Improve Your Understanding of College Level Statistics Problems

A representative of a potential supplier claims that at least 90% of the equipment manufactured and sold by the company has no defects. You have obtained a sample of equipment usage records from a firm that uses these machines. Of 121 machines in the records, 105 machines have no recorded defects. What is the probability of this event if the supplier’s claim is correct (i.e. The p -value). On the basis of this data, do you believe the representative’s claim?
My thoughts:
The significance level of the test: α = 0.1 (10%)
The percentage of 121 machines that have no recorded defects is 86.7%.
I am not sure how to arrive at the p-value from this data.

What's the formula to find the average of a set of numbers?

I'm having problems to understand the definition of the level of significance α. I thought I knew what $\mathrm{\alpha <!--\; \alpha \; -->}$ is but I realized I don't.When I stated to study statistics by myself I read this introductory book and everything was fine, the definition is very clear. He says on page 290:
You’re probably wondering, how small does a p-value have to be for us to achieve statistical significance? If we agree that a p-value of 0.0001 is clearly statistically significant and a p-value of 0.50 is not, there must be some point between 0.0001 and 0.50 where we cross the threshold between statistical significance and random chance. That point, measuring when something becomes rare enough to be called “unusual,” might vary a lot from person to person. We should agree in advance on a reasonable cutoff point.
Statisticians call this cutoff point the significance level of a test and usually denote it with the Greek letter $\mathrm{\alpha <!--\; \alpha \; -->}$ (alpha).
For example, if $\mathrm{\alpha <!--\; \alpha \; -->}$=0.05 we say we are doing a 5% test and will call the results statistically significant if the p-value for the sample is smaller than 0.05. Often, short hand notation such as P<0.05 is used to indicate that the p-value is less than 0.05, which means the results are significant at a 5% level.
Now, I'm studying about statistical inference, a more advanced subject, and I realized there are some concepts that don't exactly have the same definition as I studied before. The level of significance is an example.
I'm reading this book and on page 352 he introduces the Neyman-Pearson lemma as a method to find the UMP test.
Example:
On the basis of a random sample of size 1 from the p.d.f. $f(x;\mathrm{\theta <!--\; \theta \; -->})=\mathrm{\theta <!--\; \theta \; -->}{x}^{\mathrm{\theta <!--\; \theta \; -->}-<!--\; -\; -->1},0<x<1(\mathrm{\theta <!--\; \theta \; -->}>1)$
For ${\mathrm{\theta <!--\; \theta \; -->}}_1>{\mathrm{\theta <!--\; \theta \; -->}}_0$ , the cutoff point is calculated by:
... $C=(1-\mathrm{\alpha <!--\; \alpha \; -->}{)}^{\frac{1}{{\mathrm{\theta <!--\; \theta \; -->}}_0}}$
For ${\mathrm{\theta <!--\; \theta \; -->}}_1<{\mathrm{\theta <!--\; \theta \; -->}}_0$ , we have:
... $C={\mathrm{\alpha <!--\; \alpha \; -->}}^{\frac{1}{{\mathrm{\theta <!--\; \theta \; -->}}_0}}$
So in this second book, the cutoff point is not necessarily α, I'm confused.
MY ATTEMPT TO UNDERSTAND WITH THE HELP OF THE ANSWERS
The alpha is predetermined, but it doesn't mean I can't have a smaller rejection region. Then I end up having a smaller rejection region using NP lemma with the same level of significance alpha. Some introductory books let the cutoff point to be α by standard (why?), that's the reason of my confusion, I can shrink the rejection region keeping the value of α. Can someone say if I'm right?

How do you find the range and interquartile range for 37, 19, 13, 11, 10?

Dantzig's unsolved homework problems
An event in George Dantzig's life became the origin of a famous story in 1939 while he was a graduate student at UC Berkeley. Near the beginning of a class for which Dantzig was late, professor Jerzy Neyman wrote two examples of famously unsolved statistics problems on the blackboard. When Dantzig arrived, he assumed that the two problems were a homework assignment and wrote them down. According to Dantzig, the problems "seemed to be a little harder than usual", but a few days later he handed in completed solutions for the two problems, still believing that they were an assignment that was overdue.
What were the two unsolved problems which Dantzig had solved?

Why do we still do symbolic math?
I just read that most practical problems (algebraic equations, differential equations) do not have a symbolic solution, but only a numerical one.
Numerical computations, to my understanding, never deal with irrational numbers, but only rational numbers.
Why do mathematicians construct the real numbers and go through the pain of using their complicated properties to develop real analysis, etc. to be able to solve just a few cases symbolically?
Why do non-mathematicians deal with real numbers and symbolic calculation if the majority of cases must use rationals and numerical computation only?

How do I calculate the $p$ value of the following?
Students' height is approximately normal with s.d = $4$ inches, sample = $10$,mean height = $68$ inches.
Calculate the $p$ value corresponding to the following null hypotheses.
$H_0$ = Avg. height is $70$ inches
$H_1$ = Avg. height is not $70$ inches

How do you find the mean of $7420, $3210, $6720, $5780, $1850, $4350?

I have two questions

1) Can we use the standard deviation to compare variability of two or more datasets?
2) Is the z-score a real measure of dispersion?

What are the measures of central tendency?

Why do radio waves spread out while higher frequency waves travel in beams?
Why is it that radio waves spread out in proportion to the square of the distance, while higher frequency electromagnetic waves, like microwaves, infrared waves, light, etc are able to propagate as beams? What fundamental property allows higher energy waves to travel differently than lower energy?

Which of the following statements are true about random sampling:
A. Four tosses of an unblased coin is an example of random sampling from a Binomial random variable with probabillity of sucess =0.5 and a sample size of 4.
B. Random sampling selects every piece of data in the entire population at least once
C. None of other answers are correct.
D. Four tosses of an unblased coin is an example of random sampling from a Binomial random variable with probabillity of sucess =0.5

How do you find the mean and median of the data set: There are 28, 30, 29, 26, 31, and 30 students in a school's six Algebra l classes?

What is the mean, median, and mode of 86, 90, 93, 85, 79, 92?

So I'm trying to test if there is statistically significant difference between the frequency of play of different recreational soccer players based on their stated reasons for play.
So for example: The average amount of times someone plays soccer per week when they state that exercise is their reason for play - compared to the frequency of someone who might say the social aspect is their reason. How might I set up such a hypothesis test.

If the standard deviation of a distribution is s = 7, what is its variance?

How do you find the mean, median and mode of -500, 350, 475, -300, -500, 450, 425, -400?