00:01
All right, so the u .s.
00:01
Department of transportation reported that 77 % of all fatally injured automobile drivers were intoxicated, and they took a random sample of 27 automobile driver fatalities in kitt carson county, colorado, and they found that 15 involved an intoxicated driver.
00:24
Do these data indicate that the population proportion of driver fatalities related to alcohol is less than 77 % in kent county? using alpha 0 .01.
00:34
So the null hypothesis, let's go ahead and work through this.
00:37
So the null is that p is equal to 0 .77.
00:46
And the alternative is that to these, let's see, let's read this question again, do these data indicate population proportion is less than 70 %? that's going to be our alternative, that p is less than 0 .77.
01:05
You might see this.
01:06
This, the null written as p is greater than or equal to 0 .77, but the main piece here is this, what we're testing for.
01:15
Is the proportion significantly less than 0 .77? is this a left -tailed, right -tailed or two -tailed test? well, it's definitely not two -tailed.
01:27
And this ends up being a left -tailed test because here's our distribution.
01:39
It doesn't have to be normal, but it's, if it were the shape.
01:43
We are looking for something on this end.
01:45
And here would be our mean of 0 .77, or the proportion of 0 .77.
01:55
And we want to find something significantly less than that value, hence this tail here, which has an area of point.
02:03
I'm over -emphasizing the size of this.
02:06
So please don't think this is the scale.
02:08
It's not just to emphasize what we're looking at.
02:12
What's the level of confidence? well, it's alpha of 0 .01.
02:18
So this is the big question.
02:20
What distribution are we going to use? i said, i had this normal one.
02:26
But what we're going to be using is not normal.
02:31
We might be tempted to use the z test for proportions.
02:37
However, it fails under the assumptions for us to use a z test for proportions.
02:44
We need to have the mean, which is n times p it needs to be greater than 10, which it is.
02:53
But we also have to have the variance, which is n times 1 minus p.
02:58
We have to have that be greater than equal to 10, which it is not.
03:01
It's 6 .21, so it fails that assumption.
03:06
And so we can't use that.
03:09
And so that means we're going to use the binomial distribution.
03:17
We use an exact binomial test.
03:18
So we're going to make sure it fits our assumptions.
03:27
The number of observations end is fixed.
03:29
Yes, it is.
03:30
It's 27.
03:32
Each observation is independent.
03:34
We are going to make that assumption.
03:36
It's a safe assumption that all 27 have no relation to another.
03:43
Each observation represents one of two outcomes.
03:46
You can either success or fail or succeed or fail.
03:53
It says in quotes, it just means the proportion.
03:55
Of that thing happening, you can either have it or not.
03:58
So in this case, we're looking for a fatality or not a fatal.
04:03
So we have two choices, one of two outcomes.
04:07
And then lastly, the probability of success p is the same for each outcome.
04:12
That's part of what we're testing, is that if we were to assume that this p is correct.
04:19
0 .77, that's going to be, yes, that's what we're going to testing for.
04:29
Let's say that's true.
04:33
So yes, it is the same free show come.
04:35
So the sample test statistic, well, it's going to be a p value and it's calculated.
04:47
And what formula are we going to use to find it? our binomial distribution, which is the following.
04:56
N choose x times p to the n times one minus p to the n minus x...