WEBVTT
1
00:00:01.340 --> 00:00:04.089 A:middle L:90%
right. This problem is about the number of square
2
00:00:04.089 --> 00:00:07.719 A:middle L:90%
feet of office space available in small cities. We
3
00:00:07.719 --> 00:00:12.250 A:middle L:90%
know it's normally distributed with a mean, uh,
4
00:00:12.740 --> 00:00:21.350 A:middle L:90%
7,750,000 square feet and a stand. Standard deviation have
5
00:00:21.870 --> 00:00:28.920 A:middle L:90%
60,000 square feet. Um, we have a second
6
00:00:28.920 --> 00:00:39.350 A:middle L:90%
small city that has a mean of 800,000 square feet
7
00:00:40.939 --> 00:00:46.579 A:middle L:90%
and a standard deviation, also of 60,000 square feet
8
00:00:49.329 --> 00:00:54.259 A:middle L:90%
. It does say that the office space square footage
9
00:00:54.270 --> 00:00:59.170 A:middle L:90%
is normally distributed. So therefore, when we complete
10
00:00:59.170 --> 00:01:04.890 A:middle L:90%
part A, which is to draw a representation,
11
00:01:04.900 --> 00:01:08.799 A:middle L:90%
a sketch of this distribution on the same graph.
12
00:01:08.810 --> 00:01:15.579 A:middle L:90%
So we've got City one or I'll call it City
13
00:01:15.590 --> 00:01:19.989 A:middle L:90%
A. I'll put in blue, and for City
14
00:01:19.000 --> 00:01:26.310 A:middle L:90%
be we're gonna put it in pink. So we're
15
00:01:26.310 --> 00:01:29.849 A:middle L:90%
going to start by drawing the bell shaped curve.
16
00:01:34.340 --> 00:01:38.709 A:middle L:90%
And keep in mind in the center is where the
17
00:01:38.239 --> 00:01:42.519 A:middle L:90%
, um, average is going to go for the
18
00:01:42.519 --> 00:01:45.230 A:middle L:90%
blue. And so let's start with the blue curve
19
00:01:45.939 --> 00:01:49.560 A:middle L:90%
. So I'm gonna actually I'm gonna actually get rid
20
00:01:49.560 --> 00:01:52.390 A:middle L:90%
of the white. I'm only going to do blue
21
00:01:56.439 --> 00:02:00.040 A:middle L:90%
. Here's the blue curve and thes center of it
22
00:02:00.049 --> 00:02:08.520 A:middle L:90%
is at 750,000 and we count by standard deviations.
23
00:02:08.800 --> 00:02:14.060 A:middle L:90%
So therefore, 6000 later, we're going to be
24
00:02:14.060 --> 00:02:16.159 A:middle L:90%
at what we call our point of inflection. And
25
00:02:16.159 --> 00:02:19.719 A:middle L:90%
our point of inflection is kind of where the curve
26
00:02:19.719 --> 00:02:23.310 A:middle L:90%
starts turning from a scooping down to a scooping up
27
00:02:23.560 --> 00:02:27.960 A:middle L:90%
. So right around here is going to be one
28
00:02:27.960 --> 00:02:31.849 A:middle L:90%
standard deviation out. So we're adding 60,000 and will
29
00:02:31.849 --> 00:02:39.620 A:middle L:90%
be at 810,000 and then we'll be about the same
30
00:02:39.620 --> 00:02:49.389 A:middle L:90%
distance out will be at 870,000 and the same distance
31
00:02:49.400 --> 00:02:57.300 A:middle L:90%
out. We're going to be at 930,000 and then
32
00:02:57.300 --> 00:02:59.500 A:middle L:90%
we're gonna go back the other way. If we
33
00:02:59.500 --> 00:03:07.000 A:middle L:90%
subtract 60,000 from the average where it 690,000 and this
34
00:03:07.000 --> 00:03:15.620 A:middle L:90%
attracted again, we are at 600 30,000 and we
35
00:03:15.620 --> 00:03:19.270 A:middle L:90%
subtract again, keeping our spacing uniform. We would
36
00:03:19.270 --> 00:03:25.289 A:middle L:90%
be at 570,000. So there is our blue curve
37
00:03:25.530 --> 00:03:29.550 A:middle L:90%
. That's the curve that is representing city A.
38
00:03:30.889 --> 00:03:31.990 A:middle L:90%
Now, when it comes time to do city be
39
00:03:32.210 --> 00:03:36.199 A:middle L:90%
now, the peak has to be a 800,000.
40
00:03:36.569 --> 00:03:39.080 A:middle L:90%
So it looks like 800,000 would be like somewhere right
41
00:03:39.129 --> 00:03:47.750 A:middle L:90%
in here. So our peak has to be right
42
00:03:47.759 --> 00:03:51.680 A:middle L:90%
there, and then everything else is going to be
43
00:03:51.680 --> 00:03:55.180 A:middle L:90%
uniform. So if you think about the distance from
44
00:03:55.189 --> 00:04:02.750 A:middle L:90%
our peek out, we went over approximately that for
45
00:04:04.740 --> 00:04:11.680 A:middle L:90%
and that far. So there's our curve scooping in
46
00:04:11.680 --> 00:04:15.000 A:middle L:90%
that direction. And then the scoop down is just
47
00:04:15.000 --> 00:04:19.019 A:middle L:90%
gonna parallel the other car because they had the same
48
00:04:19.019 --> 00:04:25.939 A:middle L:90%
exact, um, standard deviation. The curves are
49
00:04:25.939 --> 00:04:29.129 A:middle L:90%
going to look identical in shape and size. They're
50
00:04:29.129 --> 00:04:31.220 A:middle L:90%
just going to be shifted a little bit further.
51
00:04:31.339 --> 00:04:43.060 A:middle L:90%
One way or the other part. B Part B
52
00:04:43.060 --> 00:04:49.459 A:middle L:90%
is asking us to determine the probability that the number
53
00:04:49.459 --> 00:04:54.259 A:middle L:90%
of square feet available in the first city is less
54
00:04:54.259 --> 00:04:58.370 A:middle L:90%
than 800,000, so X is less than 800,000.
55
00:05:00.439 --> 00:05:02.449 A:middle L:90%
But keep in mind, we're using the first city
56
00:05:02.689 --> 00:05:08.100 A:middle L:90%
. So we're using the blue numbers. And the
57
00:05:08.100 --> 00:05:10.800 A:middle L:90%
first city, which we called City A, had
58
00:05:10.800 --> 00:05:17.120 A:middle L:90%
an average of 750,000, and in this instance,
59
00:05:17.120 --> 00:05:23.509 A:middle L:90%
we're trying to go where the probability is less than
60
00:05:23.519 --> 00:05:30.939 A:middle L:90%
800,000. So we will need a Z score and
61
00:05:30.939 --> 00:05:34.730 A:middle L:90%
to brush up on your formula, Z is equal
62
00:05:34.730 --> 00:05:40.430 A:middle L:90%
to X minus mu over stigma. So for the
63
00:05:40.430 --> 00:05:44.740 A:middle L:90%
city number one or city A. It's going to
64
00:05:44.740 --> 00:05:57.350 A:middle L:90%
be 800,000, minus 750,000 divided by the standard deviation
65
00:05:57.350 --> 00:06:03.220 A:middle L:90%
, which was 60,000. So our Z score for
66
00:06:03.220 --> 00:06:10.959 A:middle L:90%
the first city, at 800,000 would be 0.83 So
67
00:06:10.970 --> 00:06:14.529 A:middle L:90%
we know that 0.83 is appear in our bell curve
68
00:06:14.870 --> 00:06:18.279 A:middle L:90%
. So when we're talking about the office space being
69
00:06:18.279 --> 00:06:23.889 A:middle L:90%
less than 800,000, it's no different than if we
70
00:06:23.889 --> 00:06:27.329 A:middle L:90%
were to solve the problem where Z is less than
71
00:06:27.329 --> 00:06:31.490 A:middle L:90%
0.83 and Z being less than pointing three. You
72
00:06:31.490 --> 00:06:35.509 A:middle L:90%
would then look into your standard normal table in the
73
00:06:35.509 --> 00:06:38.750 A:middle L:90%
back of the book, and you're going to find
74
00:06:38.750 --> 00:06:44.990 A:middle L:90%
a value of 30.7967 as the probability that in the
75
00:06:44.990 --> 00:06:48.720 A:middle L:90%
first city we have office space less than 800,000 square
76
00:06:48.720 --> 00:06:57.790 A:middle L:90%
feet. Let's look at Part C and in part
77
00:06:57.800 --> 00:07:00.860 A:middle L:90%
C. We're using information from the second city,
78
00:07:00.870 --> 00:07:08.839 A:middle L:90%
and we want to determine the probability that the square
79
00:07:08.839 --> 00:07:15.410 A:middle L:90%
feet available in the second city is greater than 750,000
80
00:07:17.680 --> 00:07:21.639 A:middle L:90%
. So in this case, are Bell Curve is
81
00:07:21.639 --> 00:07:26.110 A:middle L:90%
going to have the peak. Since we're using the
82
00:07:26.120 --> 00:07:29.319 A:middle L:90%
data from the second city, the peak is going
83
00:07:29.319 --> 00:07:34.220 A:middle L:90%
to be a 800,000 and we're trying to find the
84
00:07:34.220 --> 00:07:43.069 A:middle L:90%
probability of being greater than 700 50,000. So we
85
00:07:43.069 --> 00:07:49.990 A:middle L:90%
will need our Z score. Z equals 750,000 minus
86
00:07:50.100 --> 00:07:58.879 A:middle L:90%
800,000 all over our standard deviation, which was 60,000
87
00:07:58.889 --> 00:08:01.660 A:middle L:90%
no matter which city you were in. So the
88
00:08:01.660 --> 00:08:13.810 A:middle L:90%
Z score turns out to be a negative 0.83 So
89
00:08:13.810 --> 00:08:16.930 A:middle L:90%
when we're talking about the square footage available in the
90
00:08:16.930 --> 00:08:22.269 A:middle L:90%
second city to be greater than 750,000, it's no
91
00:08:22.269 --> 00:08:26.220 A:middle L:90%
different than saying that Z would be greater than negative
92
00:08:26.220 --> 00:08:31.529 A:middle L:90%
0.83 Because this is taking us to the right.
93
00:08:31.539 --> 00:08:35.250 A:middle L:90%
We're going to have to use one, minus the
94
00:08:35.250 --> 00:08:41.330 A:middle L:90%
probability of Z being less than negative 10.83 And when
95
00:08:41.330 --> 00:08:43.870 A:middle L:90%
you look in your standard normal table, you will
96
00:08:43.870 --> 00:08:48.830 A:middle L:90%
arrive at an area to the left of negative 0.83
97
00:08:48.830 --> 00:08:56.080 A:middle L:90%
to be points to 033 resulting in an area to
98
00:08:56.080 --> 00:09:05.350 A:middle L:90%
the right or probability of 0.7967 So if you've noticed
99
00:09:05.490 --> 00:09:09.730 A:middle L:90%
Part B and Part C had the same result because
100
00:09:09.730 --> 00:09:13.320 A:middle L:90%
they had the same spread and the value that we
101
00:09:13.320 --> 00:09:18.429 A:middle L:90%
were, um, talking about the actual X minimum
102
00:09:18.429 --> 00:09:22.740 A:middle L:90%
or maximum was the same distance away from our average
103
00:09:22.929 --> 00:09:26.009 A:middle L:90%
. So part B. The probability that your square
104
00:09:26.009 --> 00:09:28.210 A:middle L:90%
footage was less than 800,000 in the first city was
105
00:09:28.210 --> 00:09:33.139 A:middle L:90%
0.7967 And for part C, the probability that the
106
00:09:33.139 --> 00:09:37.970 A:middle L:90%
square footage available in the second city was greater than
107
00:09:37.980 --> 00:09:41.250 A:middle L:90%
750,000 was also 0.7967