(d) Assuming that the distribution of times is approximately normal, about what percentage of times are between 25 and 49 minutes? Less than 19 min or greater than 55 min? Less than 19 min? (Round the answers to two decimal places, if needed.) When we assume that the distribution of times is approximately normal, we will use the Empirical Rule to get a more precise sense of the distribution of data values. The Empirical Rule gives us approximate percentages that are within k = 1, 2, or 3 standard deviations from the mean. If the distribution can be reasonably well approximated by a normal curve, then: • Approximately 68% of the observations are within 1 standard deviation of the mean. • Approximately 95% of the observations are within 2 standard deviations of the mean. • Approximately 99.7% of the observations are within 3 standard deviations of the mean. appraximate percentages implied by the Empirical Rule 34% 34% 2.35% 13.5% 13.5% 2.35% -3 sd -2 sd Mean 1sd 2 sd 3 sd 1 sd 1 sd In contrast to Chebyshev's Rule, dividing the "within" percentages in half is permissible, because the normal curve is symmetric. Use the Empirical Rule to determine the percentage of times are between 25 and 49 minutes. We previously determined that 25 minutes and 49 minutes are 2 standard deviations of the mean. The Empirical Rule indicates that approximately 7% of times are between 25 minutes and 49 minutes. Use the Empirical Rule to determine the percentage of times that are less than 19 minutes or greater than 55 minutes. We previously detemined that 19 minutes and 55 minutes are 3 standard deviations from the mean. The Empirical rule indicates that approximately % of times are between 19 minutes and 55 minutes. However, we were asked to determine the percentage of times that are less than 19 minutes or greater than 55 minutes. These are values that are found in the lower tail and in the upper tail of the distribution, not the middle. To find this number we will subtract the percentage of values that are between 3 standard deviations of the mean from 100%. 100% - % = % Approximately % of times are less than 19 minutes or more than 55 minutes.

MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
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ch4 q19: solve part d please

The average playing time of music albums in a large collection is 37 minutes, and the standard deviation is 6 minutes.
(a) What value is 1 standard deviation above the mean? 1 standard deviation below the mean? What values are 2 standard deviations
away from the mean?
(b) Without assuming anything about the distribution of times, at least what percentage of the times are between 25 and 49 minutes?
(c) Without assuming anything about the distribution of times, what can be said about the percentage of times that are either less than
19 minutes or greater than 55 minutes?
(d) Assuming that the distribution of times
Less than 19 min or greater than 55 min? Less than 19 min?
approximately normal, about what percentage of times are between 25 and 49 minutes?
Step 1
(a) What value is 1 standard deviation above the mean? 1 standard deviation below the mean? What values are 2 standard deviations
away from the mean?
To find the value that is some number of standard deviations above or below the mean, we will add or subtract some multiple of the
standard deviation, s, to the sample mean, x. The average playing time was given to be 37 minutes with a standard deviation of
6 minutes, so we have x
37
37 and s = 6
To find a value above the mean indicates we will use laddition
addition. Finding a value below the mean indicates we will use
subtraction
subtraction. Finding values that are away from the mean includes values that are above and below the mean,
indicating we will use addition and subtraction
addition and subtraction
Step 2
We know the mean is x = 37 and the standard deviation is s = 6. To find values above the mean indicates addition, values below the mean
indicates
ction, and both addition and subtraction will be used
find values away from
mean.
Find the value that is 1 standard deviation above the mean.
1 standard deviation above the mean = x + s
= 43
43
Find the value that is 1 standard deviation below the mean.
1 standard deviation below the mean = x - s
= 37 - 6
= 31
31
Find the values that are 2 standard deviations away from the mean.
2 standard deviations above the mean = x + 25
= 37 + 26
6
49
2 standard deviations below mean = x - 25
25 v
25
Step 3
(b) Without assuming anything about the distribution of times, at least what percentage of the times are between 25 and
49 minutes? (Round the answer to the nearest whole number.)
When we assume nothing about the shape of the distribution, we will use Chebyshev's Rule to get a sense of the distribution of data
values. Chebyshev's Rule will tell us the minimum percentage
observations that are within k standard deviations of the mean when
kz 1.
We previously determined that 25 is two standard deviations below the mean and 49 is two standard deviations above the mean. Thus, we
will use k = 2
2 in Chebyshev's Rule to determine the minimum percentage of times that are between 25 and 49 minutes.
1
10 -*
2
= 75
75 %
Thus, at least75
75 % of times are between 25 and 49 minutes.
Step 4
(c) Without assuming anything about the distribution of times, what can be said about the percentage
than 19 minutes or greater than 55 minutes? (Round the answer to the nearest whole number.)
times that are either less
Again, we are not assuming anything about the shape of the distribution, so we will continue using Chebyshev's Rule. We will determine
the minimum percentage of times that are between 19 minutes and 55 minutes and then determine the maximum percentage of times
that are less than 19 minutes or greater than 55 minutes. First, we need to know how many standard deviations 19 minutes and
55 minutes are from the mean so we can determine the value for k
We previously determined that 43 minutes and 31 minutes are 1 standard deviation from the mean and that 49 minutes and 25 minutes
are 2 standard deviations from the mean.
25
31
37
43
49
?
X- 3s
X- 2s
X-5
X+5
X+ 2s
X+3s
Perhaps
9 minutes and 55 minutes are 3 standard deviations from the mean. Let's investigate.
3 standard deviations above the mean
x + 3s = 37 + 36
55
55
3 standard deviations below the mean
x - 3s = 37 -
= 19
19
We showed that 19 minutes and 55 minutes are
are
3 standard deviations from the mean.
We showed that 19 minutes and 55 minutes are
are 3 standard deviations from the mean.
Step 5
We determined that 19 minutes and 55 minutes are 3 standard deviations from the mean. Thus, we will use k = 3 3 in Chebyshev's
Rule to determine the minimum percentage
times that are between 19 and 55 minutes. (Round your final answer to the nearest whole
number.)
100(1 -% - 100 1 -
(3
89
However, we are asked to find the percentage of values that are less than 19 minutes and more than 55 minutes. These are values that are
found in the lower tail and in the upper tail of the distribution, not the middle. To find this number we will subtract the minimum percentage of
values that are between 3 standard deviations of the mean from 100%. (Round your answers to the nearest whole number.)
100% - 89
89 % = 11
Thus, there will be no more than 11
11 % of times less than 19 minutes or more than 55 minutes.
(d) Assuming that the distribution of times is approximately normal, about what percentage of times are between 25 and 49 minutes? Less
than 19 min or greater than 55 min? Less than 19 min? (Round the answers to two decimal places, if needed.)
When we assume that the distribution of times is approximately normal, we will use the Empirical Rule to get a more precise sense of the
distribution of data values. The Empirical Rule gives us approximate percentages that are within k = 1, 2, or 3 standard deviations from the
If the distribution can be reasonably well approximated by a normal curve, then:
• Approximately 68% of the observations are within 1 standard deviation of the mean.
• Approximately 95% of the observations are within 2 standard deviations of the mean.
• Approximately 99.7% of the observations are within 3 standard deviations of the mean.
approximate percentages implied by the Empirical Rule
34%
34%
2.35%
13.5%
13.5%
2.35%
-3 sd
-2 sd
-1 sd
Mean
1 sd
1sd
1 sd
2 sd
3 sd
In contrast to Chebyshev's Rule, dividing the "within" percentages in half is permissible, because the normal curve is symmetric.
Use the Empirical Rule to determine the percentage of times are between 25 and 49 minutes.
We previously determined
approximately
minutes and 4
minutes
stan
eviations of the mean. The Empirical Rule indicates
% of times are between 25 minutes and 49 minutes.
Use the Empirical Rule to determine the percentage of times that are less than 19 minutes or greater than 55 minutes.
We previously determined that 19 minutes and 55 minutes are 3 standard deviations from the mean. The Empirical rule indicates that
approximately
% of times are between 19 minutes and 55 minutes.
However, we were asked to determine the percentage of times that are less than 19 minutes or greater than
are found in the lower tail and in the upper tail of the distribution, not the middle. To find this number we will subtract the percentage of values
minutes. These are values that
that are between 3 standard deviations of the mean from 100%.
100% -
% =
Approximately
% of times are less than 19 minutes
more than 55 minutes.
Transcribed Image Text:The average playing time of music albums in a large collection is 37 minutes, and the standard deviation is 6 minutes. (a) What value is 1 standard deviation above the mean? 1 standard deviation below the mean? What values are 2 standard deviations away from the mean? (b) Without assuming anything about the distribution of times, at least what percentage of the times are between 25 and 49 minutes? (c) Without assuming anything about the distribution of times, what can be said about the percentage of times that are either less than 19 minutes or greater than 55 minutes? (d) Assuming that the distribution of times Less than 19 min or greater than 55 min? Less than 19 min? approximately normal, about what percentage of times are between 25 and 49 minutes? Step 1 (a) What value is 1 standard deviation above the mean? 1 standard deviation below the mean? What values are 2 standard deviations away from the mean? To find the value that is some number of standard deviations above or below the mean, we will add or subtract some multiple of the standard deviation, s, to the sample mean, x. The average playing time was given to be 37 minutes with a standard deviation of 6 minutes, so we have x 37 37 and s = 6 To find a value above the mean indicates we will use laddition addition. Finding a value below the mean indicates we will use subtraction subtraction. Finding values that are away from the mean includes values that are above and below the mean, indicating we will use addition and subtraction addition and subtraction Step 2 We know the mean is x = 37 and the standard deviation is s = 6. To find values above the mean indicates addition, values below the mean indicates ction, and both addition and subtraction will be used find values away from mean. Find the value that is 1 standard deviation above the mean. 1 standard deviation above the mean = x + s = 43 43 Find the value that is 1 standard deviation below the mean. 1 standard deviation below the mean = x - s = 37 - 6 = 31 31 Find the values that are 2 standard deviations away from the mean. 2 standard deviations above the mean = x + 25 = 37 + 26 6 49 2 standard deviations below mean = x - 25 25 v 25 Step 3 (b) Without assuming anything about the distribution of times, at least what percentage of the times are between 25 and 49 minutes? (Round the answer to the nearest whole number.) When we assume nothing about the shape of the distribution, we will use Chebyshev's Rule to get a sense of the distribution of data values. Chebyshev's Rule will tell us the minimum percentage observations that are within k standard deviations of the mean when kz 1. We previously determined that 25 is two standard deviations below the mean and 49 is two standard deviations above the mean. Thus, we will use k = 2 2 in Chebyshev's Rule to determine the minimum percentage of times that are between 25 and 49 minutes. 1 10 -* 2 = 75 75 % Thus, at least75 75 % of times are between 25 and 49 minutes. Step 4 (c) Without assuming anything about the distribution of times, what can be said about the percentage than 19 minutes or greater than 55 minutes? (Round the answer to the nearest whole number.) times that are either less Again, we are not assuming anything about the shape of the distribution, so we will continue using Chebyshev's Rule. We will determine the minimum percentage of times that are between 19 minutes and 55 minutes and then determine the maximum percentage of times that are less than 19 minutes or greater than 55 minutes. First, we need to know how many standard deviations 19 minutes and 55 minutes are from the mean so we can determine the value for k We previously determined that 43 minutes and 31 minutes are 1 standard deviation from the mean and that 49 minutes and 25 minutes are 2 standard deviations from the mean. 25 31 37 43 49 ? X- 3s X- 2s X-5 X+5 X+ 2s X+3s Perhaps 9 minutes and 55 minutes are 3 standard deviations from the mean. Let's investigate. 3 standard deviations above the mean x + 3s = 37 + 36 55 55 3 standard deviations below the mean x - 3s = 37 - = 19 19 We showed that 19 minutes and 55 minutes are are 3 standard deviations from the mean. We showed that 19 minutes and 55 minutes are are 3 standard deviations from the mean. Step 5 We determined that 19 minutes and 55 minutes are 3 standard deviations from the mean. Thus, we will use k = 3 3 in Chebyshev's Rule to determine the minimum percentage times that are between 19 and 55 minutes. (Round your final answer to the nearest whole number.) 100(1 -% - 100 1 - (3 89 However, we are asked to find the percentage of values that are less than 19 minutes and more than 55 minutes. These are values that are found in the lower tail and in the upper tail of the distribution, not the middle. To find this number we will subtract the minimum percentage of values that are between 3 standard deviations of the mean from 100%. (Round your answers to the nearest whole number.) 100% - 89 89 % = 11 Thus, there will be no more than 11 11 % of times less than 19 minutes or more than 55 minutes. (d) Assuming that the distribution of times is approximately normal, about what percentage of times are between 25 and 49 minutes? Less than 19 min or greater than 55 min? Less than 19 min? (Round the answers to two decimal places, if needed.) When we assume that the distribution of times is approximately normal, we will use the Empirical Rule to get a more precise sense of the distribution of data values. The Empirical Rule gives us approximate percentages that are within k = 1, 2, or 3 standard deviations from the If the distribution can be reasonably well approximated by a normal curve, then: • Approximately 68% of the observations are within 1 standard deviation of the mean. • Approximately 95% of the observations are within 2 standard deviations of the mean. • Approximately 99.7% of the observations are within 3 standard deviations of the mean. approximate percentages implied by the Empirical Rule 34% 34% 2.35% 13.5% 13.5% 2.35% -3 sd -2 sd -1 sd Mean 1 sd 1sd 1 sd 2 sd 3 sd In contrast to Chebyshev's Rule, dividing the "within" percentages in half is permissible, because the normal curve is symmetric. Use the Empirical Rule to determine the percentage of times are between 25 and 49 minutes. We previously determined approximately minutes and 4 minutes stan eviations of the mean. The Empirical Rule indicates % of times are between 25 minutes and 49 minutes. Use the Empirical Rule to determine the percentage of times that are less than 19 minutes or greater than 55 minutes. We previously determined that 19 minutes and 55 minutes are 3 standard deviations from the mean. The Empirical rule indicates that approximately % of times are between 19 minutes and 55 minutes. However, we were asked to determine the percentage of times that are less than 19 minutes or greater than are found in the lower tail and in the upper tail of the distribution, not the middle. To find this number we will subtract the percentage of values minutes. These are values that that are between 3 standard deviations of the mean from 100%. 100% - % = Approximately % of times are less than 19 minutes more than 55 minutes.
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