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A random variable Y is normally distributed with a mean of 2

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A random variable Y is normally distributed with a mean of 2 [#permalink] New post 14 Jan 2016, 07:38
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A random variable Y is normally distributed with a mean of 200 and a standard deviation of 10.

Quantity A
Quantity B
The probability of the event
that the value of Y is
greater than 220
\(\frac{1}{6}\)


A)The quantity in Column A is greater.
B)The quantity in Column B is greater.
C)The two quantities are equal.
D)The relationship cannot be determined from the information given.




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[Reveal] Spoiler: OA

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Re: A random variable Y is normally distributed with a mean of 2 [#permalink] New post 14 Jan 2016, 10:03
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Solution

The mean is 200 and the SD is + 10 or - 10 from the mean. As such , we do have a probability that ranging from 190 to 210. In total 20 numbers on our number line.

From this, we do have \(\frac{1}{20}\) of probability. This quantity is \(<\)\(\frac{1}{6}\)

The best answer is \(B\)

Alternative solution

Image



Another approach to this problem is to draw a normal curve, or “bell-shaped curve,” that represents the probability distribution of the random variable Y, as shown.

The curve is symmetric about the mean 200. The values of 210, 220, and 230 are equally spaced to the right of 200 and represent 1, 2, and 3 standard deviations, respectively, above the mean. Similarly, the values of 190, 180, and 170 are 1, 2, and 3 standard deviations, respectively, below the mean. Quantity A, the probability of the event that the value of Y is greater than 220, is equal to the area of the shaded region as a fraction of the total area under the curve. i.e. the probability of the event that the value of Y is greater than 220 must be less than 5%, or \(\frac{1}{20}\) and this is certainly less than \(\frac{1}{6}\) The correct answer is \(B\).
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Re: A random variable Y is normally distributed with a mean of 2 [#permalink] New post 13 Dec 2016, 12:40
Hey dears
unfortunately i tried hard to understand any of your approaches but, it doesn't work(
if there is more explanation please !
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Re: A random variable Y is normally distributed with a mean of 2 [#permalink] New post 08 Sep 2018, 20:48
Could you also say that the shaded region is 2.5% chance of the event happening since its between M+2D and M+3D. If so, the answer would be A. Thanks for the clarification.
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Re: A random variable Y is normally distributed with a mean of 2 [#permalink] New post 09 Sep 2018, 01:33
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Runnyboy44 wrote:
Could you also say that the shaded region is 2.5% chance of the event happening since its between M+2D and M+3D. If so, the answer would be A. Thanks for the clarification.



No 2.5% chance mean Quantity A is \(\frac{2.5}{100}\) and Quantity B is \(\frac{1}{6}\).
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Re: A random variable Y is normally distributed with a mean of 2 [#permalink] New post 10 Sep 2018, 22:56
May I know why "the probability of the event that the value of Y is greater than 220 must be less than 5%, or 1/20"?
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Re: A random variable Y is normally distributed with a mean of 2 [#permalink] New post 11 Sep 2018, 05:18
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Runnyboy44 wrote:
May I know why "the probability of the event that the value of Y is greater than 220 must be less than 5%, or 1/20"?



You have to rememer this graph.

Attachment:
norm.png
norm.png [ 100.5 KiB | Viewed 9579 times ]



You can only have probability in ranges of numbers in this case. This is very important as opposed to discreet events having probability like a coin toss or a dice roll. When we talk about events that are continuous they are represented as the graph above.

For example probability of getting a number between mean + 2 standard devation (\(\mu + 2\sigma\)) and mean + 3 standard devation (\(\mu + 3\sigma\)) is 2.14%.

Or if you picked the numbers 100 times you would get a number between \(\mu + 2\sigma\) and \(\mu + 3\sigma\) is 2.14 times.
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Re: A random variable Y is normally distributed with a mean of 2 [#permalink] New post 11 Sep 2018, 06:30
Runnyboy44 wrote:
May I know why "the probability of the event that the value of Y is greater than 220 must be less than 5%, or 1/20"?



This problem is based on normal distribution .Its better to remember those percentages as question in gre will be direct basing on them.
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Re: A random variable Y is normally distributed with a mean of 2 [#permalink] New post 11 Sep 2018, 11:01
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220 will be two SD above the mean.

So the probability will be 2% = 1/50

So we are comparing 1/50 and 1/6.

B is the answer :)
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Re: A random variable Y is normally distributed with a mean of 2 [#permalink] New post 12 Sep 2018, 13:02
:-D
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Re: A random variable Y is normally distributed with a mean of 2 [#permalink] New post 14 Sep 2018, 00:57
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sandy wrote:
Runnyboy44 wrote:
May I know why "the probability of the event that the value of Y is greater than 220 must be less than 5%, or 1/20"?



You have to rememer this graph.

Attachment:
norm.png



You can only have probability in ranges of numbers in this case. This is very important as opposed to discreet events having probability like a coin toss or a dice roll. When we talk about events that are continuous they are represented as the graph above.

For example probability of getting a number between mean + 2 standard devation (\(\mu + 2\sigma\)) and mean + 3 standard devation (\(\mu + 3\sigma\)) is 2.14%.

Or if you picked the numbers 100 times you would get a number between \(\mu + 2\sigma\) and \(\mu + 3\sigma\) is 2.14 times.


Thank you for the explanation. If one understands this principle, there is nothing hard
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Re: A random variable Y is normally distributed with a mean of 2 [#permalink] New post 14 Feb 2019, 11:32
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Carcass wrote:
A random variable Y is normally distributed with a mean of 200 and a standard deviation of 10.

Quantity A
Quantity B
The probability of the event
that the value of Y is
greater than 220
\(\frac{1}{6}\)



Here's what a normally distributed population looks like:
Image
The values on the bottom of the diagram represent the number of standard deviations from the mean.


Since the mean of the population of y-values is 200, we can add that to our diagram
Image


Since the standard deviation is 10, we can see that a value of 220 is TWO UNITS OF STANDARD DEVIATION above the mean
Image
Our diagram tells us that 2% of the y-values will be greater than 220

In other words, P(selected y-value is greater than 220) = 2% = 2/100 = 1/50

We get:
QUANTITY A: 1/50
QUANTITY B: 1/16

Quantity B is greater

Answer: B

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Brent
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Re: A random variable Y is normally distributed with a mean of 2   [#permalink] 14 Feb 2019, 11:32
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