00:01
That the mean amount purchased by a typical customer at a particular grocery store is $29 .50.
00:05
These are units or dollars.
00:07
And the standard deviation sigma is $6.
00:10
And we're going to assume the distribution of amounts purchased follows the normal distribution.
00:14
And we're going to sample 76 randomly selected customers.
00:18
And given that, we're going to find some probabilities.
00:21
So the first question, we're going to find the likelihood that the sample mean is at least $31 .51.
00:27
So the probability that the mean of those 76 is greater than $31 .50.
00:33
And it says at least, so you could put this, x bar is greater than or equal to, but as our distribution is continuous, the difference between greater than or equal to and just greater than is zero.
00:50
Because a particular value has a probability of zero.
00:53
So it doesn't really matter.
00:55
So the way we're going to do this, so here's our distribution.
00:57
It's 29.
00:58
So here's our mean.
00:59
By the way, that's the mean of the sampling distribution about the sampling as well.
01:02
This is the population mean.
01:04
That's the same as the mean about the sampling distribution.
01:07
And 31 .5 is going to be roughly here.
01:10
There's 31 and a half.
01:13
We want to find 31 .5 or more, $31 .50 or more.
01:19
So what we're going to do is do one minus the probability that the mean is less than 31 .5, $31 .50.
01:27
And what we're going to do is we're going to transform this $31 .50 into a z distribution with the following, into our standard normal random variable with the following transform.
01:39
X bar minus mean sub x bar over sigma over root n.
01:45
So i'm going to leave 31 .50 blank, but it's just, you put that in here in x bar.
01:49
Actually, i'll substitute it in just so you can see how the formula works.
01:55
Minus the mean, $29 .50 all over six over root 76.
02:12
And what we get is this z score, 2 .91 we'll say, around two decimal places.
02:19
And then if you, i use a spreadsheet to find the probability associated with being less than 31 .5, but you could use the back of your textbook as well.
02:29
You get the same value.
02:31
So 0 .99819, that's this area that's unshaded.
02:34
We want a compliment to that, so we do one minus that value.
02:38
Because this norm s dist function in your spreadsheet gives you the left area, but we don't want that, we want the right.
02:44
So one minus that value is 0 .0018 if you go to four decimal places...