00:01
So this problem tells us that the distribution of weights for widgets is bell -shaped.
00:06
It says that their weights have a mean of 53 ounces and a standard deviation of 4 ounces.
00:10
So let's go ahead and jot those two things down.
00:13
So again, the mean is 53 and the standard deviation is 4.
00:19
It says use the standard deviation rule, aka the empirical rule, and answer the three questions.
00:25
So we'll start with part a.
00:27
Part a says that 99 .7 % of the widgets lie between what 2 .4? points.
00:34
So let's first begin by thinking, okay, so 99 .7 % encompasses the middle of the data here, and we want the point that is the lower limit for this data and the upper limit for this data.
00:49
They say to use the empirical rule, i have the empirical rule on the screen here in this chart.
00:56
The empirical rule says that between negative or within one standard deviation of the mean lays 60.
01:03
8 % of data within two standard deviations of the mean lay 95 % of data, and within three standard deviations of the mean lay 99 .7 % of data.
01:13
So we know that our two standard deviations, or our two z scores for 99 .7, are going to be negative 3 and positive 3, since that encompasses the middle 99 .7%.
01:27
So we'll jot down negative 3 and positive 3.
01:31
But now that we have our two z scores, we need to use them to actually find the actual weight that we need.
01:39
So we can use the z equation that i have on the screen, which says z is equal to x minus the mean divided by the standard deviation.
01:47
So we have the z score, we have the mean, we have the standard deviation, and we just need to solve for x.
01:53
Let's go ahead and plug in that equation starting with negative 3.
01:56
So using the equation, z, which is negative 3, is equal to x, which we're supposed.
02:01
Solving four, minus the mean, which is 53, divided by the standard deviation, which is four.
02:08
So now to solve for x, we can put our z score over one, and then, it's a sloppy one, let me fix that, over one, and then cross multiply these two fractions.
02:18
That gives us x minus 53 is equal to negative 12, and then finally we get x is equal to 41.
02:27
Now let's do the same for positive three.
02:29
Z is equal to x, minus, the mean divided by the standard deviation.
02:35
We'll solve for x by cross multiplying, giving us x minus 53 is equal to positive 12, which comes out to 65.
02:45
So 99 % of the weights lie between 41 ounces and 65 ounces.
02:54
Now for part b, part b asks us to find the percentage of weights that lie between 49 and 65 ounces.
03:02
So what we need to do here is find the probability of x being greater than 49, but less than 65.
03:11
So we'll have to find a z score for both 49 and 65 again using the red equation, except this time we know x, x is 49 and 65, and we're instead solving for z.
03:23
So let's begin with 49.
03:25
Z is equal to x, which is 49, minus the mean, which is 53, divided by our standard deviation, which is four...