Improve Your Understanding of College Level Statistics Problems

If a chi-square test of Independence ends in a non-significant result it means that the variables depend on each other.
True or False

How to interpret the result according to the table below?
$\begin{array}{cc}& \text{Chi-square Tests}\\ & \begin{array}{|cccc|}\hline & \text{ Value }& \text{ df }& \text{ Asymptotic Significance (2-sided) }\\ \text{ Pearson Chi-Square }& & 2& .034\\ \text{ Likelihood Ratio }& & 2& .033\\ \text{ Linear-by-Linear Association }& & 1& .380\\ \text{ N of Valid Cases }& & & \\ \hline\end{array}\end{array}$
a) Since 0.034<0.05, the Ho hypothesis is rejected
b) Since 0.033<0.05, the Ho hypothesis is rejected
c) Since 0.034<0.5, the Ho hypothesis is rejected
d) Since 0.380>0.05, the Ho hypothesis is accepted
e) Since 0.033<0.5, the Ho hypothesis is rejected

A sample is used both to construct a 95% confidence interval for a population proportion p and to run a significance test with null hypothesis $H_0:P=0.07$ and significance level $\mathrm{\alpha <!--\; \alpha \; -->}=0.05$.
Is it possible that $p=0.7$ falls outside of the 95% confidence itnerval, yet $H_0$ is not rejected?
I'm assuming that since a two-sided test at significance level α gives roughly the same result as a $100(1-<!--\; -\; -->\mathrm{\alpha <!--\; \alpha \; -->})\mathrm{\%<!--\; \%\; -->}$ confidence interval, this is not possible.
Also, in terms of the sample size required for a significance test, does it just have to statisfy the normal condition $n{p}_0\ge <!--\; \ge \; -->10$ and $n(1-<!--\; -\; -->p_0)\ge <!--\; \ge \; -->10$?

The occurrence of false positives [in some experiment] is 40%
What does this mean? Does it mean that 40% of positives are false? Or 40% of all tested patients are specifically both false and positive? Or say 40% of those that are false are positive?

How does American obesity compare to obesity worldwide?

A new cream that advertises 52% of women over the age of 50 will experience skin improvement from using their product. A study was conducted in which a sample of 170 women over the age of 50 used the new cream for 6 months. Of those 170 women, 76 of them reported skin improvement (as judged by a dermatologist). Is this evidence that the cream will improve the skin of less than 52 % of women over the age of 50? Perform a hypothesis test using the 0.05 level of significance.
Calculate the standard deviation for the sample proportion

What is the rationale behind ROC curves?
I am not sure how ROC curves work. I see that the X-Axis is the false positive rate while the Y axis is the true positive rate.
1) I don't understand how for a given statistical learning model, you could have the true positive and false positive rate to vary from 0 to
1. Are you changing parameters in the model to make it so?
2) What about true negatives and false negatives? How are these represented in the curve?

How does hypothesis testing related to confidence intervals?

How can the median be the greatest number in a set?

How do you find the first and third quartiles of 14, 4, 13, 18, 8, 15, 8, 3, 11, 8?

What is the difference between median and mean?

I have a problem.
I want to find if there is a statististical significant difference between two groups which I have put through a practical test.
The outcome was 1 or 0 (success/fail).
Group 1 was subject to extra training material before the "exam". Group 2 was not.
Each group had one attempt in three different scenarios: easy - medium - hard.
Lets take the medium scenario as example:
The number of participants in each group is 10
Group 1 scored: 7 successes, 3 failures --> success rate 0.70 Group 2 scored: 2 successes, 8 failures --> success rate 0.22
Now what I have done, is try to use the Z-test for two population proportions in an attempt to reject the null hypothesis, buuuuut, I think my sample size is to small for the z-test?
Im not sure what test to use at this point, can anyone give me a clue?
Thanks for reading

My brain doesn't work right anymore.
I have a traumatic brain injury and one of the many areas I have trouble with is math - an area that was once naturally easy and intuitive for me. I remember standard deviations, generally, and have read up on what they indicate broadly and how they are reached. When I was tested for my brain injury, there were some tests in which my abilities were worse than 3 SDs below the mean! [I am a professional with a master's degree, and was one of those students who aced everything with ease. This information on my current condition, however, matches pretty well, I think, my current state, post injury. I struggle greatly with many varied kinds of things.]
So far as I can recall, 1 SD is a significant enough deviation that I don't recall anyone ever talking about anything greater than 2 SDs above or below the mean. So my question is how "bad," may I ask, are scores such as: 0.7 SD below the mean, 1.7 SD below the mean, and a score of 3.2 SDs below the mean. I am asking for general but concrete advice on how to interpret these numbers generally, of course, and hold no one there responsible. I just want to better understand the significance of my scores, mathematically. The brain injury staff mince and avoid answering my questions about this, so maybe someone here will help me understand this information.

If wave packets spread, why don't objects disappear?
If you have an electron moving in empty space, it will be represented by a wave packet. But packets can spread over time, that is, their width increases, with it's uncertainty in position increasing. Now, if I throw a basketball, why doesn't the basketball's packet spread as well? Wouldn't that cause its uncertainty in position to increase so much to the point it disappears?
EDIT: I realize I wasn't clear what I meant by disappear. Basically, suppose the wave packet is spread over the entire Solar System. Your field of vision covers only an extremely tiny part of the Solar System. Therefore, the probability that you will find the basketball that you threw in your field of vision is very small.

Why is standard deviation calculated differently for finding Z scores and confidence intervals?
Suppose that as a personnel director, you want to test the perception of fairness of two methods of performance evaluation. 63 of 78 employees rated Method 1 as fair. 49 of 82 rated Method 2 as fair. A 99% confidence interval for $p_1-<!--\; -\; -->p_2$ (where $p_1$ = 63/78 and $p_2$ = 49/82) is as follows:
$p_1-<!--\; -\; -->p_2\pm <!--\; \pm \; -->2.58\sqrt{\frac{p_1(1-<!--\; -\; -->p_1)}{n_1}+\frac{p_2(1-<!--\; -\; -->p_2)}{n_2}}$
$0.029⩽<!--\; ⩽\; -->p_1-<!--\; -\; -->p_2⩽<!--\; ⩽\; -->0.391$
At the 0.01 level of significance the Z score is
$Z=\sqrt{\frac{p(1-<!--\; -\; -->p)}{n_1}+\frac{p(1-<!--\; -\; -->p)}{n_2}}$
(where $(x_1+x_2)/(n_1+n_2)=0.70$ but sometimes a different formula $p=(p_1+p_2)/2$ is also used)
Z = 2.90
Both tests indicate that there is evidence of a difference.
But you could also find the Z score using the standard deviation formula in the first method to be 2.993. Why are the Z scores different? Where do the formulas for finding the standard deviation come from?

You will be required to select two different sampling methods, each method you will choose a random sample of 20 sentences from throughout the book by following the method you selected and count how many words contained in each sentence that was selected per your (random) sampling method. Use two different sampling methods, chosen from simple random sampling, stratified sampling, cluster sampling, or systematic sampling.
1. Explain which of your two methods was easier to use and why.
2. Of the two methods, you did not use, explain which of them would have been the most difficult to use and why.
3. Do you think either of your methods produced biased results? Explain.

I'm trying to understand the functional form of the Generalized Pareto distribution (GPD). In the "Definition" section location parameter μ does not appear in the function, whilst in the "Characterization" section it does.
1. how the GPD form can be reconciled with GPD form presented in other sources. Derivation of the GPD from the Generalized Extreme Value (GEV) distribution would suggest that $\frac{x-<!--\; -\; -->\mathrm{\mu <!--\; \mu \; -->}}{\mathrm{\sigma <!--\; \sigma \; -->}}$ should not appear in that form in the expression of the GPD.

Explain what chi-squared test of independence is. What is the underlying assumption? What is the test statistic and what is its asymptotic distribution? How the hypothesis is formulated and what are the expected outcomes? What is the criterion for rejecting the null hypothesis? How is the p value evaluated? Provide a hypothetical formulation of test of independence.