Lesson: Normal Distributions

Sorry for taking so long!
To properly explain those two questions, I wanted to create some graphics.

My solutions can be found here:

- http://gre.myprepclub.com/forum/the-random-variable-x-is-normally-distri...

- http://gre.myprepclub.com/forum/qotd-13-the-figure-above-shows-the-stand...

Question link: http://gre.myprepclub.com/forum/a-random-variable-y-is-normally-distribu...

As you've stated, only 5 percent of data points are more than 2 standard deviations away from the mean.

This means 2.5% of the data points are more than 2 standard deviations ABOVE the mean.
And 2.5% of the data points are more than 2 standard deviations BELOW the mean.

In other words, the probability is 2.5/100 that a data point is more than 2 standard deviations ABOVE the mean

So, P(Y is greater than 220) = 2.5/100

Since 1/6 > 2.5/100, the correct answer is B

Question link: http://gre.myprepclub.com/forum/the-figure-above-shows-the-probability-d...

First of all, all probabilities range from 0 to 1, inclusive. Since 1/0.76 is greater than 1, this is an invalid probability.

Here's how we need to look at it.

P(1 < x < 4) = P(1 < x < 2 OR 2 < x < 3 OR 3 < x < 4)
= P(1 < x < 2) + P(2 < x < 3) + P(3 < x < 4)
= 0.30 + 0.32 + 0.16
= 0.76

No, you would be required to perform that kind of calculation.

I should note that your question is considerably different from the linked question you posted.

In the linked question, we're told that the random variable x is NORMALLY distributed.
In the question above, the test scores are not necessarily normally distributed.

Here's my solution.
If a score of 700 is in the 75th percentile, then 75% of all scores are LESS THAN 700
If a score of 450 is in the 25th percentile, then 25% of all scores are LESS THAN 450

There are infinitely many scenarios that meet these two conditions. Here are two such scenarios:

Scenario 1:There are 100 test scores, and the scores are:
25 scores of 200
1 score of 450 [so, 450 is in the 25th percentile]
49 scores of 600
1 score of 700 [so, 700 is in the 75th percentile]
19 scores of 750
1 score of 800 [so, 800 is in the 95th percentile]
14 scores of 900
In this case, the 95th percentile score is 800
So, we get:
QUANTITY A: 800
QUANTITY B: 800
Here, the two quantities are EQUAL

Scenario 1:There are 100 test scores, and the scores are:
25 scores of 200
1 score of 450 [so, 450 is in the 25th percentile]
49 scores of 600
1 score of 700 [so, 700 is in the 75th percentile]
19 scores of 750
1 score of 850 [so, 850 is in the 95th percentile]
14 scores of 900
In this case, the 95th percentile score is 850
So, we get:
QUANTITY A: 800
QUANTITY B: 850
Here, Quantity B is GREATER

Answer: D

Cheers,
Brent

If a question does not specifically state that a population is normally distributed, then we cannot assume that it is normally distributed.

Question link: https://gre.myprepclub.com/forum/the-random-variable-x-is-normally-distr...
Unfortunately, the fact that the strategy worked was purely coincidental.
The strategy won't work, because we need to use the fact that the values are normally distributed.

Here's an example to show why:

Let's say set X = {1, 2, 3, 4, 5, . . . . . 100} (i.e., all positive integers from 1 to 100)
In this case, 31 is in the 30th percentile, since 30 of the 100 values are less than 31

Now, let's say set Y = {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, . . . 1, 31} (i.e., there are 99 ones and 1 thirty-one)
In this case, 31 is in the 99th percentile, since 99 of the 100 values are less than 31

As you can see, if a set is NOT normally distributed, it's hard to make strong conclusions about the values in the set.

Does that help?

Cheers,
Brent

It's a VERY tricky question.
Here's my full solution: https://gre.myprepclub.com/forum/the-random-variable-x-is-normally-distr...

Please let me know if that helps.

Cheers,
Brent

Question link: https://gre.myprepclub.com/forum/a-random-variable-y-is-normally-distrib...

It depends on how precise we want to be.

On page 149 of The GRE Official Guide (3rd edition), we're told that 2% of of the values in a normal distribution are more than 2 units of standard deviation from the mean.

Other resources like to be more precise and say that 2.5% of of the values in a normal distribution are more than 2 units of standard deviation from the mean.

Regardless of whether we use 2% or 2.5%, the correct answer is still B (since both of those values are still less than 1/16)

Cheers,
Brent

I had a similar discussion above in the comment section.

Both are fine. It really depends on how precise we want to be.

On page 149 of The GRE Official Guide (3rd edition), we're told that 2% of of the values in a normal distribution are more than 2 units of standard deviation from the mean. Other resources like to be more precise and say that 2.5% of of the values in a normal distribution are more than 2 units of standard deviation from the mean.

Rest assured, the GRE test-makers won't create a question that exploits these minor differences.

Cheers,
Brent

Question link: https://gre.myprepclub.com/forum/the-graph-represents-the-normally-distr...

Yes, normal distributions are definitely symmetrical about the mean. However, as you can from my solution, if we are shading a portion of the distribution, the shaded portion need not be symmetrical about the mean.

Does that help?

Cheers,
Brent

Question link: https://gre.myprepclub.com/forum/the-random-variable-x-is-normally-distr...

That's a perfect analysis of my solution!

Cheers,
Brent

Question link: https://gre.myprepclub.com/forum/the-graph-represents-the-normally-distr...

If we say Mean + SD = 600, then we're suggesting that a score of 600 is 1 standard deviation above the mean.

Likewise, if we say Mean - SD = 500, then we're suggesting that a score of 500 is 1 standard deviation below the mean.

Unfortunately, there's nothing in the question that tells us that this is so.
That said, the misleading diagram certainly LOOKS like this is so, but we can't trust the diagram.

Does that help?

Cheers,
Brent

Thanks for that!

Link: https://gre.myprepclub.com/forum/the-scores-for-the-500-students-who-too...

You're correct about context.

The given information, tells us about the students who scored at least 92 points and students who scored at or below 56. So, these are the only scores we need to concern ourselves with.

A score of 92 is 1 standard deviation ABOVE the mean
A score of 56 is 2 standard deviations BELOW the mean

So, these are the only scores we need to consider.

Cheers,
Brent

Question link: https://gre.myprepclub.com/forum/the-scores-for-the-500-students-who-too...

This is the nature of all normal distributions.

For example, in any normal distribution:
- about 50% of the population is ABOVE the mean and 50% is BELOW the mean.
- about 2% of the population is more than 2 standard deviations BELOW the mean
- about 68% of the population is less than 1 standard deviation from the mean
etc.

One of the characteristics of all normal distributions is that 16% of the population is more than 1 standard deviation ABOVE the mean.
See the graph at https://gre.myprepclub.com/forum/the-scores-for-the-500-students-who-too...

Cheers,
Brent

Question link: https://gre.myprepclub.com/forum/the-graph-represents-the-normally-distr...

There are two times in which we can assume a diagram is drawn to scale:
1) The question is a data interpretation question
2) We are explicitly told the diagram is drawn to scale.

Neither of these conditions are met in this question, we we can't make any assumptions.

Cheers,
Brent

Question link: https://gre.myprepclub.com/forum/the-scores-for-the-500-students-who-too...

The math you used in your solution suggests that you're using the same strategy I used in my video solution. The only difference is that you didn't formally write the equations m + d = 92 and m - 2d = 56

So, I think your approach is perfectly valid.

Cheers,
Brent

Question link: https://brainly.com/question/13031225

This question is beyond the scope of the GRE.
For GRE questions involving normal distributions, you need only know the percentiles for 1, 2 and 3 standard deviations from the mean.
The linked question above requires additional information that requires a SCIENTIFIC calculator (which is not available on the GRE)

Cheers,
Brent

Tricky!!
I needed to make a few graphics, so I posted the question to GRE Prep Club.
You'll find my solution here: https://gre.myprepclub.com/forum/the-scores-on-a-given-history-test-are-...

Cheers,
Brent

Question link: https://gre.myprepclub.com/forum/the-75th-percentile-on-a-test-correspon...

With that change, the question would never be an official question, since we'd need a scientific calculator to answer it.

That said, our solution would begin by finding the mean.
From the given info, we know that 25% of scores are LESS than 450, and 25% of scores are GREATER than 700
Given the symmetry of the normal distribution, we know that the mean lies in the middle of 450 and 700 at 575
If the mean is 575, we need to know the number of standard deviations (away from the mean of 575) that correspond to the 75th and 25th percentiles. This part would require a scientific calculator.
Once we know those values, we can determine the standard deviation of the set.
Once we know the standard deviation, we can answer the (new) question.

Note: I edited your question (you had a Q in it, which I assumed stood for "question")

There are a couple of different ways to solve this question.

ALGEBRA
Take: -|x| = |x|
Add |x| to both sides of the equation to get: 0 = 2|x|
Divide both sides by 2 to get: 0 = |x|
Here, we can see that x = 0, which means the two quantities are equal.
Answer: C

NUMBER SENSE
If x ≠ 0, then -|x| is NEGATIVE, and |x| is POSITIVE, which means it's impossible for -|x| to equal |x|.
So, it must be the case that x = 0
Answer: C

PS: In the future, please provide a link to the question (if possible)