Answer:
a) Standard error = 2
b) Range = (76.08, 83.92)
c) P=0.69
d) Smaller
e) Greater
Step-by-step explanation:
a) When we have a sample taken out of the population, the standard error of the mean is calculated as:
where n is te sample size (n=25) and σ is the population standard deviation (σ=10).
Then, the standard error of the classroom average score is 2.
b) The calculations for this range are the same that for the confidence interval, with the difference that we know the population mean.
The population standard deviation is know and is σ=10.
The population mean is M=80.
The sample size is N=25.
The standard error of the mean is σM=2.
The z-value for a 95% confidence interval is z=1.96.
The margin of error (MOE) can be calculated as:
Then, the lower and upper bounds of the confidence interval are:
The range that we expect the average classroom test score to fall 95% of the time is (76.08, 83.92).
c) We can calculate this by calculating the z-score of X=79.
Then, the probability of getting a average score of 79 or higher is:
The approximate probability that a classroom will have an average test score of 79 or higher is 0.69.
d) If the sample is smaller, the standard error is bigger (as the square root of the sample size is in the denominator), so the spread of the probability distribution is more. This results then in a smaller probability for any range.
e) If the population standard deviation is smaller, the standard error for the sample (the classroom) become smaller too. This means that the values are more concentrated around the mean (less spread). This results in a higher probability for every range that include the mean.
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</span><span>Suppose you have a triangle with sides {6,7,8} — how do you find the height?This is a question some GMAT test takers ask. They know they would need the height to find the area, so they worry: how would I find that height. The short answer is:fuhgeddaboudit! Which height?First of all, the “height” of a triangle is its altitude. Any triangle has three altitudes, and therefore has three heights. You see, any side can be a base. From any one vertex, you can draw a line that is perpendicular to the opposite base — that’s the altitude to this base. Any triangle has three altitudes and three bases. You can use any one altitude-base pair to find the area of the triangle, via the formula A = (1/2)bh.
In each of those diagrams, the triangle ABC is the same. The green line is the altitude, the “height”, and the side with the red perpendicular square on it is the “base.” All three sides of the triangle get a turn. Finding a heightGiven the lengths of three sides of a triangle, the only way one would be able to find a height and the area from the sides alone would involve trigonometry, which is well beyond the scope of the GMAT. You are 100% NOT responsible for knowing how to perform these calculations. This is several levels of advanced stuff beyond the math you need to know. Don’t worry about that stuff.In practice, if the GMAT problem wants you to calculate the area of a triangle, they would have to give you the height. The only exception would be a right triangle — in a right triangle, if one of the legs is the base, the other leg is the altitude, the height, so it’s particularly easy to find the area of right triangles. Some “more than you need to know” caveatsIf you don’t want to know anything about this topic that you don’t absolutely need for the GMAT, skip this section!a. Technically, if you know the three sides of a triangle, you could find the area from something called Heron’s formula, but that’s also more than the GMAT will expect you to know. More than you needed to know!b. If one of the angles of the triangle is obtuse, then the altitudes to either base adjacent to this obtuse angle are outside of the triangle. Super-technically, an altitude is not a segment through a vertex perpendicular to the opposite base, but instead, a segment through a vertex perpendicular to the line containing the opposite base.In the diagram above, in triangle DEF, one of the three altitudes is DG, which goes from vertex D to the infinite straight line that contains side EF. That’s a technicality the GMAT will not test or expect you to know. Again, more than you needed to know!c. If the three sides of a triangle are all nice pretty positive integers, then in all likelihood, the actual mathematical value of the altitudes will be ugly decimals. Many GMAT prep sources and teachers in general will gloss over that, and for the purposes of easy problem-solving, give you a nice pretty positive integer for the altitude also. For example, the real value of the altitude from C to AB in the 6-7-8 triangle at the top is:Not only are you 100% NOT expected to know how to find that number, but also most GMAT practice question writers will spare you the ugly details and just tell you, for example, altitude = 5. That makes it very easy to calculate the area. Yes, technically, it’s a white lie, but one that spares the poor students a bunch of ugly decimal math with which they needn’t concern themselves. Actually, math teachers of all levels do this all the time — little white mathematical lies, to spare students details they don’t need to know.So far as I can tell, the folks who write the GMAT itself are sticklers for truth of all kinds, and do not even do this “simplify things for the student” kind of white lying. They are more likely to circumvent the entire issue, for example, by making all the relevant lengths variables or something like that. Yet again, more than you needed to know! What you need to knowYou need to know basic geometry. Yes, there is tons of math beyond this, and tons more you could know about triangles and their properties, but you are not responsible for any of that. You just need to know the basic geometry of triangles, including the formula A = (1/2)b*h. If the triangle is not a right triangle, you have absolute no responsibility for knowing how to find the height — it will always be given if you need it. Here’s a free practice question for you.</span>
Answer:
(x-y) (a+x-y)
Step-by-step explanation:
(y-x)=-(x-y)
-a(y-x) = a(x-y)
(x-y)^2 = (x-y)(x-y)
(x-y)(a + x - y)
