobal: studies to to verify. But that's kind of our initial Check to see. John Cristobal: if that population value can be rejected. John Cristobal: right? If our sample value is outside, it's not different enough, right? It's not in that space where it might be unusual to get a result.
26:11
John Cristobal: We're kind of in that body where we anticipated the majority of value would be found. We do not reject, right? We're going to keep John Cristobal: That population value, we're not rejecting it. We're going along, that must still be true. John Cristobal: must still be valid for our population. Okay? So that's the decision part. So I'm kind of working with John Cristobal: Step 3. Finding those critical values. John Cristobal: using those distribution that in that inverse process.
26:39
John Cristobal: Given an area, finding the cutoff value, the critical value. John Cristobal: and then using that to compare with our test statistic, our sample data. John Cristobal: So that was a critical value map. So then. John Cristobal: Similar when we want to do the p-value method.
26:58
John Cristobal: Okay, that was critical values. And now for our p-value method.
27:04
John Cristobal: with our p-value method. The key statement here is that we're going to compare
27:12
John Cristobal: Your significance level.
27:21
John Cristobal: that alpha to our area.
27:32
John Cristobal: From.
27:35
John Cristobal: the test statistic.
27:41
John Cristobal: Okay. John Cristobal: So when we start to see. John Cristobal: in that calculation.
27:50
John Cristobal: Whether we have… that left tail test that right tail test, so on. John Cristobal: right? We can have our test statistic.
28:07
John Cristobal: that T value. John Cristobal: that Z score from our step number 2, right? Those values that we computed. John Cristobal: here.
28:19
John Cristobal: depending on which distribution we're using right, and we'll calculate that area for a left tail test. John Cristobal: Left side, right tail test. John Cristobal: Right? Two-tailed test, you get your… John Cristobal: test statistic area. You double it right to get those 2 tails that balance. John Cristobal: And so this area.
28:41
John Cristobal: That's going to be the p-value
28:45
John Cristobal: So that's what's being computed with a P value. John Cristobal: Whether it be the normal distribution. John Cristobal: P. Distribution that's doing T distribution. John Cristobal: Or, again, we can go further. John Cristobal: with the chi-square distribution or standard deviation. But once you get that p-value, you're going to make a comparison. John Cristobal: Okay. John Cristobal: And so then, when it comes to making that decision.
29:11
John Cristobal: When we see that our p-value is less than alpha.
29:22
John Cristobal: Right? So the probability of getting a sample value attested to the sample data John Cristobal: in the space that probability here is smaller than our significance level. John Cristobal: The p-value is less than alpha. That's when we Reject.
29:40
John Cristobal: the null hypothesis. Okay.
29:45
John Cristobal: That's our decision.
29:49
John Cristobal: We're less likely to get a value here than make an error. John Cristobal: Okay.
29:56
John Cristobal: Then the opposite. If our p-value is bigger than Alpha.
30:05
John Cristobal: Right.
30:08
John Cristobal: We're more likely to get that value than make an error John Cristobal: Right? We're more likely to make an error John Cristobal: Then get a value there, okay? That's when we do not reject.
30:22
John Cristobal: They're null hypothes Okay.
30:29
John Cristobal: email string. So it kind of falls into place. John Cristobal: is very similar. I'll let you kind of look that through. John Cristobal: for the critical value. John Cristobal: Okay, when that critical value is Below… John Cristobal: or our test is even below the critical value. John Cristobal: We're gonna reject, right?
30:48
John Cristobal: That area is smaller. John Cristobal: That could that test statistic is. John Cristobal: Below… is beyond your critical value. John Cristobal: Okay.
31:00
John Cristobal: This case, our test statistic is outside of that critical zone. John Cristobal: We're not gonna reject. Okay. John Cristobal: And from there, once you make that decision. John Cristobal: Okay, and again, you can use either one of those two methods. John Cristobal: the critical value method. John Cristobal: kind of looking to see where John Cristobal: your test statistic was in relationship to your critical value. Looking at that critical zone to make your decision.
31:27
John Cristobal: or the p-value, which we'll kind of see a lot more often with our software systems.
31:34
John Cristobal: Okay, once you make that decision, though.
31:40
John Cristobal: Then we're going to do our summary right? So this is the part that kind of takes a little more time. This is now our wording, our summary. Okay? Or sometimes I'll call it the conclusion.
31:50
John Cristobal: Right? I know we've made a decision. So that's like a conclusion. But now, in this case. John Cristobal: with our summary, our conclusion. I want to go back to using John Cristobal: the wording, that kind of particular application statement. John Cristobal: of the original problem.
32:12
John Cristobal: Okay. John Cristobal: And There's going to be a lot of examples that we can read in the text John Cristobal: You can probably find online some examples as well. But in this case we kind of start to use these kind of phrases that John Cristobal: They're… is sufficient, or my other example, say, like there's enough. John Cristobal: Evidence.
32:36
John Cristobal: And if we're rejecting… that null hypothesis right? John Cristobal: to reject, And we go along with that null hypothesis, putting it into words, right? That.
32:50
John Cristobal: that.
32:52
John Cristobal: Population mean.
32:55
John Cristobal: or proportion, standard deviation.
33:02
John Cristobal: Okay.
33:06
John Cristobal: is… I'll put dot dot dot. Now, that's From the problem comes in. John Cristobal: Okay. John Cristobal: Or likewise.
33:15
John Cristobal: If we're not gonna reject. John Cristobal: there is not sufficient, or I say, not enough evidence based on our sample.
33:28
John Cristobal: To reject.
33:31
John Cristobal: I'll put my dot dot dot here. Continue with that wording from from the problem. And again, that this part of the summary takes some time. John Cristobal: to kind of get used to that wording. John Cristobal: how we phrase that, how we kind of understand. But there's always again, based on like our decision.
33:50
John Cristobal: that null hypothesis that assumed value for our population. John Cristobal: All right. John Cristobal: So with that said. John Cristobal: Let's go through an example or 2. Looking at some of the information, we can write up some more examples as well for you to look through.
34:08
John Cristobal: And as we go through, I'll switch over to
34:12
John Cristobal: stack crunch as well for these calculations. Right? And so I'll have these handouts available to view as well.
34:22
John Cristobal: There's our 5 steps, those procedures. John Cristobal: Okay. John Cristobal: the different test statistics that we can use to calculate. Again, we can also do one for the mean mu John Cristobal: with the Z distribution. If we happen to know the population center is in Sigma. John Cristobal: But, as you can see, most commonly we have our sample data. Let me try to zoom in here. John Cristobal: Mine was a little bit small.
34:48
John Cristobal: for review. All right. So let's go through this example, though. So I said in this 1st problem that a survey in a recent newspaper article stated at 38%. John Cristobal: of Americans said that they will not eat carbs this Thanksgiving. Okay, so this is an example I had done John Cristobal: around Thanksgiving time. Right? John Cristobal: And again, with all the dieting trends and whatnot, maybe people are not going to eat car Right.
35:12
John Cristobal: And so the dietician thinks that this percent is far too high, like more people are saying they're not gonna eat carbs. But that's too high. This dietician thinks. John Cristobal: that the percent that don't eat carbs is going to be much less, or not much less, but at least different, less than oxidative farming than that 30%. John Cristobal: right? So with that percent. John Cristobal: I'm already starting to think about proportions. John Cristobal: Okay.
35:36
John Cristobal: And so in this case, a random sample of 150. So there's our N, a random sample of 150 residents was selected. 48 said that they're not going to be carbs. So 40 people out of the 150 have that characteristic that they're not going to be carbs.
35:53
John Cristobal: using 5% significance as our alpha. John Cristobal: Let's test the dietitian's claim. John Cristobal: that the proportion of Americans that will not eat carbs is lower, will be lower than that 38%, okay?
36:08
John Cristobal: So as I go through, that 38%, Okay, that value John Cristobal: This is going to be our P sub 0. That's the percent that we're assuming to be true for the proportion. The dietician, this researcher, this person. John Cristobal: has an idea that that proportion should be lower as we're trying to compare to. So that's what we're assuming to be true John Cristobal: Okay. John Cristobal: looking through some of these other values, right? A sample of 150 people, because there's our N.
36:39
John Cristobal: of those 48 John Cristobal: are not gonna eat carbs. So there's our X, the number of successes, the number of people that have that characteristic. John Cristobal: RP hat. John Cristobal: our sample proportion X over N. We'll compute that 48 over 150.
36:56
John Cristobal: We're gonna test. Okay. John Cristobal: At the 5% significance level, store value for alpha.
37:05
John Cristobal: 5%.
37:08
John Cristobal: or point 0 5.
37:10
John Cristobal: Okay.
37:15
John Cristobal: So I know it's a lot of information to kind of go through. John Cristobal: But let's start to look at this example. Then. John Cristobal: As we start to look at these pieces, right? Looking at our 5 steps. The 1st step that we're gonna go through step one. Okay, I'm gonna go. John Cristobal: Roman numerals or hypotheses.
37:37
John Cristobal: So for those hypotheses. John Cristobal: Starting off… the null hypothesis, H. Sub 0, always equality. John Cristobal: our proportion. We're talking about some of our parameter. P.
37:51
John Cristobal: We're going to assume that it's going to be that 38%. I'm going to convert John Cristobal: right away. So that's what we're assuming to be true going by. John Cristobal: That article… That's what the population is is. John Cristobal: Agreed upon at the moment. John Cristobal: But now the dietitian. John Cristobal: has a different idea, an alternate hypothesis, right? So for our alternative hypothesis, A sub 1, Same. John Cristobal: Parameter P, that proportion.
38:22
John Cristobal: But now they think that it's going to be lower.
38:26
John Cristobal: then that 30% lower. John Cristobal: less than right. So this alternative hypothesis John Cristobal: is that that percent? That proportion should be smaller than John Cristobal: 30% or as a dozen point three eight. The proportions.
38:42
John Cristobal: The common having rounding is to keep 3 decimal places. Okay. John Cristobal: Which is gonna show when we start to deal with our… Test statistic next. John Cristobal: And so with our test statistic, step 2,
39:02
John Cristobal: As soon as we're doing proportions. Okay. John Cristobal: Our test statistic is going to be a Z score. John Cristobal: And it's going to be that p hat. John Cristobal: minus E sub 0. John Cristobal: All over this big square root. John Cristobal: P sub 0, then null hypothesis proportion times this complement, 1 minus P sub 0. John Cristobal: all over and best sample size. Slide that up a little bit.
39:31
John Cristobal: Okay. John Cristobal: And so with that, Set up. John Cristobal: Okay.
39:38
John Cristobal: As I mentioned up here, we can calculate that test or that sample proportion. John Cristobal: X over N, 48. John Cristobal: over 150. And off to the side here, I'm using my scientific calculator, 48 over 150.
39:59
John Cristobal: And we're getting thankfully a nice John Cristobal: simple terminating decimal 0 point 3 2. John Cristobal: Had we gotten some long decimal? John Cristobal: We can keep that, or again, the common habit, we round to 3 dozen places. John Cristobal: Okay.
40:16
John Cristobal: We have our value from 150, and so they started to calculate.
40:21
John Cristobal: I'm gonna take my sample portion. John Cristobal: 0 point 3 2. John Cristobal: minus… The null hypothesis proportion 0 point 3 8.
40:31
John Cristobal: Now we're gonna divide that all by the square root. John Cristobal: big fraction inside.
40:43
John Cristobal: null hypothesis, population proportion point 3 8.
40:48
John Cristobal: Time is compliment, 1 minus… John Cristobal: that same 0 point 3 8. If you want to do that calculation ahead, you can write one minus point 3 8. John Cristobal: 100% minus 38%. You're gonna get 62%. John Cristobal: 0 point 6 2. John Cristobal: You divide by the sample size, which was 150. John Cristobal: and as a helpful. John Cristobal: Same here. You always subtract 1st don't worry about that division. John Cristobal: and Desmos, we can kind of share that screen.
41:21
John Cristobal: And we can see… Okay. John Cristobal: that calculation. Let me share that real quick.
41:30
John Cristobal: if we can't.
41:33
John Cristobal: Back.
41:37
John Cristobal: Alright. So now we should see my Desmos.
41:42
John Cristobal: calculations.
41:46
John Cristobal: doing. Oh, it's not square. Yeah, there's my John Cristobal: T hat in my sample portion, as you can see, but not calculated as Z. We're gonna need a fraction division in the numerator, 0.32 minus 0.38. John Cristobal: P hat minus P sub 0. Now we have our big square root in the denominator. John Cristobal: fraction inside of their division is going on there 0 point 3 8 times. John Cristobal: one minus 0 point 3 8. John Cristobal: all over 150.
42:14
John Cristobal: And there's… our long test statistic, the Z score. John Cristobal: And as we're kind of used to doing, let me copy that down real quick.
42:27
John Cristobal: when it comes to the Z score. I'm gonna stop sharing now.
42:38
John Cristobal: I don't know, M decimal, right? So there's my unrounded value. And then for our test statistic. John Cristobal: that Z score, when it comes to normal distribution, our Z scores here, we're going to round to two decimal places John Cristobal: All right, so round into two decimal places, one, two. That three says that one can do the same, negative 1.51, right? So there's…
43:04
John Cristobal: or test statistic. John Cristobal: And so, even going back before right kind of that idea. John Cristobal: was these scores was kind of considered unusual. John Cristobal: Z scores that are bigger than 2, right? Less than negative 2. John Cristobal: That's when we started to get to some unusual quantities.
43:25
John Cristobal: a Z score of negative 1.5 1, right? That's not too bad. It's like only one and a half 1.5, right? One and a half standard deviations below the mean, which isn't unusual. But that's kind of tying into what that Z score represents for us. That standard deviation idea so on. John Cristobal: But there's our test statistic. John Cristobal: hypotheses. John Cristobal: Using the… parameter symbols for proportion had been for a mean mu, right? So on.
43:54
John Cristobal: And then our test statistic when it comes to proportion is going to be a Z score.
43:59
John Cristobal: Alright, so now with that said, Next up. John Cristobal: We're going to do our p-value or critical value. So now, in this case, now I'm going to start to use
44:10
John Cristobal: Sacra. Okay.
44:18
John Cristobal: because if we're looking for the critical value.
44:22
John Cristobal: that Z score cut off. We're gonna need the stat crunch for that normal John Cristobal: distribution calculator, or likewise, thankfully, StatCrunch has John Cristobal: tools similar to our confidence enrolled right tools built in for hypothesis testing right away. Okay, so I'm gonna start to share screen again with our stat crunch calculator this time.
44:46
John Cristobal: And here we are. Okay, so now, looking at StatCrunch.
44:51
John Cristobal: similar to what we had seen with our confidence intervals.
44:57
John Cristobal: Now… John Cristobal: We're going to look at either Z stats. That's going to be when we're dealing with the mean, and we know the population's derivation very rare, more commonly T stats. John Cristobal: one sample, we have the mean, right? With a summary, we'll see that in another example. Or, in this case, since we're dealing with proportions, very similar to what we do with confidence intervals, right? John Cristobal: those statements.
45:19
John Cristobal: We're going to do proportion tests. We only have one sample, one population we're dealing with, okay? And we're going to go with summary, very similar to what we saw with confidence rules. If we're doing a test for standard deviation or variance. John Cristobal: right? Standard Basic on the hidden is using variance, and we'll see that work there. Alright. So with our example, though.
45:40
John Cristobal: Similar to what you saw with confidence intervals, we're doing proportions. We have proportion stats. We have one sample, one population that we're dealing with.
45:50
John Cristobal: We're gonna go with summary.
45:53
John Cristobal: And so what we had talked about before, when we had confidence intervals, right, we highlighted that dot. John Cristobal: Now we're going to stick to hypothesis tests. John Cristobal: Right there in front of us from before. And so now we start to input, right? So our number of successes, our X values, right? John Cristobal: your number of observations, your sample size. That's your N. John Cristobal: And then we'll put in from our null hypothesis.
46:16
John Cristobal: that quantity. Okay, so let me start to fill in, though, for our example, our number of successes was 48.
46:22
John Cristobal: Our number of observations, our sample size was 150. John Cristobal: We are doing a hypothesis, our null hypothes John Cristobal: that 38% 0 point 3 8 right right as a decimal. John Cristobal: Nathan Miller, And then our alternative hypothesis. I know I did John Cristobal: The calculator here uses H sub a right. We have that less than we think that dietician, I should say. thinks that claims that that percentage is too high, so it should be less than point 3 8.
46:49
John Cristobal: I'll have it show the critical value, that way it's doing that work to get the critical value if you want to do it. John Cristobal: our significance level 5%, 0 5. So that's good. You can change that as necessary. Show the critical value.
47:04
John Cristobal: I'm not worried about the graphing. We'll compute.
47:09
John Cristobal: And there we go. So right. So now in this case, here's our results.
47:15
John Cristobal: There's our proportion excesses, right? So there's
47:19
John Cristobal: null hypothesis, alternative hypothesis, right? So it's copying that down for step one. John Cristobal: We're looking for proportion. There's the count. That's our number of successes. X total your sample size N. John Cristobal: sample proportion. There's our P. Hat. John Cristobal: Standard error, that's the standard deviation for music, but we don't need that, right? Our Z-stat, that is our test statistic.
47:42
John Cristobal: Right? Only a few digits shown here, but again, with rounding, we see the same results. John Cristobal: that critical Z scores. There's our critical value. John Cristobal: There's our cutoff, I think, negative, right? It's a left tail test. We expect negatives there. John Cristobal: and now our p-value, right? So whether we're doing the critical value method. John Cristobal: There's our critical value. John Cristobal: Don't forget my negative. There's a highlighting right?
48:07
John Cristobal: p-value method like so. John Cristobal: Right? So taking those values, we have all that data there, all information. John Cristobal: Then now we can use that to finish up our. John Cristobal: work with the hypothesis test. Okay. John Cristobal: Alright, so I'm gonna go back to the notes.
48:23
John Cristobal: There's our information. John Cristobal: All right.
48:27
John Cristobal: And we'll borrow that result there, okay? So using StatCrunch, remember, that's gonna be under the Stat menu.
48:37
John Cristobal: And then we're gonna either look for, depending on what we're working on, in this case, this was proportion stats.
48:48
John Cristobal: Right? Proportion… stats.
48:53
John Cristobal: We're doing one population or one sample that we've identified.
48:59
John Cristobal: And we're gonna go with summary.
49:03
John Cristobal: later, or in other problems. John Cristobal: you can work on, or maybe if you're given raw data, in particular when it comes to, like, the mean or standard deviation, right, having a list of data instead of having to calculate John Cristobal: your sample mean sample standard deviation which you can similar. We did in Chapter 3. Okay. John Cristobal: What we can do instead, put that data into a column, and then have it refer back to that column.
49:26
John Cristobal: But we're gonna go with sample with summary, and we'll input that information. Okay? And so, based on that work going through those calculations. John Cristobal: inputting that data there, right? And again, for the mean, it'll look very similar. We can see a quick kind of run through John Cristobal: without work later. John Cristobal: But… looking at that result, right, that Z stat, just kind of highlighting here, this is our test statistic.
49:58
John Cristobal: Then it has your critical Z.
50:03
John Cristobal: This is your critical value.
50:10
John Cristobal: And then, of course. John Cristobal: directly, p-value.
50:17
John Cristobal: Nothing really extra to say there, but I'll put the highlighter there, right, that p-value. John Cristobal: Okay. John Cristobal: And so, from our values, what we saw, that Z stat negative John Cristobal: 1.513942, right? There's my unrounded value. John Cristobal: That same result we saw back in step 2.
50:42
John Cristobal: You can call it a Z sub p hat if you wish, right?
50:46
John Cristobal: Now that's step 2, the critical value.
50:51
John Cristobal: Negative 1.6 4. John Cristobal: 4, 8, 5, 3, 6. And you might remember that from
51:02
John Cristobal: Our confidence intervals, right? We have that 5%, that 90%.
51:09
John Cristobal: Confidence level, that Z, that Z score. John Cristobal: That's Z sub L for two. John Cristobal: 1.645 or 1.65. John Cristobal: Textbooks rounded differently, use that value differently. And then our p-value, that's zero point. John Cristobal: 0, 6, 5. John Cristobal: Okay, so there's our factor P. I'll put in blue, 0.065, so on. Okay. John Cristobal: So there's those values. These are the quantities from stat crunch. John Cristobal: unrounded, rounded values, so on.
51:44
John Cristobal: Okay. John Cristobal: And so when it comes to making our decision step 4 there.
51:55
John Cristobal: Right? Looking at our p-value.
52:02
John Cristobal: Looking at this value for alpha. John Cristobal: the significance level.
52:09
John Cristobal: 20, 0, 5. We can't put a 0 when we can. John Cristobal: comparing those right working. John Cristobal: Left to right. John Cristobal: Same digit in the ones place, same digit in the tenths place, okay? Get that to focus.
52:25
John Cristobal: All right.
52:29
John Cristobal: Comparing those, then, now getting into the hundreds place. John Cristobal: 6 is bigger than 5. So our p-value is greater than alpha. John Cristobal: There's that comparison. So for our decision. when your p-value is bigger than Alpha, you're going to say we do not, or we fail to. John Cristobal: Reject.
52:52
John Cristobal: that null hypothesis.
52:56
John Cristobal: Okay, so there's our decision. John Cristobal: Based on that information, those values, computed using StatCrunch.
53:05
John Cristobal: We're not rejecting the null hypothesis. John Cristobal: Okay, had the p-value been less than alpha. John Cristobal: How we got like point 0 1 5 instead. John Cristobal: Then we would say we reject right. But now, because P. Value is bigger. we do not reject. John Cristobal: Now, this is the part… that students always haven't.
53:25
John Cristobal: questions and trouble some time to develop. This is our summary, right? Our summary for our conclusion. So John Cristobal: Based on…
53:39
John Cristobal: our sample data, Right? John Cristobal: We can't reject that null hypothesis, so even though this dietician thought that that percentage should be lower, right, we're not rejecting that value. So in this case, for our summary. John Cristobal: Is that… Alright, there… is not enough, or I'll go back to that word sufficient. Either way is fine.
54:04
John Cristobal: Evidence.
54:07
John Cristobal: Alright, based on our sample.
54:10
John Cristobal: to reject, But… those proportion.
54:22
John Cristobal: Oh. John Cristobal: Americans!
54:31
John Cristobal: that will not Eat carbs.
54:36
John Cristobal: Carbohydrates.
54:41
John Cristobal: this Thanksgiving right, or during Thanksgiving.
54:49
John Cristobal: is right, because we're still going along with what that population and proportion
54:57
John Cristobal: was assumed to be right at that 38%. John Cristobal: Right? We cannot support that claim.
55:04
John Cristobal: that that…
55:09
John Cristobal: proportion.
55:14
John Cristobal: Right at that percentage. John Cristobal: From this mark that claim of the dietitian that.
55:23
John Cristobal: that proportion, that percentage.
55:28
John Cristobal: should be lower.
55:33
John Cristobal: Right, that's kind of what we're identifying, the more formal John Cristobal: conclusion. Summary, though, is always based on that null hypothesis. We don't have enough evidence. We don't have enough data.
55:45
John Cristobal: To reject, to go against, That assume the value. John Cristobal: for your population. John Cristobal: Okay. John Cristobal: And a lot of that work right? John Cristobal: Coming from Stat Crunch, similar to what we do with our Confidence interval. But there's that. John Cristobal: longer process. Right? Starting with, read through the problem. John Cristobal: gathering up those values. John Cristobal: Hypotheses first.
56:12
John Cristobal: test statistic.
56:17
John Cristobal: Statcrunch finds that same test statistic. So even if you want to jump into Statcrunch right away, we can do so. John Cristobal: Watch those values that you're inputting. John Cristobal: Then, once we have those results, though. John Cristobal: right? That decision, that comparison, your p-value to alpha. We can draw that distribution diagram and see that that
56:39
John Cristobal: Test this statistic. John Cristobal: is outside that critical region based on our critical value.
56:46
John Cristobal: So we have that same conclusion that we don't reject. John Cristobal: And then there's our conclusion summary. John Cristobal: kind of going through. Okay, what it means to not reject. John Cristobal: then no hypothesis. That's why we have people sleep Okay.
57:01
John Cristobal: All right.
57:03
John Cristobal: And so then, briefly, going through looking at
57:11
John Cristobal: that stat crunch here. I can put up some other examples for you to review as well.
57:18
John Cristobal: without me talking, kind of working through. But John Cristobal: looking at the information. How do we dealt with a problem involving the mean? John Cristobal: Close that down for now. We can go stat. John Cristobal: key stats. When you only have sample data, sample standardization in particular. John Cristobal: one sample with summary. John Cristobal: And again, you're gonna put your sample mean that X bar.
57:44
John Cristobal: right? Your standard deviation there. S.
57:49
John Cristobal: Okay, your sample standard deviation. S. Your sample size, that N.
57:56
John Cristobal: And then now we're doing a hypothesis test for the mean mute. So notice the symbol we're using here. That's the mean whether right? We're seeing that the average John Cristobal: The mean amount is this quantity, the average rent.
58:09
John Cristobal: the average amount of trash that we're producing, whatever it is. The alternative, maybe, is different than that quantity. We don't think it's equal to that quantity. John Cristobal: It's gotten smaller, it's less than, it's greater than. Right, I mean, whatever value that is. The amount of rent we're paying per month. Okay, your mortgage payment, your insurance costs, whatever it is. Okay. John Cristobal: So instead of doing confidence interval, now we're doing hypoth
58:36
John Cristobal: Again, I'd have it, I'll show the critical value. Significant cell, right, that is your alpha
58:43
John Cristobal: what we're gonna use. John Cristobal: and you can compute. John Cristobal: Okay.
58:49
John Cristobal: So when you put in that information, you can do so.
58:53
John Cristobal: Then, if we happen to jump into the next set of problems. John Cristobal: If we have 2 populations that we're comparing. John Cristobal: Okay, we can look in the text. Look online to see how we write, how we notate the hypotheses. There we're going through very similar set of operations. But now, whether it be proportions. John Cristobal: the mean. John Cristobal: or even standard deviation or variance. If we have 2 populations, 2 samples that we're comparing.
59:23
John Cristobal: Right? We go to 2 samples instead. John Cristobal: My same thing with proportion to sample. John Cristobal: and we'll go for 2 samples with summary. I'll do like the mean work. John Cristobal: set up here. And now again, from our 2 samples, you'll see. John Cristobal: sample one. This 1st group is sample, mean sample, standard deviation, sample size. John Cristobal: Then for the second group you're comparing right.
59:47
John Cristobal: sample mean for group 2 derivation group 2 sample size. It could be the same. John Cristobal: Or we have different samples where we have 50 of one group. John Cristobal: Maybe we're able to get like 65 the next group to compare. John Cristobal: We don't make sure to keep that open. We don't pull our variances. That's a special case when we start to assume
1:00:10
John Cristobal: that the standard deviation for our populations are the same, but we never really know that. So I don't pool. But now notice with our hypothesis. John Cristobal: Tess. John Cristobal: We're looking at that difference. We start off with that. That difference is 0. John Cristobal: that there's no difference that both those means are the same for our 2 populations. John Cristobal: And then from there we can start to identify. Maybe that
1:00:31
John Cristobal: They're not equal to each other, not equal to, or if we think that the mean for population one is bigger, that difference thing, however, is attracting a big number minus a smaller number, then that number's gonna then that difference gonna be greater than 0. John Cristobal: right? Mu one's gonna be greater than mu 2. John Cristobal: or if we think You want it smaller.
1:00:53
John Cristobal: then mu 2 population. One mean is smaller than population. Two's mean. Then we have that less than 0. John Cristobal: Okay, you can also do a confidence interval calculation similar to what we did before for one population. John Cristobal: And then similar business there. Okay, so I'll leave that with the discussion, because those actions are very similar.
1:01:12
John Cristobal: on how we deal with true population. Okay. John Cristobal: Alright. So with that said, there's our work. John Cristobal: for using stack crunch.
1:01:23
John Cristobal: dealing with this 5 steps for our hypothesis test. Let me go back to our camera here.
1:01:43
John Cristobal: Alright. So here we go.
1:01:46
John Cristobal: looking at those values, finishing up that process with hypothesis testing. Hopefully, that was clear. John Cristobal: Sorry for the little bobble at the end. If you have any questions, always reach out, let me know. All right. Take care, y'all.
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