Question 1203867: The mean price paid is $1400 and the standard deviation is $110.
What is the approximate percentage of buyers who paid between $1290 and $1400?
%
What is the approximate percentage of buyers who paid between $1290 and $1510?
%
What is the approximate percentage of buyers who paid between $1180 and $1400?
%
What is the approximate percentage of buyers who paid less than $1180?
%
What is the approximate percentage of buyers who paid less than $1070?
%
What is the approximate percentage of buyers who paid between $1070 and $1400?
%
Found 2 solutions by Theo, ikleyn:
Answer by Theo(13342)   (Show Source):
Answer by ikleyn(53762)  (Show Source):
You can put this solution on YOUR website! .
The mean price paid is $1400 and the standard deviation is $110.
(a) What is the approximate percentage of buyers who paid between $1290 and $1400?
(b) What is the approximate percentage of buyers who paid between $1290 and $1510?
(c) What is the approximate percentage of buyers who paid between $1180 and $1400?
(d) What is the approximate percentage of buyers who paid less than $1180?
(e) What is the approximate percentage of buyers who paid less than $1070?
(f) What is the approximate percentage of buyers who paid between $1070 and $1400?
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Tutor @Theo solved this problem using calculator.
He presented the answers referring to the calculator' snapshots.
But now his video-format is not maintained, so his answers are invisible,
and, therefore, his solutions are now practically dead and useless.
Meanwhile, this problem can be solved without using a calculator, because all these cases fall
under the empirical rules of normal distribution.
Moreover, from the problem, it is obviously clear that this problem is indented
to be solved using these empirical rules.
So, I came to provide a solution in a way as it should be done.
The Empirical Rule, or the 68-95-99.7 Rule, states that for a normal distribution,
- approximately 68% of data falls within one standard deviation of the mean,
- 95% falls within two standard deviations, and
- 99.7% falls within three standard deviations.
This rule provides a quick way to understand the spread of data around the mean in symmetrical normal distributions.
(a) Interval from $1290 to $1400 is from one standard deviation left of the mean to the mean,
so it is half of the complete interval "one standard deviation from the mean".
Therefore, due to the empirical rule and the symmetry of standard distribution, the answer in this case
is half of 68%, i.e. 34%, approximately.
(b) Interval from $1290 to $1510 is precisely "one standard deviation from the mean" in both directions.
Therefore, due to the empirical rule, the answer in this case is half of 68%,
i.e. 68%, approximately.
(c) Interval from $1180 to $1400 is from two standard deviations left of the mean to the mean,
so it is half of the complete interval "two standard deviation from the mean".
Therefore, due to the empirical rule and symmetry of standard distribution, the answer in this case
is half of 95%, i.e. 47.5%, approximately.
(d) In this case, we want to find the area under the normal distribution curve on the left
from two standard deviations from the mean.
So, we subtract half of 95% from 50%, which corresponds to the mean.
We get then the answer 50% - 47.5% = 2.5%.
(e) In this case, $1070 is in 3 standard deviation from the mean.
So, we want to find the area under the normal distribution curve on the left from three standard
deviations from the mean.
Therefore, the answer for case (d) is 50% minus half of 99.7%, i.e. 50% - 49.85% = 0.15%, approximately.
(f) The interval from $1070 t0 $1400 is three standard deviation on the left from the mean.
So, the answer in this case is half of 99.7%, i.e. about 49.85%, approximately.
Thus, all questions are answered and the problem is solved completely.
It is done mentally, using the empirical rules for normal distribution,
precisely in a way as it should be done and as it was expected, due to the design of this problem.
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There is another (= one more) reason why I produced and placed my solution here.
@Theo' posts used visual plots to support his solutions.
These plots were integral inseparable part of his solutions.
But some time ago, Theo left this forum and stopped supporting web-site with his plots.
As a result, you see now some colored spots in his posts, where his plots should be.
Due to this reason, @Theo's post lost their educational meaning and value.
Therefore, I create my posts with my own mathematical solutions
to replace @Theo' solutions and provide meaningful mathematical content.
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