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

Just noticed a few sales on the books I wrote a few years ago. I had forgotten about them since sales were minimal. I was just pleased to put out my work on algebra. Now looking at completing a long standing project involving math for kids K through 5.  Check out the link below for the books.

http://www.lulu.com/spotlight/KKauffman1969

As the school year is under way in many areas, don't forget my self help algebra books on lulu.com. Working on a kids book that has been stagnant lately, but getting back to it. Thanks for checking!

http://www.lulu.com/spotlight/KKauffman1969

Remember that the t distribution is more wide than the standard normal distribution, but at n gets larger and larger, the t distribution become approximately normal in nature.

The t-test is used when population standard deviation is not known. When we know that population is normally distributed, then we can use Z and also when population standard deviation is known, we can use Z