Improve Your Understanding of College Level Statistics Problems

Systematic random sampling is easier to execute than simple random sampling and usually provides comparable results. The exception is the presence of some kind of ____ in the listing of the members of the population.

Joanne needs to test boxes of pasta to see if they contain the correct amount of product. Each of the 500 boxes in a recent batch has a unique serial number from 1001 to 1500 stamped on it. She will pick a random sample of 50 boxes by generating a random integer between 1001 and 1011 to select the first box, and then selecting every tenth number in sequence. Her method is called
A) cluster sampling.
B) convenience sampling.
C) multistage sampling.
D) stratified sampling.
E) systematic sampling.

A service agency wishes to assess clients' views of quality service over the past year. Computer records identify 1000 clients over the past 12 months and a decision is made to select 100 clients to survey. The 100 clients to survey are identified through assigning 1-1000 to the clients and a software package was used to randomly select 100 of them. Sampling Techniques:
A. Simple Random Sampling
B. Systematic Random Sampling
C. Stratified Sampling
D. Cluster Sampling

Given that out of $1000$ individuals where $60$ use a drug. I am given the probability of a false positive is $0.009$ and the probability of a false negative is $0.10$. I'm trying to find the true positive and true negative.
Let $D$ be the even that the user uses a drug and $D^C$ the event the user is not a drug user. I know that $P(D)=0.06$ and $P(D^C)=0.994$ as well as $P(+|D^c)=0.009$ and $P(-<!--\; -\; -->|D)=0.10$. So that means that $P(+|D^C)=0.009⟹<!--\; ⟹\; -->0.009\ast <!--\; \ast \; -->990=8.91$ people. Also $P(-<!--\; -\; -->|D)=0.10⟹<!--\; ⟹\; -->60\ast <!--\; \ast \; -->.10=6$ people. Does it follow that $P(-<!--\; -\; -->|D^C)=981.09/990=0.991$ and $P(+|D)=54/60=0.90$?

How does positive charge spread out in conductors?
I know that when there are excess positive charges in a conductor, for example, a metal sphere, the positive charges will spread out over its surface. However, I am confused about how this excess charge spreads out over the surface, if protons cannot move and only electrons can move.
Can someone please inform me on how the excess positive charge spreads out over the surfaces of conductors?

What is central tendency?

How can we describe or model the spread of ice during the freezing of the surface of a still lake?
Premise: I live in a place where it is unquestionably wintertime in every aspect of the phrase. I was driving down the street and I drove by a lake. There was a recent and sudden temperature drop (from above freezing but still quite cold to well below freezing) overnight. The water had not yet frozen over.
It occurred to me while I was driving that the process of beginning the freezing process of the water at the interface of the water and the ambient air seemed to be counterintuitive. While I half (and admittedly naively) expected the ice to begin forming at the outside edges of the lake and continue to freeze inwards in a uniform fashion, the ice seemed to be beginning on one side of the lake only. I drove back by later on during the day, and the lack was nearly frozen over sans a bit of the side that was originally left unfrozen during my first observation.
The details are rather unimportant. I’m well aware of the fact that this was hardly a controlled environment, so the question remains to be how can we best describe or predict the nature of and changes within the interfaces of a substance and its surroundings during the a phase change of the substance? In particular, what’s prompting this question is the fact that there seemed to be a ‘preferred’ origin of the systematic spread of freezing. I’m curious as to why that might be and whether it is predictable. It could well just be dependent on the particular properties of the lake/general substance, but if so I’m curious as to what quantities influence these ready and obviously more rapid transitions.
My first intuition was that perhaps some parts of the lake are deeper and so there is a greater rate of heat transfer between the cold air/water interface and the volume of water below, resulting in that surface freezing more slowly in the long term. But, that doesn’t seem consistent with the fact that one of the edges of the lake remained entirely unfrozen until the end- most likely that edge was not a vertical drop and so would have fallen victim to earlier freezing as well. Presuming that the lake is a still body of water - are there any thoughts?

A town with a population 10,000 has suffered an outbreak of dragon pox, with 3% of the population being infected. There is a test to diagnose the disease. If you have the dragon pox, the test will correctly register positive 97% of the time and will yield a false negative 3% of the time. If you do not have the dragon pox the test will register that correctly 97% of the time and will yield a false positive 3% of the time. Given that you tested positive for the disease, what is the probability that you actually have dragon pox?

Geometric interpretation of the multiplication of complex numbers?
I've always been taught that one way to look at complex numbers is as a Cartesian space, where the real part is the x component and the imaginary part is the y component.
In this sense, these complex numbers are like vectors, and they can be added geometrically like normal vectors can.
However, is there a geometric interpretation for the multiplication of two complex numbers?
I tried out two test ones, 3+i and −2+3i, which multiply to −9+7i. But no geometrical significance seems to be found.
Is there a geometric significance for the multiplication of complex numbers?

If we look at the solutions for the first problem here and the first problem here, we see both problems are one-tailed tests for the upper-tail. However, in the first paper the rejection region is Z > 1.65, but in the second paper the rejection region is Z > 1.96.
How can this be if both problems have a significance level of 0.05 and both are upper-tailed tests? Shouldn't they both have a rejection region of Z > 1.65? Is the second paper incorrect?

What is the mean, median, mode, and range of 68.4, 65.7, 63.9, 79.5, 52.5?

Risk analysis is typically a two-step process: qualitative risk analysis and quantitative risk analysis.
As a systems analyst, for which sorts of project management decisions would you use the results from qualitative risk analysis? From the quantitative risk analysis?

Divide the population of U.S. workers into two categories: those who have a job in the goods-producing industries, and those who have a job in the service-producing industries. Among all workers, sixteen percent have a job in the goods-producing industries. For these workers, the mean of annual earnings equals $\mathrm{\$<!--\; \$\; -->}42,419$, with a standard deviation of $\mathrm{\$<!--\; \$\; -->}35,572$. Among all workers, the remaining eighty-four percent have a job in the service-producing industries. For these workers, the mean of annual earnings equals $\mathrm{\$<!--\; \$\; -->}36,976$, with a standard deviation of $\mathrm{\$<!--\; \$\; -->}31,247$. Find the mean and standard deviation of annual earnings among all workers in the population.

Is this the correct answer:

Pop mean $(.16\ast <!--\; \ast \; -->42,419)+(.84\ast <!--\; \ast \; -->36,976)=37,846.88$
Pop StdDev $(.16\ast <!--\; \ast \; -->35,572)+(.84\ast <!--\; \ast \; -->31,247)=31,939$

5% of the population have the disease (D). A test is available that has a 10% false positive and a 10% false negative rate.
part (B) of the question asks what is the probability of having the disease given that you test positive. I'm quite confident the answer to this is ~ 32%
Part (D) - Suppose that there is no cost or benefit from testing negative but a benefit B from true positives which detect the disease and a cost C to false positives.
i) What is the expected value of the testing programme?
ii) What benefit-cost ratio would you require to proceed with the testing programme?
I'm not quite sure to how to answer either of part D. My somewhat educated guesses are
the expected value = 0.045B - 0.095C
and the benefit to cost ratio required to proceed with the testing programme is 2.11 (or greater)

Aurelia, another student, suggested they conduct a significance test to investigate the claim.
She tested the hypotheses $H_0:p=16\text{ versus }{H}_a:p>16$ at the significance level of α=0.05.
She obtained a test statistic of 1.83 with a p-value of 0.033.
Do the results of the significance test agree with the results of Clarke’s confidence interval in part (a)?
Explain your reasoning.

What is the mean, median, and mode of 45, 50, 47, 52, 53, 45, 51?

A researcher obtains a Pearson correlation of r = 0.43 for a sample of n = 20 participants. For a two-tailed test, which of the following accurately describes the significance of the correlation?​
a.)The correlation is not significant with either $\mathrm{\alpha <!--\; \alpha \; -->}$ = .05 or $\mathrm{\alpha <!--\; \alpha \; -->}$ = .01.
b.)​The correlation is significant with either $\mathrm{\alpha <!--\; \alpha \; -->}$ = .05 or $\mathrm{\alpha <!--\; \alpha \; -->}$ = .01.
c.)The correlation is equal to zero.
d.)The correlation is significant with $\mathrm{\alpha <!--\; \alpha \; -->}$ = .05 but not with $\mathrm{\alpha <!--\; \alpha \; -->}$ = .01.