Improve Your Understanding of College Level Statistics Problems

What are some examples when practical significance would outweigh statistical significance?

Three numbers have a mean of 14. If two of the numbers are 12 and 17, what is the other number?

Compare and contrast stratified random sampling with proportionate stratified random sampling.

You are analyzing results for cupcake restaurant. The study aims to determine if the cupcake sales have increased. What would be the most appropriate significance level for hypothesis tests for this type of study?
1.0
0.05
0.01
0.001

Calculation of $p$-value: Fabric water properties, $H_0:\mathrm{\mu <!--\; \mu \; -->}\le <!--\; \le \; -->55\mathrm{\%<!--\; \%\; -->}$ versus $H_A:\mathrm{\mu <!--\; \mu \; -->}>55\mathrm{\%<!--\; \%\; -->}$;
$n=15,\stackrel{¯<!--\; ¯\; -->}{x}=59.81\mathrm{\%<!--\; \%\; -->},s=4.94\mathrm{\%<!--\; \%\; -->}$
I understand that the hypothesized mean is $55$ samples mean is $59.81$ sample standard deviation is $4.94$ sample size is $15$. And I use the $t$-test, I have gotten $3.77106$. In this circumstance, how can I get the $p$-value, and what is different between critical value and $p$-value?

What is the mode of 1, 4, 7, 2, 7, 3, 8, 4, 2, 4, 9, 5, 2, 1?

What do quantitative and qualitative measurements mean in science?

Basically, I am having a hard time understanding the difference between the p-value, the significance level and the size of a test.
In doing questions about hypothesis testing, I am somehow able to get the right answers when asked to derive these quantities. But at the end of the day, it seems to me that all three of them represent the probability of rejecting the null hypothesis?
Any enlightenment would be much appreciated.

Is there a simple test for uniform distributions?
I have a function that (more or less) is supposed to select a small number m of random numbers from the range [1,n] (for some n≫m) and I need to test that it work. Is there an easy to implement test that will give good confidence that it is working correctly? (For reference the full on chi squared test is not simple enough.)
Edit:
m in [5,500], n is small enough I can use normal int/float types for most math.I can get as many sets as I'm willing to wait for (>100s per second)The reason I'm looking for something simple is that the code I'm checking isn't that complicated. If the test code is more complex than the code under test, then the potential for the test being wrong becomes a real problem.
Edit 2:
I took another look at Chi-squared and it's not as bad as I remembered... as long as I have a fixed number of buckets and a fixed significance threshold. (Or a canned CDF, and I don't think I do.)

Use Bayes' Theorem to prove that the rate of false positives is accurate (86%) in the following passage.
What I have done so far is list the following
$P(A)=0.99$
$P(B)=0.05$
As Bayes' Theorem is as follows
$P(A|B)=\frac{P(B|A)P(A)}{P(B)}$
I need $P(B|A)$. I'm stuck on how to find this. Help would be greatly appreciated.
Is it possible to prove this using Bayes'? If not, is there some other sort of mathematics that I can use here?
Roughly 1 per cent of the population suffer from mild cognitive impairment, which might, but doesn’t always, lead to dementia. Suppose that the test is quite a good one, in the sense that 95 per cent of the time it gives the right (negative) answer for people who are free of the condition. That means that 5 per cent of the people who don’t have cognitive impairment will test, falsely, as positive. That doesn’t sound bad. It’s directly analogous to tests of significance which will give 5 per cent of false positives when there is no real effect, if we use a p-value of less than 5 per cent to mean ‘statistically significant’.
But in fact the screening test is not good – it’s actually appallingly bad, because 86 per cent, not 5 per cent, of all positive tests are false positives. So only 14 per cent of positive tests are correct. This happens because most people don’t have the condition, and so the false positives from these people (5 per cent of 99 per cent of the people), outweigh the number of true positives that arise from the much smaller number of people who have the condition (80 per cent of 1 per cent of the people, if we assume 80 per cent of people with the disease are detected successfully).

1. Twenty schools in a province are selected for external moderation. This is an example of:
A. Cluster sampling
B. Stratified sampling
C. Random sampling
D. Systematic sampling

Light of laser doesn't spread?
How exactly can a combination of monochromaticity and coherent light make a light not want to spread out? When it's all one wavelength and it's all amplified it should make the light brighter and thus more energetic. I can't seem to understand why the light doesn't spread normally.

A company produces millions of 1-pound packages of bacon every week. Company specifications allow for no more than 3 percent of the 1-pound packages to be underweight. To investigate compliance with the specifications, the company’s quality control manager selected a random sample of 1,000 packages produced in one week and found 40 packages, or 4 percent, to be underweight.
Assuming all conditions for inference are met, do the data provide convincing statistical evidence at the significance level of $\mathrm{\alpha <!--\; \alpha \; -->}=0.05$ that more than 3 percent of all the packages produced in one week are underweight?
(A) Yes, because the sample estimate of 0.04 is greater than the company specification of 0.03.
(B) Yes, because the p-value of 0.032 is less than the significance level of 0.05.
(C) Yes, because the p-value of 0.064 is greater than the significance level of 0.05.
(D) No, because the p-value of 0.032 is less than the significance level of 0.05.
(E) No, because the p-value of 0.064 is greater than the significance level of 0.05.
The answer is (B) and I was trying to understand why. My calculation was:
$H_a:p>0.03$
$z=\frac{\stackrel{^<!--\; ^\; -->}{p}-<!--\; -\; -->p_0}{\sqrt{\frac{p_0(1-<!--\; -\; -->p_0)}{n}}}$
$z=\frac{0.04-<!--\; -\; -->0.03}{\sqrt{\frac{(0.03)(0.07)}{1000}}}$
But I get a ridiculously high number. I'm confused about how to get the p-value in this case.
A two-sided t-test for a population mean is conducted of the null hypothesis $H_0:\mathrm{\mu <!--\; \mu \; -->}=100$. If a 90 percent t-interval constructed from the same sample data contains the value of 100, which of the following can be concluded about the test at a significance level of $\mathrm{\alpha <!--\; \alpha \; -->}=0.10$?
(A) The p-value is less than 0.10, and $H_0$ should be rejected.
(B) The p-value is less than 0.10, and $H_0$ should not be rejected.
(C) The p-value is greater than 0.10, and $H_0$ should be rejected.
(D) The p-value is greater than 0.10, and $H_0$ should not be rejected.
(E) There is not enough information given to make a conclusion about the p-value and $H_0$.
Here the answer is D, but again I am confused. How can I find the p-value in this case?

The summary of that regression shows that one of my variable has a big p-value (0.705). Should I include that variable when writing the the y hat equation?

Over 6 days, Dan jogged 7.5 miles, 6 miles, 3 miles, 3 miles, 5.5 miles, and 5 miles. What is the mean distance that Dan jogged each day?

What is the range of the data: 0.167, 0.118, 0.541, 0.427, 0.65, 0.321?

The following risks are given:
1. $X_3$
2. $X_2\sim <!--\; \sim \; -->Bin(15,\frac{1}{3})$
3. $X_3\sim <!--\; \sim \; -->15I,\text{where}I\sim <!--\; \sim \; -->Bernoulli(\frac{1}{3}$
Do any docaiou makers with (increasing) ulility function agree aboul prceferring risk$X_1$ to $X_2$? given an exponential utility function which risk does a decision makerpreder: $X_1$ or $X_3$?
The last task is related to eponential ordering of risk denoted by $\le <!--\; \le \; -->e$.

Suppose that for every α∈(0,1) we have a size α test with rejection region Rα. Then,
$\text{p-value}=inf\{\mathrm{\alpha <!--\; \alpha \; -->}:T(X^n)\in <!--\; \in \; -->R_{\mathrm{\alpha <!--\; \alpha \; -->}}\}$
That is, the p-value is the smallest level at which we can reject $H_0$.
what is this exponent $n$. What does $X^n$ mean?

What is the geometric mean between 3 and 18?