00:01
Once again, welcome to new problem.
00:05
This time we're dealing with probabilities.
00:08
And if you think about the normal distribution, in a normal distribution of data, we can have the middle section.
00:18
The middle section has the mean, and mu is the mean of the population.
00:27
So we have the mean of the population, being in the middle.
00:32
And then we also have the what you call the empirical rule.
00:39
So the empirical rule pretty much says that one standard deviation from the mean, the z score is one, and then two standard deviations from the mean, the z score is two, and then three standard deviations from the mean, the z score is three.
00:56
The same thing happens on the left side.
00:59
Of the distribution where the z score is negative 1, we also have a z score of negative 2 and a z score of negative 3 below the mean.
01:10
A typical z score is x minus mu over sigma.
01:15
We could also have if we're dealing with samples, we could also have a t score, which means that in this case we have we have x by minus mu sub x that's a sampling distribution all over s of a radical n this is the standard error of the distribution that you're looking for so that's the standard error within one standard deviation of the mean this is 34 % of the data and this one too is 34 % of the data and then within two standard deviations of the mean we have what we call so 68 % of the data we have 95 % of the data and then within three standard deviations of the mean we have 99 .7 % of the data so we have a new problem and in this particular problem we're dealing with annual income and the mean annual income is the same as $45 ,000.
02:35
That's the mean annual income is $45 ,000 and the standard deviation for the income is also $5 ,000.
02:46
And this happens to be a city, so this is a city.
02:52
And of course the income does follow a normal distribution.
02:57
The income does follow a normal distribution.
03:00
So we have a mean of $45 ,000 and a standard deviation of $5 ,000.
03:08
And then the first question is determine the probability, determine the probability that that a randomly selected person has an income x is less than $43 ,000.
03:45
So we have an income x is less than $43 ,000.
03:49
And then in the second part, we want to determine the so have a have a mean income, a mean income of x is less than $43 ,000.
04:04
So what we're seeing is that these are income distribution.
04:12
So this are mean sampling distribution.
04:18
This is sampling distribution of sample mean.
04:23
And the means represent the income.
04:28
And in the second case is that determine the probability that the probability that the main income is between $47 ,000 and $43 ,000.
04:58
So $47 ,000 and $43 ,000.
05:03
And then in part c, we have a poster with a sample of 43 people interviewed.
05:16
So the poster has a sample of 43 people interview.
05:23
Determine if the sample, sample, sample may income x bar is less than is less than $44 ,700.
05:43
So this one the mean income was between $43 ,000 and $47 ,000.
05:50
So this is the problem we're dealing with.
05:53
We're just going to jump right into it.
05:56
So in part a, in part a of the problem, we're going to look at the distribution.
06:03
In this sense, we have a mean that happens to be $45 ,000.
06:11
And then we also have a standard deviation that happens to be $5 ,000 for the distribution.
06:18
And if you go back, you see that we're looking for x bar is less than $43 ,000.
06:24
So our x bar is $43 ,000 right here.
06:29
So the probability that we're looking for is this section.
06:34
So we're going to say p of x bar is less than $43 ,000.
06:41
And so this becomes the same as if we're looking for less than $43 ,000, we're going to say that the probability that x bar minus $45 ,000.
07:08
Over $5 ,000...