Learn Significance Test with Examples & Questions

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

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?

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?

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 $H_0$?

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

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.)

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?

Summarize the four types of a hypothesis test. Be sure to include any formula and important tips and terminology.
1. Hypothesis
2. Model:
3. Mechanics:
4. Conclusion:

Suppose we are testing 20 drugs and use a significance value of 0.01 in each test. Suppose at least one out of the 20 tests rejects the null hypothesis. What is the upper bound for the probability that declaring the corresponding drug useful is a type I error (i.e. reject null hypothesis when one should not)?
I am confused on where to start. Any help is appreciated

According to shopper data, 97% of households purchase toilet paper. An analyst believes this is too low. To investigate, the analyst selects a random sample of 500 households and finds that 98% of them purchase toilet paper. We found that we do not have convincing evidence at the α = 0.05 level to conclude that the true proportion of all households that purchase toilet paper differs from 0.97.
The power of this test to reject p = 0.97 if 0.98 is true is 0.21. If the analyst were to decrease the significance level of the test, how would the power be affected?
It would increase.
It would decrease.
It would remain the same.
It would double.

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?

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

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 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.

What is the p value if the test static is 1.319 and the significance level is 0.01?

A colleague of yours is completing a final report on the causes of the frequency of cyberbullying. In this report, she is asked to identify the causes that most strongly impacted the frequency of cyberbullying. She conducts an OLS regression.
What statistic do you advise her to use in her discussion? Why?