Repeated student samples. Of all freshman at a large college, 16% made the dean’s list in the current year. As part of a class p
roject, students randomly sample 40 students and check if those students made the list. They repeat this 1,000 times and build a distribution of sample proportions. (a) What is this distribution called?
(b) Would you expect the shape of this distribution to be symmetric, right skewed, or left skewed? Explain your reasoning.
(c) Calculate the variability of this distribution.
(d) What is the formal name of the value you computed in (c)?
(e) Suppose the students decide to sample again, this time collecting 90 students per sample, and they again collect 1,000 samples. They build a new distribution of sample proportions. How will the variability of this new distribution compare to the variability of the distribution when each sample contained 40 observations?
a) p-hat (sampling distribution of sample proportions)
d) Standard error
e) If we increase the sample size from 40 to 90 students, the standard error becomes two thirds of the previous standard error (se=0.667).
a) This distribution is called the <em>sampling distribution of sample proportions</em> <em>(p-hat)</em>.
b) The shape of this distribution is expected to somewhat normal, symmetrical and centered around 16%.
This happens because the expected sample proportion is 0.16. Some samples will have a proportion over 0.16 and others below, but the most of them will be around the population mean. In other words, the sample proportions is a non-biased estimator of the population proportion.
c) The variability of this distribution, represented by the standard error, is:
d) The formal name is Standard error.
e) If we divided the variability of the distribution with sample size n=90 to the variability of the distribution with sample size n=40, we have:
If we increase the sample size from 40 to 90 students, the standard error becomes two thirds of the previous standard error (se=0.667).
The information given in the question to solve it is insufficient.
The complete question related to this found at GMAT club forum is stated below:
Steve works at an apple orchard and is paid by the bushel for the apples he harvests each day. For the first 42 bushels Steve harvests each day, he is paid y dollars per bushel. For each additional bushel over 42, he is paid 1.5y. How many bushels of apples did Steve harvest yesterday?
1. Yesterday, Steve was paid $180 for the apples he harvested.
2. Today, Steve was paid $240, and he harvested 10 more bushels of apples than he harvested yesterday.
1st 42 bushels Steve harvests each day:
Payment of 1 bushel = y dollar
Payment of ≤ 42 bushel = 42× y = $42y
Payment for Additional bushel after the 1st 42 bushels = 1.5y per bushel
Yesterday's Payment = $180
Today's payment = $240
Let the total number Steve harvested yesterday = p
Today he harvested 10 more bushels than he did yesterday
Total number Steve harvested today = p + 10
Now looking at the information in the question:
First statement did not tell us the amount he harvested that got him that pay, so it's insufficient. Second statement alone two is not sufficient to determine yesterdays pay.
In both statement, we were not told if he harvested more or less than 42 yesterday. Neither were we told if the additional 10 bushels was after he had harvested more than 42 bushels.
Let's assume the additional 10 bushels was after he had harvested more than 42 bushels (that is he harvested 42 yesterday) .
Difference in payment = 240-180 = 60
For the additional 10bushels, he was paid $60
10(1.5y) = 60
15y = 60
y = 60/15 = 4
If y = $4 per bushel and he was paid $180, the number he harvested will be:
4 × number of bushels = $180
4p = 180
p = 180/4 = 45 bushels
p = 45bushels isn't possible because the amount for the first bushels should either be less than or equal to 42
If we make another assumption, let's say he harvested more than 42 yesterday (42 + an additional number), we would get a different answer.
Thus the information given in the question to solve it is insufficient.
8x(10)^9 - 2 x (10)^1 ( we use 9 as the power because greater number used as power gives the bigger number and the next smaller number 8 as base. Similarly we use smaller number for power to get a smaller for the greatest difference)
= 8,000,000,000 -20
For smaller number
_x10^blank - _ x 10^_
1x(10)^3 - 9 x (10)^2 ( we use 3 as the power because smaller number used as power gives the smaller number and the next smallest number 1 as base. Similarly we use next smaller number for power to get the next smaller for the smallest difference)
= 1000- 900
We fill in the blanks keeping in mind that we do not have to repeat the numbers from 1-9 and also the numbers should have such an arrangement that they show the smallest and largest possible differences.