00:01
The monthly income of residence of a certain city is normally distributed.
00:06
So that means for each part of this problem, we could draw that bell -shaped curve.
00:11
The mean is $2 ,500 as a monthly income, and the standard deviation is $500.
00:23
So let's begin part a.
00:26
In part a, we now know that the mayor of daisy city earns 1 ,550 a month.
00:35
So let's draw that bell -shaped curve.
00:38
And the mean is at the center.
00:42
So the mayor's salary will be over here at 1 ,750.
00:47
And we want to know what percentage of daisy city's residents have incomes that are more than the mayors.
00:54
So we are interested in this area right here.
00:58
So what we're going to do is we're going to find the probability that x is greater than 1 ,750.
01:06
And in order to do that, we're going to need our z score that is associated with 1 ,750.
01:14
And we find our z score by using the formula x minus mu over sigma.
01:19
So that means we'll take 1 ,750, we'll minus 2 ,500, and we'll divide that difference by the standard deviation of 500.
01:30
I'm just going to move this down.
01:35
And in doing so, we will get a z score of negative 1 .50.
01:42
So the probability that x is greater than 1 ,750 is comparable to the probability that your z score is greater than negative 1 .50.
01:53
So at this point, we'll rely on our standard normal distribution that is found in the back of your book.
02:01
And down the left side of that chart, you're going to look up the units place and the tenths place.
02:07
And across the top, you're going to find the hundreds place.
02:10
And where those two line up, we're going to get an area.
02:14
And that area represents the area from that particular z score into the left tail.
02:21
So the area in the left is .0068.
02:27
But our area of interest is at the right.
02:31
So therefore, we'll take one minus the area to the left, which will provide us with the area to the right.
02:40
And we get 0 .9 -332.
02:43
But the question didn't say, what is the probability? it asked what percentage.
02:49
So we now have to convert that into a percentage, and that means we're going to move the decimal point two places to the right.
02:57
So we could say 93 .32 % of the residents of daisy city earn more than the mayor.
03:23
Let's tackle part b.
03:26
In part b, once again, we're going to draw our bell -shaped curve, and at the center of the curve will be that average of 2 ,500.
03:36
And in this case, we want to know what percentage of residents is exempt from city taxes.
03:44
And any resident that earns less than 1 ,465, so 1 ,465 will be to the left of center, and less than that would be referencing this area.
03:58
And those individuals will end up being exempt from city tax.
04:03
So once again, we're going to find our z score by using the formula x minus mu over sigma.
04:10
So we'll do 1 ,465 minus 2 ,500, and we'll divide by 500.
04:17
And this time we get a z score of negative 2 .70.
04:22
So we'll determine the probability that x is less than 1 ,465 by, by comparing it to the probability that z is less than negative 2 .70.
04:36
So once again, we're going to rely on our standard normal distribution table.
04:43
We'll look for the units place and the tenths place down the left side.
04:48
We'll look underneath the .00 for the hundreds place, and we are going to get an area of .0035.
05:00
Once again, what comes out of that chart describes the area to the left of that z score.
05:06
So to the left of this z score, the area under our standard normal distribution curve is going to be 0 .035.
05:16
And that happens to be the area of interest.
05:19
So we could say that this is equal to 0 .0035.
05:24
So now we have to convert that to a percent because once again the directions.
05:29
Or the question is asking what percentage of residents.
05:33
So when we move our decimal point two places to the right, we could say 0 .35 % of the residents are exempt from city tax.
05:59
Let's look at part c.
06:01
In part c, it's slightly different.
06:03
Once again, we are going to start with that bell -shaped curve, and 2 ,500 is going to.
06:11
To be in the center.
06:13
And keep in mind that that corresponds to a z score of zero.
06:18
And this time, our area of interest is going to be the middle 95%.
06:23
So that means we want the middle 95%.
06:29
So as a decimal, 95 % would be written as 0 .95.
06:35
And because of the symmetric nature of this bell, then the remaining two parts have to add up to 5 % or 0 .05, so the left tail would be 0 .025, and the right tail would have an area of 0 .025...