When solving an SSA triangle, you could possible have 0,1 or 2 triangles

Suppose in triangle ABC, you know side a = 10, b = 16 and angle A is 30 degrees

By the law of sines, 

SinB/16 = Sin30/10

10*sinB = (1/2)!6

SinB = 0.8, therefore B = 53.1 degrees. But sine is also positive in the second quadrant, so there is a possible second triangle with B = 126.9 degrees. This can work because C would equal 23.1 degrees in this triangle and C = 96.9 in the first triangle

If angle A was 60 degrees and B came out to 53.1 or 126.9 then only 1 triangle exists since (126.9 + 60 = 186.9) angle C + angle A > 180.

If you try to solve for an angle an get Sin > 1 or < -1 then there are no solutions.

suppose we want a linear approximation for sin(0.3). Look at the function sinx, where x = 0.3 We want to pick a value for a to approximate x, a value for which we know the sin. We know sin0 = 0, so choose a = 0, which is close to x =0.3.

Now use the formula L(x) (linear approximation as function of x) = f(a) + f'(a)(x-a)

f(0) = 0
f'(x) = cosx, so f'(a) = f'(0) = 1

L(x) = 0 + 1(0.3 - 0)

       = 0 + 0.3

      =0.3

The value of sin(0.3) = .295. so the approximation is close.  Note that the value of x = 0.3 is in radians

Roger's statement isn't exactly correct, as it doesn't make sense that a movie would make 3.8735 million dollars without any money put into the production of the movie. A movie cannot even be produced with money budgeted.

Looking at the definition of a linear equation of the regression line, y = a + bx or sometimes written as y = b0 +b1(x). The y-intercept is b0, which is the value when x is 0.

In this case by definition x being the budget and y being the revenue, yes with 0 dollars for budget, the revenue is 3.8735.

But we have to look at practicality with our conclusion. Like I mentioned above, this makes no sense in the context of the problem. So, we cannot make this conclusion. The y-intercept in this case is something we shouldn't even consider because you will never have a budget of 0 dollars.

With the most recent US Presidential election in the books, let me examine a few numbers closely. The two most populous states, (California and New York) gave Hillary Clinton the advantage in the popular vote. Donald Trump had more votes in the other 48 states overall. The four states with the most population make up 1/3 of the total US population, while the other 46 states account for only 67% of the population. This is why the electoral college is in place, so a few states can't determine the election when the vast majority of the other states show support for the other candidate. Each state has a proportional amount of electoral votes depending on the population in the state. Each state has its fair stake in the outcome of the election. It's a process that has worked and will continue to work, even if sometimes (on a rare occasion) the popular vote is for the candidate that ultimately loses.

For the hypothesis test, we need to find the test statistics

t = (x-bar - Mu)/standard error

standard error = sample standard deviation/square root of n

From here we compare the test statistic to the critical value t or you can compute the p-value and compare to alpha level of the test at .05

If the p-value < alpha, then reject Ho. Notice in the case of your problem, there is a rejection of the null hypothesis in both cases.

For the confidence interval it's mean +/- tcritical*standard error.

To calculate the Q1 (25th percentile) take n times .25 and that value is the data value that is Q1. If n(.25) does not come out even, round up to the next integer. Remember the data values must be sorted in order from lowest to highest for this. Q3 is found by taking n(.75) and round up if necessary and that is the value in order of Q3, IQR is interquartile range and is Q3-Q1. We use that to find out if there are any outliers. Any data value less than Q1 - 1.5(IQR) or greater than Q3 + 1.5(IQR) is an outlie

Mean (x + y) = Expected value (X + Y) = E(x) + E(y) which means the mean is meanx + meany, so is meanx = 75 and meany =70, then the mean (x +y) =  75 + 70 = 145

The Var(x + y) = Var(x) + Var(y) if x and y are independent.
Var(x) = 6^2 = 36
Var(y) = 8^2 + 64
Var(x+y) = 100. The standard deviation is the square root of the variance, so standard deviation = 10

The mean of the difference is the difference of the means E(X - Y) = mean (x - y) = meanx - meany

So the difference of the means is 75-70 = 5

The standard deviation of the difference is the square root of the Var(x-y)

Var(x - y) = Var(x) + Var(y) if x and y are independent, same as for Var(x +y) . Therefore the answer is 10