WEBVTT
1
00:00:01.090 --> 00:00:03.580 A:middle L:90%
right and problem in the 48 there are six parts
2
00:00:03.580 --> 00:00:06.610 A:middle L:90%
and all six parts do with the fact that the
3
00:00:06.620 --> 00:00:12.380 A:middle L:90%
average is 28 and the standard deviation will be Southern
4
00:00:13.009 --> 00:00:15.439 A:middle L:90%
. So in part A, you're asked to determine
5
00:00:15.439 --> 00:00:18.570 A:middle L:90%
what the what is the probability that X is less
6
00:00:18.570 --> 00:00:25.690 A:middle L:90%
than 28. We also knew that the variable was
7
00:00:25.699 --> 00:00:29.480 A:middle L:90%
normally distributed 28. Since it was the average would
8
00:00:29.480 --> 00:00:31.899 A:middle L:90%
be right in the center of the bell and the
9
00:00:31.899 --> 00:00:36.649 A:middle L:90%
probability that we have picking X value less than 28
10
00:00:37.039 --> 00:00:39.850 A:middle L:90%
which would represent half the bell, which would be
11
00:00:39.850 --> 00:00:44.799 A:middle L:90%
0.5 for Part B. We're looking to do the
12
00:00:44.799 --> 00:00:51.780 A:middle L:90%
probability that X takes on a value between 28 38
13
00:00:52.740 --> 00:00:55.899 A:middle L:90%
. So again, I would highly recommend drawing that
14
00:00:55.909 --> 00:00:59.780 A:middle L:90%
bell shaped curve again. 28 is in the center
15
00:01:00.090 --> 00:01:03.640 A:middle L:90%
and 38 would be over here. So we're going
16
00:01:03.640 --> 00:01:07.540 A:middle L:90%
to utilize Z scores in order to solve this problem
17
00:01:07.599 --> 00:01:11.370 A:middle L:90%
. And the formula for Z scores is that Z
18
00:01:11.370 --> 00:01:18.370 A:middle L:90%
equals X minus mu divided by sigma. So the
19
00:01:18.370 --> 00:01:26.189 A:middle L:90%
Z score associated with 28 would be 28 minus 28
20
00:01:26.040 --> 00:01:30.569 A:middle L:90%
all over seven or zero, which we should have
21
00:01:30.569 --> 00:01:34.099 A:middle L:90%
known that just because the center of the bell does
22
00:01:34.099 --> 00:01:38.219 A:middle L:90%
represent a Z score of zero and the Z score
23
00:01:38.219 --> 00:01:42.040 A:middle L:90%
associated with 38 would be found by doing 38 minus
24
00:01:42.040 --> 00:01:47.060 A:middle L:90%
28 divided by seven, and you do get an
25
00:01:47.060 --> 00:01:56.500 A:middle L:90%
answer of 1.43 So when we posed the question,
26
00:01:56.510 --> 00:02:00.780 A:middle L:90%
what's the probability of being between 28 38? It's
27
00:02:00.780 --> 00:02:04.640 A:middle L:90%
no different than the question that says, What's the
28
00:02:04.640 --> 00:02:12.680 A:middle L:90%
probability that Z is between zero and 1.43? You
29
00:02:12.680 --> 00:02:15.849 A:middle L:90%
can then calculate the probability that Z is less than
30
00:02:15.849 --> 00:02:23.110 A:middle L:90%
1.43 and from that subtract the probability that Z is
31
00:02:23.110 --> 00:02:25.370 A:middle L:90%
less than zero. You would then go to your
32
00:02:25.370 --> 00:02:29.050 A:middle L:90%
standard normal table in the back of your textbook.
33
00:02:29.539 --> 00:02:35.580 A:middle L:90%
You would get a value of 0.9236 for the probability
34
00:02:35.580 --> 00:02:38.080 A:middle L:90%
that Z is to the left of negative one of
35
00:02:38.090 --> 00:02:43.439 A:middle L:90%
positive 1.43 and the probability that Z is less than
36
00:02:43.439 --> 00:02:49.879 A:middle L:90%
zero would be 00.5000 for an answer for Part B
37
00:02:50.300 --> 00:02:59.620 A:middle L:90%
two B 20.4 to 36 in part C, making
38
00:02:59.620 --> 00:03:02.050 A:middle L:90%
a little bit more difficult as we go along in
39
00:03:02.050 --> 00:03:07.300 A:middle L:90%
part C. We're asking, What's the probability that
40
00:03:07.310 --> 00:03:16.669 A:middle L:90%
ex falls between 24 40? So again, I
41
00:03:16.669 --> 00:03:22.319 A:middle L:90%
highly recommend drawing that bell shaped curve, putting 28
42
00:03:22.330 --> 00:03:25.129 A:middle L:90%
in the center because that is our population mean and
43
00:03:25.129 --> 00:03:30.500 A:middle L:90%
we want to be between 24 and 40. So
44
00:03:30.500 --> 00:03:35.150 A:middle L:90%
we're going to find the Z score associated with each
45
00:03:35.840 --> 00:03:39.099 A:middle L:90%
and the Z score for 24. We'll do Z
46
00:03:39.099 --> 00:03:46.229 A:middle L:90%
equals 24 minus 28. Divide by the standard deviation
47
00:03:46.240 --> 00:03:54.849 A:middle L:90%
of seven, and we get negative 0.57 and then
48
00:03:54.849 --> 00:03:59.750 A:middle L:90%
the Z score associated with 40 would be 40 minus
49
00:04:00.240 --> 00:04:05.000 A:middle L:90%
28. Divide by seven, which yields a value
50
00:04:05.000 --> 00:04:13.740 A:middle L:90%
of 1.71 So when we say, what's the probability
51
00:04:13.930 --> 00:04:17.019 A:middle L:90%
that X is between 24 40? It's no different
52
00:04:17.019 --> 00:04:21.930 A:middle L:90%
than saying What's the probability that Z is between negative
53
00:04:21.939 --> 00:04:30.470 A:middle L:90%
400.57 and positive 1.71? We would then find the
54
00:04:30.470 --> 00:04:38.199 A:middle L:90%
probability that Z is less than 1.71 and subtract from
55
00:04:38.199 --> 00:04:41.800 A:middle L:90%
that the probability that Z is less than negative.
56
00:04:41.810 --> 00:04:46.029 A:middle L:90%
0.57 If you used the table in the back of
57
00:04:46.029 --> 00:04:48.269 A:middle L:90%
your book, the standard normal table you will find
58
00:04:48.269 --> 00:04:51.860 A:middle L:90%
the probability that Z is less than 1.71 to be
59
00:04:51.860 --> 00:04:59.379 A:middle L:90%
0.9 five 64 and the probability that Z is less
60
00:04:59.379 --> 00:05:02.550 A:middle L:90%
than negative 0.57 you will find to be point to
61
00:05:03.139 --> 00:05:09.389 A:middle L:90%
eight four three. Thus, the answer to Part
62
00:05:09.389 --> 00:05:16.290 A:middle L:90%
C would be 0.67 to one in Part D.
63
00:05:20.199 --> 00:05:27.029 A:middle L:90%
We're looking to calculate the probability The X Files between
64
00:05:27.040 --> 00:05:38.589 A:middle L:90%
30 and 45. So here's our picture. We
65
00:05:38.589 --> 00:05:46.730 A:middle L:90%
have 28 here, so 30 and 45. So
66
00:05:46.740 --> 00:05:49.189 A:middle L:90%
because both of those values or above 28 both of
67
00:05:49.189 --> 00:05:53.829 A:middle L:90%
those thescore should be positive values. So we'll find
68
00:05:53.829 --> 00:05:59.029 A:middle L:90%
the Z score associated with 30 by doing 30 minus
69
00:05:59.040 --> 00:06:05.100 A:middle L:90%
28. Divide by seven. And when we do
70
00:06:05.100 --> 00:06:10.360 A:middle L:90%
that, we will get a value of approximately 0.29
71
00:06:11.240 --> 00:06:15.750 A:middle L:90%
and the Z score associated with 45 would be found
72
00:06:15.750 --> 00:06:18.259 A:middle L:90%
by doing 45. Divide by 20 of minus 28
73
00:06:18.259 --> 00:06:26.220 A:middle L:90%
divide by seven and you get approximately a 2.43 We
74
00:06:26.220 --> 00:06:29.649 A:middle L:90%
can put both those on the bell at the 30
75
00:06:29.649 --> 00:06:32.790 A:middle L:90%
mark. We're gonna have the 300.29 and at the
76
00:06:32.790 --> 00:06:38.629 A:middle L:90%
45 mark, the 2.43 so when the problem is
77
00:06:38.629 --> 00:06:42.319 A:middle L:90%
asking you was the probability that the X values between
78
00:06:42.329 --> 00:06:46.120 A:middle L:90%
30 and 45. We can also say the problem
79
00:06:46.120 --> 00:06:51.149 A:middle L:90%
would be What's the probability that Z is between 0.29
80
00:06:51.540 --> 00:06:57.139 A:middle L:90%
and 2.43? We're going to rewrite this then as
81
00:06:57.139 --> 00:07:02.639 A:middle L:90%
the probability that Z is less than 2.43 minus the
82
00:07:02.639 --> 00:07:09.660 A:middle L:90%
probability that Z is less than 0.29 and using what's
83
00:07:09.670 --> 00:07:13.160 A:middle L:90%
in the back of your book. The probability that
84
00:07:13.160 --> 00:07:18.550 A:middle L:90%
Z is less than 2.43 would be 0.99 to 5
85
00:07:19.040 --> 00:07:25.089 A:middle L:90%
. From that, you're going to subtract 0.6141 for
86
00:07:25.089 --> 00:07:32.449 A:middle L:90%
an answer to Part D of 0.3784 in party.
87
00:07:33.540 --> 00:07:44.610 A:middle L:90%
Very similar style problem. Part E is asking us
88
00:07:44.899 --> 00:07:49.269 A:middle L:90%
what's the probability that your ex value is between 19
89
00:07:49.269 --> 00:07:57.970 A:middle L:90%
and 35. So again, we're gonna have our
90
00:07:57.970 --> 00:08:01.149 A:middle L:90%
bell shaped curve. We've got 28 in the center
91
00:08:01.579 --> 00:08:05.430 A:middle L:90%
, so 19 would be to its left and 35
92
00:08:05.430 --> 00:08:09.230 A:middle L:90%
would be to its right. We're going to find
93
00:08:09.230 --> 00:08:13.769 A:middle L:90%
the corresponding Z scores, so Z equals 19 minus
94
00:08:13.779 --> 00:08:18.269 A:middle L:90%
28/7. And when you do that, you get
95
00:08:18.269 --> 00:08:28.350 A:middle L:90%
a negative 1.29 and Z equals 35 minus 28/7,
96
00:08:28.839 --> 00:08:33.139 A:middle L:90%
which gets you a Z score of nice clean one
97
00:08:33.799 --> 00:08:35.950 A:middle L:90%
. So we're gonna place those on our bell.
98
00:08:37.340 --> 00:08:39.830 A:middle L:90%
So this is a Z score of one and this
99
00:08:39.830 --> 00:08:43.570 A:middle L:90%
is Aziz Score of negative 1.29 Someone were asking,
100
00:08:43.570 --> 00:08:48.750 A:middle L:90%
What's the probability that X is between 19 and 35
101
00:08:48.340 --> 00:08:52.870 A:middle L:90%
? It's no different than asking What's the probability that
102
00:08:52.879 --> 00:09:00.440 A:middle L:90%
Z is between negative 1.29 and one? We can
103
00:09:00.440 --> 00:09:03.009 A:middle L:90%
then solve this problem by calculating or looking up the
104
00:09:03.009 --> 00:09:07.299 A:middle L:90%
probability that Z is less than one and subtract from
105
00:09:07.299 --> 00:09:11.549 A:middle L:90%
it the probability that Z is less than negative 1.29
106
00:09:13.340 --> 00:09:16.970 A:middle L:90%
So from your standard normal table, the probability that
107
00:09:16.970 --> 00:09:22.990 A:middle L:90%
Z is less than one is 0.8413 And the probability
108
00:09:22.990 --> 00:09:31.620 A:middle L:90%
that Z is less than negative 1.29 is 0.985 for
109
00:09:31.620 --> 00:09:37.950 A:middle L:90%
an answer to Part E of 0.74 to 8.
110
00:09:46.720 --> 00:09:56.070 A:middle L:90%
And then finally, for part F, the problem
111
00:09:56.070 --> 00:10:00.850 A:middle L:90%
is asking you to determine what's the probability that X
112
00:10:01.639 --> 00:10:05.710 A:middle L:90%
is less than 48. So again, one more
113
00:10:05.710 --> 00:10:11.909 A:middle L:90%
time. Here's the image. Here's 28 so 48
114
00:10:11.919 --> 00:10:15.669 A:middle L:90%
would be to the right. We're going to calculate
115
00:10:15.679 --> 00:10:18.830 A:middle L:90%
the Z score, the easy score. Oops,
116
00:10:18.830 --> 00:10:28.570 A:middle L:90%
sorry about that. The Z score would be found
117
00:10:28.570 --> 00:10:35.049 A:middle L:90%
by doing 48 minus 28. Divide by seven,
118
00:10:35.580 --> 00:10:39.730 A:middle L:90%
and you get a Z score value of 2.86 So
119
00:10:39.730 --> 00:10:43.110 A:middle L:90%
when you're asking, what's the probability that X is
120
00:10:43.110 --> 00:10:46.899 A:middle L:90%
below 48? It's no different than asking. What's
121
00:10:46.899 --> 00:10:54.470 A:middle L:90%
the probability that Z is less than 2.86? And
122
00:10:54.470 --> 00:10:58.549 A:middle L:90%
looking up in your chart, you would get an
123
00:10:58.559 --> 00:11:01.450 A:middle L:90%
answer of 0.9979