The sample data distribution tends to resemble the population distribution more closely than the sampling distribution. A random sample of data from a population should be representative of the population, and its distribution should be similar to the population distribution.

 Suppose that we draw all possible samples of size n from a given population and then get the mean, standard deviation, proportion or other statistic for each sample. The probability distribution of this statistic is called a sample distribution.

Now suppose we take all possible samples of a certain size from a population. Once we obtain the samples, we get the mean for each sample. This is called the sampling distribution of the sample mean.

The price ceiling which is the highest the price can be is lower than the price equilibrium, so when the price is $5, you see the supply is 10 and the demand is 30. This means there is a higher demand than what is in supply, therefore there is a shortage of 20.

The price floor is the lowest the price can be and since it's still $5, the supply and demand is the same as in the first part. But since the price floor can be increased, there is no surplus or shortage, it can be moved to create equilibrium or surplus. Basically it isn't fixed, whereas with the price ceiling at $5, there is no way to rectify the situation of the shortage.

1.The distance that one professional golfer can drive a golf ball has a normal distribution with a mean of 258 yards and a standard deviation of 6 yards. What proportion of his drives exceed 280 yards in length? How many yards should this golfer drive a ball so that the distance is among his longest 25%

(X > 280) , we need to get the Z score, which is Z =(x -mean)/standard deviation

Z = (280 - 258)/6
Z = 3.67
1-Z(3.67) ,.... approximately 0, less then .0001

Z for upper 25th percent is .67

Mean + Z*standard deviation
258 +.67(6) = 262.02

2.The average number of pounds of meat a person consumes a year is 212.3 pounds. Assume that the standard deviation is 20 pounds. If a sample of 50 individuals is selected, find the probability that the mean of the sample will be less than 210 pounds per year.

 mean = 212.3, standard deviation is 20
n = 50

P(x-bar < 210)
Z = (x-bar -mean)/(standard deviation/square root(n))
Z = (210 - 212.3)/(20/square root(50))
Z = -0.81

Z(-0.81) = .2090

answer is .2090

3. Suppose that a presidential candidate is favored by 51% of all eligible voters. What is the probability that in a random sample of 100 registered voters, less than 49% will favor that candidate?

p = .51
1-p = .49

Z = (p^ - p)/(sqrt(p(1-p)/n))

Z = (.49 - .51)/(sqrt(.51*.49)/100)
Z = -0.4

Z(-0.4) = .3446

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.