Improve Your Understanding of College Level Statistics Problems

How do ripples form and why do they spread out?
When I throw a rock in the water, why does only a small circular ring around the rock rises instead of the whole water body, and why does it fall outwards and not inwards or why fall out in any direction instead of just going back down like ball which after getting thrown up falls back to its initial position?

The subscribers to a magazine are numbered. Then, a sample of these people is selected using random numbers. What is the sampling method used in this study?
1. Simple random sampling
2. Systematic sampling
3. Cluster sampling
4. Stratified sampling

In a multiple regression, which of the following tests can be used to determine if the relationship between the dependent variable and the set of independent variables is significant?
The Coefficient Of Determination R-Squared for each coefficient of the X variables & an F-
Statistic Test for the overall regression
1. F-Statistic tests for each coefficient & a t-test for the overall regression
2. The Coefficient Of Determination R-Squared and/or an F-Statistic Test for the overall regression & t-tests for each coefficient of the X variables
3. t-tests for each coefficient and for the overall regression

Could Miller-Rabin primality test give false negative, for example when test prime number and gives it as composite?

What is the mean, median, and mode of 14, 15, 15, 16, 17, 20, 21, 21, 21?

Spread of the energy levels and sharp energy eigenvalues of the Schrodinger equation of the H-atom
Solving the Schroedinger equation for the H-atom (or any other system, say a particle in a box, or harmonic oscillator or anything), we obtain the energy eigenvalues are sharp with no spread. However, in reality, there must be a spread for the excited states which follows from uncertainty principle, and the spread of an energy level $\mathrm{\Delta <!--\; \Delta \; -->}E\sim <!--\; \sim \; -->\frac{\mathrm{\hslash <!--\; \hslash \; -->}}{\mathrm{\tau <!--\; \tau \; -->}}$ where $\mathrm{\tau <!--\; \tau \; -->}$ is the lifetime of the particle in that state. Why is this effect not captured in the calculation of energy levels by solving the Schroedinger equation?
Is there any other way to compute the spread other than using the uncertainty principle and match them?

Organizations typically use both quantitative and qualitative risk analysis techniques when analyzing the risk to the organization and determining the appropriate counter-measures. Compare and contrast quantitative and qualitative risk analysis. Describe a situation when a qualitative risk analysis method is most appropriate, and describe a situation when a quantitative risk analysis method is most appropriate.

How do you find the mean of 16, 19, 17, 18, 15, 14, 31, 31, 41?

Data can be described as being quantitative or qualitative. If qualitative, it can be described as nominal or ordinal. In order to be ordinal, you must be able to order the data in a meaningful way; therefore, being able to order pins from least to greatest does not make them ordinal, I know.
I am wondering though, why can't pins be described as ordinal given they can be ordered in terms of complexity, by the number of repeating digits, etc? EDIT: In this case, the order would be meaningful in a study about links between simple pin numbers and stolen credit cards for instance.

I am doing a two sample hypothesis problem that goes like this:
Research has shown that good hip range of motion and strength in throwing athletes results in improved performance and decreased body stress. The article “Functional Hip Characteristics of Baseball Pitchers and Position Players” (Am. J. Sport. Med., 2010: 383–388) reported on a study involving samples of 40 professional pitchers and 40 professional position players. For the pitchers, the sample mean trail leg total arc of motion (degrees) was 75.6 with a sample standard deviation of 5.9, whereas the sample mean and sample standard deviation for position players were 79.6 and 7.6, respectively. Assuming normality, test appropriate hypotheses to decide whether true average range of motion for the pitchers is less than that for the position players (as hypothesized by the investigators).
Using the information from the question I was able to get a t value of −2.63, from what I understand from here I'm supposed to compare this -2.63 to a critical value I get from the t table, or I can get find the p-value (I got 0.0043 from the normal table) and compare it to alpha which is the significance level. For both of these I don't really understand on which situations I'm supposed to be using for which method, It would be great if someone could explain this to me too. But regardless the main issue I have is that both of these requires a significance level to figure out, which wasn't given to me. Since I didn't know how to proceed I went to look at the answer to this question, which said this:
Because the one-tailed P-value is .005<.01, conclude at the .01 level that the difference is as stated.
The alternative hypothesis was chosen because the P value was lower than the significance level of .01 but if they had just chosen a smaller significance level(e.g. 0.001), then the alternative hypothesis would have been rejected
>So here my question is how did they know that a significance level of .01 was the level they are supposed to compare to?
Did they just choose it because there was no significance level given in the question?

Why is the mode not used very much as a measure of central tendency?

State the significance test to be used in each of the following:
a) Want to test for association between garden (male/female) and farming type (commercial/subsistence)
b) Want to compare maize yield among three fertilizer types (Urea,CAN and Manure)
c) Want to compare maize yield between Urea and CAN

$H_0:\mathrm{\mu <!--\; \mu \; -->}=0$ against$H_A:\mathrm{\mu <!--\; \mu \; -->}>0.$
Had the following question on my exam today and I'm just wondering if I did this correctly.
From a normally distributed population with mean $\mathrm{\mu <!--\; \mu \; -->}$ and variance ${\mathrm{\sigma <!--\; \sigma \; -->}}^2$ a sample has been drawn:
$\mathbf{X}=(0.05,4.35,-<!--\; -\; -->0.48,-<!--\; -\; -->0.63,1.17,2.01).$.
a) Test $H_0:\mathrm{\mu <!--\; \mu \; -->}=0$ against $H_A:\mathrm{\mu <!--\; \mu \; -->}>0$ on a 5% significance level under the assumption that σ2 is unknown.
c) Do the same test, given that $\mathrm{\sigma <!--\; \sigma \; -->}=\mathrm{1.5.}$.
Solution: I know that $H_0$ is to be rejected if $T\ge <!--\; \ge \; -->c,$, where T is the teststatistic and $F_{t_{n-<!--\; -\; -->1}}=1-<!--\; -\; -->\mathrm{\alpha <!--\; \alpha \; -->},$, where $\mathrm{\alpha <!--\; \alpha \; -->}$ is the significance level.
So lets first compute them for a), where we need to use a t-distribution since σ is unknown, so we have that $c=F_{t_5}^{-<!--\; -\; -->1}(0.95)\approx <!--\; \approx \; -->\mathrm{2.015.}$. Also, the estimator for ${\mathrm{\sigma <!--\; \sigma \; -->}}^2$ is
$s^2=\frac{1}{n-<!--\; -\; -->1}(\underset{k=1}{\overset{n}{\sum <!--\; \sum \; -->}}{X}_k^2-<!--\; -\; -->\frac{1}{n}{\left(\underset{k=1}{\overset{n}{\sum <!--\; \sum \; -->}}{X}_k\right)}^2)=\frac{1}{5}(24.961-<!--\; -\; -->\frac{1}{6}(41.861))=3.405,$
so, $s=\sqrt{3.405}=\mathrm{1.845.}$. We also have that $\stackrel{¯<!--\; ¯\; -->}{X}=6.47/6=\mathrm{1.078.}$.
The teststatistic is
$T=\frac{\stackrel{¯<!--\; ¯\; -->}{X}-<!--\; -\; -->{\mathrm{\mu <!--\; \mu \; -->}}_0}{s/\sqrt{n}}=\frac{1.078-<!--\; -\; -->0}{1.845/\sqrt{6}}=\frac{1.078}{0.753}=1.43\le <!--\; \le \; -->c,$
which means that we can not reject H0.
For b) we do an identical approach but wtith $s=\mathrm{\sigma <!--\; \sigma \; -->}=1.5$ and we use a z-test instead. Here we have that $c={\mathrm{\Phi <!--\; \Phi \; -->}}^{-<!--\; -\; -->1}(0.95)=\mathrm{1.644.}$ The teststatistic in this case is
$Z=\frac{\stackrel{¯<!--\; ¯\; -->}{X}-<!--\; -\; -->{\mathrm{\mu <!--\; \mu \; -->}}_0}{\mathrm{\sigma <!--\; \sigma \; -->}/\sqrt{n}}=\frac{1.078}{1.5/\sqrt{6}}=1.76\ge <!--\; \ge \; -->c,$
thus we reject $H_0.$.
Is this correct?
What my professor does in the solutions is he just makes a 95% confidenceinterval and just arrives to the same conclusions as I've done. Is his method correct without using the teststatistics?

The mean of eight numbers is 41. The mean of two of the numbers is 29. What is the mean of the other six numbers?

Q:/ You examine the correlation between two continuous variables X and Y, and find that this correlation is close to zero. A hypothesis test on this correlation would have you retain the null hypothesis and conclude there is no correlation between X and Y. This means that there is no relationship between X that can be used in a predictive model. Is this True/False?
A:/ Is this true or false and why?

In statistics, why do you reject the null hypothesis when the p-value is less than the alpha value (the level of significance)
This is a question that I've always wondered in statistics, but never had the guts to ask the professor. The professor would say that if the p-value is less than or equal to the level of significance (denoted by alpha) we reject the null hypothesis because the test statistic falls in the rejection region. When I first learned this, I did not understand why were comparing the p values to the alpha values. After all, the alpha values were brought in arbitrarily. What is the reason for comparing them to the alpha values and where do the alpha values of 0.05 and 0.10 come from? Why does the statement $p_{\text{value}}\le <!--\; \le \; -->\mathrm{\alpha <!--\; \alpha \; -->}$ allow you to reject $H_0$?

The purpose of measures of central tendency is to describe the variability within a data set. Is this true or false? Why?

How do you find the mean, median and mode of -2, 2, 2, 3, 4, 4, 4?

Intend to calculate the first and third quartiles of a lognormal distribution with mu and sigma (two lognormal parameters) equal to -0.33217492 and 0.6065058. The expected value and the standard deviation gave me 0.8622153003153145 and 0.6191622375133721.
I intend to calculate the first and third quartiles and found this formula:
$F^{-<!--\; -\; -->1}(p)=exp(\mathrm{\mu <!--\; \mu \; -->}+\mathrm{\sigma <!--\; \sigma \; -->}{\mathrm{\Phi <!--\; \Phi \; -->}}^{-<!--\; -\; -->1}(p))\text{ with }0<p<1$
to calculate the quartiles but I didn't quite understand how to apply it and how to calculate the p-value. How can I find out the first and third quartiles of this distribution?