Improve Your Understanding of College Level Statistics Problems

Suppose a coin with probability $p$ of heads is repeatedly flipped until $10$ heads appear. Call this number $X$. Find the p-value of $H_0:p=0.5$ versus the alternate $H_a:p\ne 0.5$ using the test statistic $|X-20|$ if $X=27$.

I have 12 balls with mean weight of 500g, and S.D 25g. I have 5 balls with mean weight 200g and S.D 10g. If I combine all of them, is the mean just $\frac{8000}{17}$ i.e.
$8000=500\cdot <!--\; \cdot \; -->12+200\cdot <!--\; \cdot \; -->10$
and divided by 17 since there are 17 balls. What is the new S.D.? Is it just
$\sqrt{25\cdot <!--\; \cdot \; -->12+10\cdot <!--\; \cdot \; -->5}$

Daily anxiety was measured on a scale from 1 (not at all anxious) to 5 (very anxious) in a random sample of 2000 city dwellers from across the U.S. They found that M = 4.13, 95% CIs [4.06, 4.20].
How would you interpret these results? What conclusions would you draw about the precision of the point estimate?
What statistical decision would have been made in this scenario if the researchers employed Null Hypothesis Significance Testing instead of the New Stats?

How do you find the median of 2.8, 3.9, 5, 7.7?

Calculate the Pearson Correlation Coefficient for these two variables and b) test whether the correlation is significantly different from 0. Run the test at a 5% level of significance. Give each of the following for part b to receive full credit:
1) the appropriate null and alternative hypotheses;
2) the appropriate test;
3) the decision rule;
4) the calculation of the test statistic; and
5) your conclusion including a comparison to alpha or the critical value. You MUST show your work to receive full credit. Partial credit is available.
ID Length of Stay (x) Total Charge (y)
1)5 6
2)7 5
3)9 7.2
4)10 8
5)12 9.4
6)15 7.9

What is the average of the 3 temperatures -50°C, -20°C, and 15°C?

One-sided significance test: essay - key question
For an essay on the one-sided significance test, I have considered the following structure:
First, generally describe the one-sided test (such as an introduction, setting up the hypotheses, decision rule), then describe the errors more accurately, then write an example and finally make a comparison between the two-sided significance test and the alternative test with the one-sided significance test.
Is that a good structure or could we improve it? The structure should be such that the level is always higher, it should be divided into three requirement areas.
In addition, I am looking for a key question. I thought something like, "How do we know how much we're wrong if we don't reject $H_0$?", and describe in the answer the probability of error? Or is that wrong? Or is that not suitable as a key question?

Statistical significance: Test for different (small) sample sizes?
I recently finished my simulations which I now need to statistically analyse. I am comparing mean firing rates (FR) of neurons in the same population (Pop1) before and after I change the connectivity probability (Pconn).
I have n=140 observations/simulations for the network with the natural connectivity state, but I have only n=5 observations/simulations for the network with varied connectivity.
(I am arbitrarily changing the Pconn, while measuring the FR of a population before and after the change. n=140 observations correspond to BEFORE and n=5 observations correspond to AFTER the change)
I am quite unsure in which statistical test should I use to determine the significance of my results, since the sizes of samples vary so much. Can anyone suggest a solution?
Thanks in advance

A store-bought pregnancy test produces a correct result 99% of the time when a woman is actually pregnant but is only 97% accurate when a woman is not pregnant. What are the chances of a false positive and a false negative respectively?
So I have tried to solve it and is the false positive 3%? SO, false positive is when the result shows positive despite her not being pregnant. And for her not being pregnant , the test is 97% accurate. Then, the false positive should be 100-97%?

Simple random sampling:
is used when there is a need to emphasize one or more types of population items.
requires both input and output parameters to be set when using a random number generator.
is generally used with replacement sampling.
is a probabilistic sampling method.

How to calculate the false negative, the true negative, the false positive and the true positive to form a confusion matrix. A numeric example would clear all doubts I think.

For the set (3, 4, 5, 8, x), the mean, the median and the mode all have the same value. What is the value of x?

Wire manufactured by a company is tested for strength. The test gives a correct positive result with a probability of 0.85 when the wire is strong, but gives an incorrect positive result (false positive) with a probability of 0.04 when in fact the wire is not strong.
If 98% of the wires are strong, and a wire chosen at random fails the test, what is the probability it really is not strong enough?

What is the most common measure of central tendency?

How can an electron be fired at a target when uncertainty principle says it will spread out around axis of motion?
Consider an electron fired at a target. Taking the axis of motion to be $x$, and position to be $(x,y,z)$ then
$\mathrm{\Delta <!--\; \Delta \; -->}y=\mathrm{\Delta <!--\; \Delta \; -->}z=0$
Therefore by the uncertainty principle
$\mathrm{\Delta <!--\; \Delta \; -->}{p}_y=\mathrm{\Delta <!--\; \Delta \; -->}{p}_z=\mathrm{\infty <!--\; \infty \; -->}$
So the electron will spread out in $y$ and $z$ and never hit its target (unless that target was very close i.e. on the same scale as the initial effective size of the electron). How can it hit a target in reality?
I realise there is a possible duplicate to this question: How can particles travel in a straight line? The accepted answer for the duplicate is "because $\mathrm{\hslash <!--\; \hslash \; -->}$ is really small so the particles don't spread out much before hitting the target". I'm not sure this stands though. The time-dependent Schrodinger equation is:
$i\mathrm{\hslash <!--\; \hslash \; -->}\frac{\mathrm{\delta <!--\; \delta \; -->}}{\mathrm{\delta <!--\; \delta \; -->}t}\mathrm{\psi <!--\; \psi \; -->}(r,t)=[-<!--\; -\; -->\frac{{\mathrm{\hslash <!--\; \hslash \; -->}}^2}{2\mathrm{\mu <!--\; \mu \; -->}}{\mathrm{\nabla <!--\; \nabla \; -->}}^2+V(r,t)]\mathrm{\psi <!--\; \psi \; -->}(r,t)$
In the absence of external potential $V$ and dividing both sides by $\mathrm{\hslash <!--\; \hslash \; -->}$
$i\frac{\mathrm{\delta <!--\; \delta \; -->}}{\mathrm{\delta <!--\; \delta \; -->}t}\mathrm{\psi <!--\; \psi \; -->}(r,t)=-<!--\; -\; -->\frac{\mathrm{\hslash <!--\; \hslash \; -->}}{2\mathrm{\mu <!--\; \mu \; -->}}{\mathrm{\nabla <!--\; \nabla \; -->}}^2\mathrm{\psi <!--\; \psi \; -->}(r,t)$
So if we change $\mathrm{\hslash <!--\; \hslash \; -->}$ (or $\mathrm{\mu <!--\; \mu \; -->}$), sure the particle will spread out less as a function of time, but won't it also move slower in exactly the same proportion to its lack of spreading out, so it still won't hit the target?

A researcher suspects that the mean birth weights of babies whose mothers did not see a doctor before delivery is less than 3000 grams. The researcher states the hypotheses as H:x = 3000 grams H, x < 3000 grams.
These hypotheses are incorrect. What are the appropriate hypotheses for performing this significance test?
$H_0:\stackrel{¯<!--\; ¯\; -->}{x}>3000\text{ grams; }{H}_a:\stackrel{¯<!--\; ¯\; -->}{x}<3000\text{ grams, where }\stackrel{¯<!--\; ¯\; -->}{x}$ = the mean birth weights of the sample of babies whose mothers did not see a doctor before delivery.
$H_0:\stackrel{¯<!--\; ¯\; -->}{x}<3000\text{ grams; }{H}_a:\stackrel{¯<!--\; ¯\; -->}{x}=3000\text{ grams, where }\stackrel{¯<!--\; ¯\; -->}{x}$ = the mean birth weights of the sample of babies whose mothers did not see a doctor before delivery.
$H_0:\mathrm{\mu <!--\; \mu \; -->}=3000\text{ grams; }{H}_a:\mathrm{\mu <!--\; \mu \; -->}\ne <!--\; \ne \; -->3000\text{ grams, where }\mathrm{\mu <!--\; \mu \; -->}$ = the true mean birth weights of all babies whose mothers did not see a doctor before delivery.
$H_0:\mathrm{\mu <!--\; \mu \; -->}=3000\text{ grams; }{H}_a:\mathrm{\mu <!--\; \mu \; -->}<3000\text{ grams, where }\mathrm{\mu <!--\; \mu \; -->}$ = the true mean birth weights of all babies whose mothers did not see a doctor before delivery.
$H_0:\stackrel{¯<!--\; ¯\; -->}{x}=3000\text{ grams; }{H}_a:\stackrel{¯<!--\; ¯\; -->}{x}\ne <!--\; \ne \; -->3000\text{ grams, where }\stackrel{¯<!--\; ¯\; -->}{x}$ = the mean birth weights of the sample of babies whose mothers did not see a doctor before delivery.

Christine's test scores are 89, 94, 88, 75, and 94. What does Christine need to score on the next test to have a median score of 90?

The shape characteristics of gravitational wells given different masses and spread of objects
I am curious as to research that calculates the shape of gravitational wells, and their limits, and affect on time, for different masses and spread of mass. The actual question is st the end.
For example: When the sun becomes a red giant the Earth's orbit is supposed to move out with the redistribution of mass, but what is the science behind this? Re-edit: Which I think was explained as the spreading density of the suns mass (it was a "TV" program. Another example, is it is said that the termination of the Sun's gravity field is 1.5 light years out. I am unaware of any distance studies to measure shape to specifically prove the theory.
The Question: Explanation of how the shape, time over distance, and extent of a gravitational well changes with the density of the mass, and the science behind this please? Re-edit: The curvature, research to verify theory, and actual simple graphical description of how the physical shape responds and changes based on contributing mass distribution. Say, does a more dense matter object cause the gravity well to more tightly curve to the surface of the matter, than to the surface of a cloud of gas of equal weight but magnitudes bigger. How does that look in physical shape over distance, how does the field terminate in shape. I'm interested in observational research on the profile. For instance does it continue the same decay equation or does it change/flatten out etc to a different equation at distance. This is more looking at verification/explanation of the conventional versus deviations. If we can only say so much at X distance from verified studies, that would be appreciated?
As we know there has been speculation based on deviations in observation of gravity on grander scales, such as across the galaxy. But I do not wish to go into those hypothesises, only the limits of what we have verified we know, which is a good starting point onto looking into this further.

What is the mean, median, and mode of 14, 15, 22, 15, 2, 16, 17, 13?