When integrating by parts with one factor as a variable and the other with a trig function, generally the variable will be set to u and the trig function set to dv

integral udv =  uv - integral vdu

let's take integral x^2*sinx

u = x^2
du = 2x dx
dv = sinx
v = -cosx

this gives us  -x^2(cosx) - integral -2xsinx

now we have to integral by parts again

u = -2x
du = -2 dx
dv = sinx
v = -cosx


2xcosx - integral 2cosx = 2xcosx -(1/2)sinx

answer is -x^2(cosx) - 2xcosx + (1/2)sinx + C

When estimating the area under a curve using the left endpoints and right endpoints, if the function is increasing the left endpoint approximation underestimates the area, while the right endpoint estimation overestimates. This is vice versa if the function is decreasing. The most accurate of the estimates is the midpoint estimation.

Suppose you know that integral (0 to 3) f(x) dx = 6 , what is the integral (0 to 3)(f(x) + 5x)?

We can break down down into the sum of integrals

integral (0 to 3)f(x) dx + integral (0 to 3) 5x dx

= 6 + [(5/2)x^2) evaluated at 3... = 6 + (5/2)(9) = 6 + 45/2 = 28.5

Remember that removing an outlier can greatly affect the correlation between two variables.

Suppose x,y are as follows

x  1  1 1  2 2  2  3 3  3  1
y  8 9 10 8 9 10 8 9 10 10

This has a relatively strong negative correlation, but if you remove the data point (1,10), all you have left is a block of data with correlation coefficient of 0.

When finding correlation coefficient r by hand, as well as regression slope and intercept, you will see many calculations involving terminology you are unfamiliar with.

This is what is all means

sum(x) = all x's added together
sum(y) = all y's added together
sum(x^2) = square each x, then sum the total
sum(y^2) = square each y, then sum the total
sum(xy) = muitiply each xy value , then sum the total
(sumx)^2 = sum of the x's then square
(sumy)^2 = sum of the y's, then square
r^2 is just taking r and squaring it
y-bar is the average of the y's, which is sum(y)/n
x-bar is the average of the x's, which is sum(x)/n

Hypothesis test procedure

1. state the Ho and Ha
2. find test statistic z or t, for proportions we can use z if np > 5 and n(1-p) or nq > 5
3. find critical value for z or t using the appropriate chart
4. make the decision on the test, if using p-value, if p-value > significance level then do not reject. If p-value < significance level of the test, reject.
5. state the conclusion
Type I error is when you reject Ho when you should not reject Ho
Type II error is when you do not reject Ho when you should reject Ho
The power of the test is the probability of not committing a type two error, which is basically the probability of making the correct decision of rejecting Ho.

When using critical value approach for decision making with a hypothesis test, proceed as follows:


left tailed test (if test statistic < critical value, then reject Ho), if not, then do not reject Ho
right tailed test (if test statistic >= critical value, then reject Ho), if not, then do not reject Ho
two-tailed test, (if test statistic falls in between the critical values, do not reject Ho, if not then reject Ho


for p-value if p-value is > = alpha level of the test, then reject Ho, if not then do not reject Ho

Suppose that a committee is studying whether or not there is waste of time in our judicial system. It is interested in the mean amount of time individuals waste at the courthouse waiting to be called for jury duty. The committee randomly surveyed 81 people who recently served as jurors. The sample mean wait time was eight hours with a sample standard deviation of four hours.
a.
x -bar =____
 Sx=
n=___
n-1 = ____
b. Define the random variables X and X (with a line over top of it)
c. Which distribution should use you for this problem?
d. Construct a 95% confidence interval for the population mean time wasted. State the confidence interval


x-bar is the sample mean which is 8
Sx is the standard deviation of x which is 4
n = sample size of 81
n-1 is the degrees of freedom which is 80
part b, x is the time an individual waited to be called for jury duty and x-bar is the sample mean, so that is the mean waiting time
c) this is t-distribution since population standard deviation is not known
part d, 95% CI, for 80 df, t value is 1.99
8 +/- 1.99(4/sqrt(81))
8 +/- 0.88 = (7.12, 8.88)
the error bound is also known as the margin of error which is the value added and subtract from the mean in the interval which is 0.8

R-square is the percent of variability explained by the model and 64% then is not explained by the model, explained by other factors and possibly due to chance. This is important because the higher the r-squared the better the data fits the model. So with a low r-squared the data isn't the best fit for the model.

First we need the hypotheses:

Ho: Mu = 10
Ha: Mu > 10

now get the test statistic t, since sample size is small and population standard deviation is not known.

t = (x-bar - Mu)/(standard deviation/square root(n))

t = (9.5 - 10)/(2.5/square root(16))

t = -0.8

We get the critical value for the test,

look up t at n-1 df for one tailed area of .05

t, 15df, .05 = 1.753

Since -0.8 < 1.753, we do not reject Ho. There is not enough evidence to support the claim that mean is greater than 10

For part b, the CI is x-bar+/- t(standard deviation/square root(n))

t for 95% interval, 15 df is 2.131

CI = 9.5 +/- 2.131(2.5/sqrt(16)) = 9.5 +/- 1.332

(8.168, 10.832)

 Steps for hypothesis test

For the first part we want to prove the claim that mean life of time of battery exceeds 400 hours, so Ho would be that the mean is 400 and Ha is that the mean is greater than 400
step 1: Ho: Mu = 400
Ha: Mu > 400
for the second step, recall that since sample size is small and standard deviation is not known, have to use the t-distribution, so test statistic is t
t = (sample mean - population mean)/(sample standard deviation/square root (n))
x-bar = 473.46
s = 210.77 as done on the calculator
n = 13
t = (473.46 - 400)/(210.77/square root(13))
t = 1.26
For the next step,we know using the p-value approach that we reject if p-value is less than alpha level of the test. So we reject Ho if p-value < .025
For the next step, I obtained the p-value from this site..
http://www.socscistatistics.com/pvalues/tdistribution.aspx
Putting in the values for t-statistic and one-tailed test, p-value is .11581
Last step, since .115811 > .025, we do not reject Ho since there is not significant evidence to conclude mean battery life is more than 400 hours.

A random sample of size 49 is taken from a population with mean 31and standard deviation of 12.
What are the expected value and the standard deviation of the sample mean ?
Describe the probability distribution of .
What is the probability that the sample mean is greater than 32?
What is the probability that the sample mean falls between 27 and 29?
What is the probability that the sample mean will be within ±3 of the population mean?

The distribution is the normal distribution which is symmetrical about the mean. Symmetric means it looks the same on both sides of the mean.
P(sample mean > 32)
We need to find a Z-score, which shows how many standard deviations away from the mean a value is.
Z = (sample mean - population mean)/(standard deviation/square root(n))
Z = (32 - 31)/(12/square root (49))
Z = 1/(12/7)
Z = 0.58

Now find Z(0.58) on the standard normal distribution chart. Since the chart shows probabilities less than, we need to take 1- Z(0.58)
Here is the chat I used for all these. http://www.regentsprep.org/regents/math/algtrig/ats7/zchart.htm
Z(0.58) = .7190
1- .7190 = .2010
The probability it falls between 27 and 29 ... P(27 < X < 29)
We need 2 Z-scores now, one for 27 and one for 29
Z = (27 -31)/(12/sqrt(49))
Z = -4/(12/7)
Z = -2.33
and
Z = (29 - 31)/(12/sqrt(49))
Z = -2/(12/7)
Z = -1.17
Now we get Z(-1.17) and Z(-2.33) and subtract them
.1210 - .0099 = .1111
Probability sample mean is within +/- 3standard deviations we simply find Z(3) and Z(-3) and subtract
So we get .9987 - .0013 = .9974

he 2003 Statistical Abstract of the United States reported the percentage of people 18 years of age and older who smoke. Suppose that a study designed to collect new data on smokers and nonsmokers uses a preliminary estimate of the proportion who smoke of .26.
a. How large a sample should be taken to estimate the proportion of smokers in the population with a margin of error of .02 (to the nearest whole number)? Use 95% confidence.

b. Assume that the study uses your sample size recommendation in part (a) and finds 520 smokers. What is the point estimate of the proportion of smokers in the population (to 4 decimals)?

c. What is the 95% confidence interval for the proportion of smokers in the population (to 4 decimals)?

The margin of error is the 1.96(sqrt(pq/n) and that has to equal .02. We know p =.26, so q = .74

So we have 1.96(sqrt(.26*.74/n) = .02

3.8416(.1924)/n = .0004

0.73912384/n = .0004

n = 1848

b) point estimate is 524/1848 = .2835

c) .26 +/- .02 = (.24, .28)

In this recession, yours truly, CEO of the Outrageous Products Enterprise, would like to make extra money to support my frequent filet-mignon-and-double-lobster-tail dinner habit. A promising enterprise is to mass-produce tourmaline wedding rings for brides. Based on my diligent research, I have found out that women's ring size normally distributed with a mean of 6.0, and a standard deviation of 1.0. I am going to order 5000 tourmaline wedding rings from my reliable Siberian source. They will manufacture ring size from 4.0, 4.5, 5.0, 5.5, 6.0, 6.5, 7.0, 7.5, 8.0, 8.5, 9.0, and 9.5. How many wedding rings should I order for each of the ring size should I order 5000 rings altogether?


 what we have to do is notice that the ring sizes or 4, 4.5, 5, 5.5, 6, 6.5, 7, 7.5, 8, 8.5, 9 and 9.5 are a certain number of standard deviations from the mean. This is how we will solve this
4 is -2 st dev obtained by taking (4 - 6)/1 = -2
4.5 is -1.5 st dev by taking (4.5 - 6)/1 = -1.5 etc
5 is -1 st dev
5.5 is -0.5 st dev
6 is the mean , so 0 st dev
6.5 is 0.5 st dev
7 is 1 st dev
7.5 is 1.5 st dev
8 is 2 st dev
8.5 is 2.5 st dev
9 is 3 st dev
9.5 is 3.5 st dev.
This is how we will figure out what percentage of each ring should be bought. We can't just say we look at standard normal distribution probabilities and take P(Z = -2) for size 4.0 because remember in a continuous distribution, there are no probabilities for exact values. So we take P(-2.5 < Z < -2.0) = .0166 (i did half standard deviations because each ring size is half st dev apart) . Now take .0166(5000) = 83
The same method is applied to each size
size 4.5 = P(-2 < Z < -1.5) = .044, multiplied by 5000 = 220
size 5 = P(-1.5 < Z < -1) = .0919, multiplied by 5000 = 460
size 5.5 = P(-1 < Z < -0.5) = .1498, multiplied by 5000 = 749
size 6 = P(-0.5 < Z < 0) = .1915, multiplied by 5000 = 958
Since this is the normal distribution, this is symmetrical, so size 6.5 = 958, size 7 = 749, size 7.5 = 460, size 8 = 220, size 8.5 = 83
For size 9, same idea = P(2.5 < Z < 3) = .0049 multiplied by 5000 = 25
size p.5 = P(3 < Z < 3.5) = .0011 times 5000 = 6
Now that gives a total of 4971, the rest are either bigger than 9.5 or less than 4, since no rings will be produced bigger than size 9.5, the other 29 should be of size 4, making size 4 total 112

Suppose you know that the heights of elementary school students in southern Kenya follow a normal probability distribution with a mean of 40.1 inches and a (population) standard deviation of 2.2 inches.

What is the probability that an elementary school student in southern Kenya is greater than or equal to 41.3 inches tall? (Round to the nearest percent.)

  z score is (x - mean)/standard deviation

  so (41.3 - 40.1)/2.2 

• and look up that on a standard normal distribution chart
• z-chart
• to that value for z (.55) is .7088
•   that is the probability of LESS than z(.55) 

So you take 1- .7088 to get .2912

It's clear that because of symmetry, the derivatives at -1/2, -1 and -2 will be the negative of those at 1/2, 1 and 2 for the function f(x) = x^2 since that is the graph of a parabola. Therefore I get f'(-1/2) = -1, f'(-1) = -2 and f'(-2) = -4


Applied the definition to get the derivative of x^2. Put in (x + h) for x in f(x) to get (x+ h)^ then subtract off f(x) which is x^2 and divide the whole thing by h. Remember to expand (x+h)^2, which is (x+h)(x+h), then simplify and put in 0 for h to get 2x, which matches the guess in part c

For the first part, suppose we want to know approximately how much data falls within 2 standard deviations from the mean using Chebyshev's theorem. This is how we do it. We square the standard deviations first to get 4. Then we take 1/4. Now simple subtract 1/4 from 1 to get 3/4. Therefore at least 75% of the data will fall within 2 st dev from the mean. In this problem we don't know the standard deviations, but we know it has to be at least 70%. So we basically are working in reverse now.
That means that 1 - .3 = .7........... therefore .3 = 1/(standard deviations)^2
.3(standard deviations)^2 = 1
divide by .3 to get (standard deviations)^2 = 3.33, therefore standard deviations = 1.83
We know that at least 70% of the data falls within 1.83 standard deviations from the mean.
If you don't understand, please let me know!
For the second part, we simply get the mean and standard deviation of the data set. I did it on my calculator and get mean = 5 and st dev = 1.777
Therefore to see the range of values that are within 1.83 st deviations, take 5 +/- 1.83(1.777)
That gives us 1.75 and 8.25. Look at the data and you see the lowest value is 2.22 and the highest is 8.11, so ALL of the data, 100% fall within the range.

The parabola in the form y = (x - h)^2 + k has a vertex of (h,k). The problem we have is y-3 = (x-1)^2, so we have to get the -3 from the left side and move to the right, so add 3 to both sides of the equation. That gives us y = (x-1)^2 + 3, so the vertex is (1,3)

Since the x^2 term is positive, the parabola opens up away from the x-axis, so there are no x-intercepts. If you are unsure of this you can put 0 in for y and solve for x, like i demonstrated and by the quadratic formula you will see there are no real number solutions for x, so no x-intercepts.

For the y-intercept, put 0 in for x and you'll see that y = 4.
The axis of symmetry is simply the line the goes through the vertex, cutting the parabola in half. The domain is all the possible x values, and since there are no restrictions on x, it's all real numbers. The range is all the possible y-values that the function takes on. As you can see on the graph, the lowest y value is at (1,3) so the range is from 3 to infinity.

 Solve: Cos(Sec^-1 u)

Recall that sec is 1/cos
Suppose to make this a little easier to understand that the problem says sec^-1(2), which means u = 2 . So we want the angle which has a sec value equal to 2. That is the same as saying 1/cos = 2 which means cos = 1/2

Cos is 1/2in the first quadrant 60 degrees

That gives us cos(60) which we know is 1/2 and 1/2 = 1/u.

Therefore the answer is simply 1/u. That makes logical sense too since sec = 1/cos and cos = 1/sec. They are inverses.

You can also do this by labeling parts of the right triangle. You know that sec^-1 u means that the adjacent side of the right triangle is 1 and the hypotenuse is u, since sec = hypotenuse/adjacent.

Therefore cos of the angle equal 1/u

Yes the CLT is a bit of a confusion here because as we know, The central limit theorem states that the sampling distribution of any statistic will be normal or nearly normal, if the sample size is large enough. That's where the problem comes in, sample size should be at least 30, some say 40. That would lead you to believe that you cannot use apply CLT here, BUT. the more closely the original population resembles a normal distribution, the fewer sample points will be required.

But what we can do is use the normal approximation to the binomial if this condition holds true. If np > 5 and n(1-p)> 5, then it can be used. We know p = .5 and if you consider 4 trials, then np < 5, as is n(1-p). BUT if you use all 12 tosses, then n = 12 and np = 6 and n(1-p) = 6. But doing it this was we would get the average number of heads expected in 12 tosses to be 6 and the average in all 12 tosses to be 8, instead of 2 in 3 tosses.

npq = (12)(.5)(.5) = 3
sqrt(npq) = 1.732
Formula is z = (x-bar - np)/(sqrt(npq)
z= (8 - 6)/(1.732) = 1.15
z(1.15) = .8749
1- .8749 = .1251
If you notice when you go through a coin tossing experiment tossing three times, there are 8 outcomes and the only way to get greater the 2 heads is if you get HHH, which is 1 out of 8, which is 12.5%.

since population standard deviation is known, we can use Z

Test stat Z = (x-bar - mean)/(standard deviation/sqrt(n))
Z = (16.2 - 15)/(5.6/sqrt(49))

If you calculate this correctly , you should get Z = 1.50

The critical values are Z = 2.43 and Z = -2.43 since this is a two-tailed test. The critical value was found looking up .9925 on the chart. Took .015/2 = .0075 and then 1-. 0075 to get .9925. It was alpha/2 because of the two-tailed test.

Since 1.5 < 2.43, accept Ho

 P-value was .133614. Using the chart it's .1336. In any event, since that value is greater than alpha for the test, accept Ho

The power of the test is probability of Type II error. IN this case it is probability of type II error when x-bar = 23.6. This is the probability of Ho not being rejected when it is false because x-bar = 23.6
B(23.6) = 1 - z(23.6 - 25.1)(14/sqrt(100))
1- z(-1.07)
Finr z(-1.07) on standard normal distribution chart to get .1423
1- .1423 = .8577
b) we know the power of the test is .9
so 1 - z(23.6 - 25.1)/(14/sqrt(n)) = .9
therefore z(-1.5/914/sqrt(n)) = .1
looking at the z-chart we know z(-1.28) = .1
so -1.28 = -1.5(14/sqrt(n))
-17.92/sqrt(n) = -1.5
cross multiply and square both sides and solve for n and you get n = 143.

Chart


                             NY             NJ             CT

Male                     100             60               50
Female                   80             50               80


1) Probability person is female and lives in NY

P(F and NY)

80 females from New York and (100 + 60 + 50 + 80 + 50 + 80) = 420 people total in the company from those three states

P(F and NY) = 80/420 = 4/21

2) Probability person is male or lives in NJ

P(M or NJ) = (100 + 60 + 50 + 50)/420

                 = 260/420

                 = 13/21

3)  P(CT if M)

Number of male and living in Connecticut = 50
Number of males (100 + 60 + 50) = 210

P(CT if M) = 50/210 = 5/21

If a random sample of 28 homes south of Center Street in Provo has a mean selling price of $145,450 and a standard deviation of $4775, and a random sample of 25 homes north of Center Street has a mean selling price of $148,300 and a standard deviation of $5900, can you conclude that there is a significant difference between the selling price of homes in these two areas of Provo at the 0.05 level?


Ho: Mean1 = Mean2
Ha: Mean 1 does not equal Mean2
x-bar1 (south of Center Street) = 145,500 s1 = 4775, n1 = 28
x-bar2 (north of Center Street) = 148,300 s2 = 5900, n2 = 25
test statistic for the test is (x1-bar - x2-bar)/sqrt(s1^2/n1 + s2^2/n2)
If you put the values into the formula and calculate correctly, you should get
t= -1.919
Make sure you understand how to get that
P-value was obtained from this site. http://www.socscistatistics.com/pvalues/tdistribution.aspx
Just enter the t-score of -1.919, 24 df two sides test and alpha = .05 and you get
.0670
b) the conclusion, since .0670 > .05, we have to accept Ho. So you fail to reject Ho since there is not significant evidence of a difference of means.
Remember when p-value > alpha, accept Ho and when p-value < alpha, reject Ho.

Say you want to get a confidence interval for the difference of two proportions.

To get p^ you take x/n, so the first two are quite simple
p1^ = 25.176 = .142
 p2^ = 32/143 = .224


 for 90% CI you use 1.645 for zcritical
formula is p1^ - p2^ +/- 1.645(sqrt(p1^q1^/n1 + p2^q2^/n2))
if you do this correctly for the data we have you get
-.082 +/- .071850965
(-.154, -.010)

A type I error is the probability of rejecting the null hypothesis when it is true, which is the alpha level of the test.

A type II error is the probability of not rejecting the null hypothesis when we should reject it, which is Beta.

By intuition, the greater the departure from Ho, the less likely that departure will be detected, therefore there is a less chance of rejecting the null hypothesis, which decreases the chance of Type II error, which means Beta is smaller

If you look at the regression equation for TVHOURS = 1.86 + 0.02 (AGE), the slope is .02 which is practically a straight horizontal line. Although a test might show that this slope is statistically significant, .02 is so small that it adds very little to the overall result, having no practical significance.

For example. The difference between hours for a person 20 years old and 30 years old, using this equation is 1.86+ 20(.02) = 2.26 hours and 1.86 + 30(.02) = 2.46 hours That comes out to just a 12 minute difference. Even for a wide difference in age from 20 to 70 it only adds 1 extra hour.. for a 50 year difference. Each increase of 1 year in age adds only 1.2 extra minutes of tv time, showing no practical significance.

Example of a problem using a contigency table

American Continental Delta United Total
Yes 48 69 68 25 210
No 52 41 62 35 190
Total 100 110 130 60 400
Now what we have to do is get the expected number in each spot. For instance we need the expected number from American and YES. We get the expected number by taking the row total times the column total divided by the total number in all.
For American and YES that would be 210(100)/400 (row total of 210)(column total of 100)/(total of 400), which equals 52.5
You have to do that for ALL of the 8 different spots on the table. I'll show you one more .. For Continental and YES it's (210)(110)/400 = 57.75
If you do that correctly this is what you will have and i'll show on another table
American YES NO
Observed = 48 52
Expected = 52.5 47.5
Continental YES NO
Observed = 69 41
Expected = 57.5 52.25
Delta YES NO
Observed 68 62
Expected 60.25 61.75
United YES NO
Observed 25 35
Expected 31.5 28.5


Now to get the test statistic you have to take the (observed - expected)^2 . Divided that by the expected and sum all the values.
For American and YES that is (48-52.5)^2/52.5
For Continental and YES that is (69 - 57.75)^2/57.75
and so on....
If you do that for all eight values you should get 8.251 (you could have a slightly different number depending on how you round)

Now you need the critical value for the test. To get the df you take (rows - 1)(columns - 1)
We have 2 rows in the table and 4 columns, so df = (4-1)(2-1) = 3
Look in any chart for Chi-square distribution for 3 df and alpha = .05 and you get 9.488
We compare our test statistic to 9.488
Remember we reject Ho and conclude there is a preference to which airline we choose if the test statistic is greater than the critical value, but 8.251 < 9.488 so we do what? We do not reject Ho, so we have no preference as to what airline to choose

For linear regression line, you need y^ = a + bx, where b = r(sy/sx) and a = y-bar - b(x-bar). you can get sy, sx, x-bar and y-bar by putting the values into a calculator. R can be gotten using Excel or by hand with a long, tedious formula

For multiple linear regression, it's best to use Excel or use this online calculator for multiple regression, simply put in the values for the x variables and the y variable and you will the result you need

http://www.xuru.org/rt/MLR.asp#Manually

Remember that a multiple linear regression equation will take on the form y^ = a + b0x1 + b1x2 and so on.

Resampling is simply mimicking the process of sampling
by choosing another sample at random from the population based on data from your sample.we sample instead from an artificial population constructed on our computer and that embodies everything we know
about the population of interest. In many, but not all, of the examples
that follow, this artificial population is the very data set from which
we seek to draw inferences. Since the data set is itself a sample of the
whole population, we are taking a sample from the sample: resampling.
This doesn’t, of course, provide more information about the population,
but it does provide us with a way of understanding the consequences of
sampling variability for drawing inferences about the population based
on our data.

Remember when determining concavity, you use the same method as finding maximum and minimum using the first derivative test, except now you used the second derivative.

For example, suppose the function if f(x) = x^3 + 2x^2 - 5x

f'(x) = 3x^2 +4x - 5

f''(x) = 6x + 4


6x + 4 = 0

6x = -4

x = -2/3

Now test a point on both sides of -2/3, we'll use -1 and 0

f''(-2) = 6(-2) + 4 = -8

f''(0) = 6(0) + 4 = 4

that means we have concave down from (-infinity, -2/3) and concave up from (-2/3, infinity)

For the gamma distribution Mean (Mu) = aB and variance (sigma squared) = aB^2
For the exponential distribution Mean (Mu) = B and variance (sigma squared) = B^2
Examine these and you can see the difference.. The means are basically the same except gamma has a and exponential does not. Variance for gamma also includes a, while exponential does not.

If a = 0 and B = 0 then Mu for exponential is 0 and sigma^2 = 0.
If a = 0 and B = B then Mu for exponential is B and sigma^2 = B^2
If a = 1 and B = B then Mu for exponential is B and sigma^2 = B^2, but note that the Gamma Distribution will also have Mu = B and sigma^2 = B^2.

That's the key, the exponential distribution is EQUAL to the Gamma distribution when a = 1 and B = B.

Remember for the normal distribution, the standard deviation of the sample mean is the standard deviation of the population divided by the square root of n.

For example, if the population standard deviation is 10 and we have a sample of size 50 from the population, the standard deviation of the sample mean is 10/sqrt(50)

Suppose you have two triangles, ABC and DEF and want to determine if they are congruent. The Side Angle Side theorem (known as SAS) says that if two sides and the included angle of one triangle are congruent to the corresponding two sides and included angle of another triangle, then the triangles are congruent. So if side AB is length 5 and AC is length 6 with angle A of 50 degrees, and side DE is also 5 and DF is 6 with angle D of 50 degrees, the triangles are congruent by the SAS theorem.

What you need to do for these is figure out how many standard deviations from the mean the value is. the farther away from the mean, the less likely it is for that value to occur. For a graduate to have a salary of 80,000, that is 10,000 above the mean, which is 2 standard deviations above the mean, since (80,000 - 70,000)/5,000 = 2. It's value minus mean, all divided by standard deviation. If you check on a z chart for normal probability distribution, you will see that .9772 will be less than a salary of 80,000 and only 2.228 percent have more than 80,000. Suppose we test that the mean salaries are indeed 70,000, the null hypothesis, against a hypothesis that the mean salaries are not 70,000 at alpha = .05. The rejection region for this would be when z > 1.96 or z <-1.96. So any value of z in between the -1.96 and 1.96 would be considered a "reasonable" value for z, which would be a reasonable outcome. Since 80,000 gives a z-value of 2, it falls just outside of 1.96 which is not a reasonable outcome.
The way to figure out the highest reasonable outcome, take the 1.96(which is 1.96 standard deviations above the mean) and multiply by the standard deviation and add to the mean.
70,000 + 1.96(10,000) = highest maximum reasonable value.
For part c, we need to take (80,000 - 70,000)/(5,000/sqrt(100)) to get the z value. The formula is (value - mean)/(standard deviation/square root of n)
If this falls between -1.96 and 1.96, it is a reasonable value. It does not, as you will see, so it's not reasonable.
d) For this, we know the z-value that will give the maximum reasonable mean salary is 1.96
so use the formula
1.96 = (x - 70,000)/(5,000/sqrt(100))
The x is the value we are solving for, which is the mean salary for a random sample.
If you do this correctly, you will get 70,980

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Suppose you have a piece of cardboard 32 inches by 48 inches and cut equal sized squares from each corner and fold up the sides of the cardboard to form a box with an open top. What length of x maximizes the volume of the box?

Volume = lwh

l = 48 - 2x
w = 32 - 2x
h = x

V= x(48 - 2x)(32 - 2x)

V= x(1536 - 96x - 64x + 4x^2)

V = 1536x - 160x^2 + 4x^3

V' = 1536 - 320x + 12x^2

set V' = 0 and solve for x.

From quadratic formula, x =  6.28 inches

y=mx + b is equation of a line. The slope of the line is "m" and the y-intercept is "b". We can get the equation of the line given the slope and y-intercept or given two points (x1, y1) and (x2, y2). If given two points, use the formula (y2 - y1)/(x2 -x1) to find the slope. Then use the slope and one of the coordinates give to find "b". Then you have the slope and y-intercept needed to write the equation.

Division with remainders is simply a division problem in which one number does not divide into the other number evenly. For example, 2 divides into 24 evenly 12 times. But 2 divides into 25, 12 times with 1 left over, which is the remainder. The answer to 25 divided by 2 would be 12 R1 or 12 1/2.

Suppose you have two triangles, ABC and DEF and want to determine if they are congruent. The Side Angle Side theorem (known as SAS) says that if two sides and the included angle of one triangle are congruent to the corresponding two sides and included angle of another triangle, then the triangles are congruent. So if side AB is length 5 and AC is length 6 with angle A of 50 degrees, and side DE is also 5 and DF is 6 with angle D of 50 degrees, the triangles are congruent by the SAS theorem.

Algebra is a type of mathematics where symbols are used to represent amounts that are unknown. These
unknown quantities are generally combined with mathematical operations (addition, subtraction, multiplication,division, square root, cube root, exponents, etc) to form statements that describe the relationship of things that change over a period of time. These statements are expressed using equations, expressions and terms. Problems can be solved by translating words into algebraic equations. The description of a problem using an equation and other mathematical concepts is known as a mathematical model. A mathematical model can be used to solve numerous types of problems in every day life

We use the change of base formula to solve logs that are not base 10. The log button on a calculator solves for base 10 logs. So if you have a problem such as log (base 5) 8, you change that to base 10, which would be log8/log5. This can be simply solved on a calculator as 1.29.

In a problem involving data that is normally distributed, the Z-score is how many standard deviations from the mean that a sample data value lies. For example, if the standard deviation is 10 and the mean is 20, then a data value of 25 is 5 units from the mean and 5/10, which is 1/2 standard deviation away from the mean. The Z-score is then 0.5.

Statistics involves the study of data that is collected. The data is analyzed, presented and interpreted in various ways such as numerically and graphically. The two primary methods used to analyze the data are descriptive statistics and inferential statistics. Mean and standard deviation are examples of descriptive statistics. Inferential statistics involves data that has errors in observation or sample variation.

The surface area of a cylinder is the area of the two bases, both circles plus the area of the part in the middle. Think of a soda can, generally cylindrical in shape. That part that you hold around middle of the can, if you could peel off the put flat, would be a rectangle. You can also think of it as the height of the cylinder times the circumference of the top or bottom. So a formula could be 2Pi(r)^2 (area of top and bottom) plus h(2Pi)(r) , area of the part around the middle.

Odds can be thought of as a chance of an event occurring. Suppose you know there are 8 possible outcomes, each equally likely. The odds or any one happening are 1 in 8. It can also be represented as a ratio, in which case it is the one chance that the event occurs and seven that is does not occur, written as 1:7, and it is said that odds are "1 to 7"

Real numbers are all rational and irrational numbers which include whole numbers, repeating decimals and non-repeating decimals. Non-real numbers are known as "imaginary" numbers. The square root of negative numbers, which will come up as an "error" on most calculators are imaginary numbers. The square root of -1, is known as "i" and imaginary numbers can be part of the complex number which has a real an imaginary part. And example of a complex number is 2 + i.

Suppose we want to solve an equation such as x^3 + 2x^2 -3x + 6 = 0. We can use the rational root theorem to find the possible rational roots and then use synthetic division to find a root. After we find a root, we will have a quadratic equation left to solve to obtain the other roots. This can be quite tedious. With a graphing calculator, you can solve this equation by seeing where the graph of the equation crosses the x-axis. These points are the solutions to the equation. If you have multiple equations, the solutions are the intersection points of the graphs.

Quotient rule for differentiation:

derivative of f(x)/g(x) = [g(x)f'(x)- f(x)g'(x)]/g(x)^2

Product rule for differentiation:

derivative of f(x)g(x) = f(x)g'(x) + g(x)f'(x)

Chain rule

derivative of f(x)^2 is

2f(x)f'(x)

in general

derivative of f(x)^n is

n(f(x))^(n-1)f'(x)

Remember that "natural log of x", written as "lnx" is really log (base e) x where "e" is approximately 2.718.  You see the used of "e" in exponential growth and decay, also interest compounded continuously A=Pe^(rt) where A is the amount, P is the initial investment, r is the interest rate and t is the time in years.

When solving a system of equations by substitution, remember to remove the one variable and replace it with a quantity equal to that variable. For example, suppose the system is x = 2y + 3 and 3x + 4y = 16.

You can solve this system by substituting 2y+3 for x in the second equation. The second equation becomes 3(2y+3) + 4y = 16. Now we can solve for y.

6y + 9 + 4y = 16

10y = 7

y = 7/10

To solve for x, substitute 7/10 for y in the first equation. That gives us

x = 2(7/10) + 3

x = 14/10 + 3

x = 4.4

Suppose you want to determine the area bounded by two curves denoted by functions f(x) and g(x).

We find the points of intersection by setting f(x) = g(x). Then we determine which curve is the upper curve in the integral and which is the lower curve in the integral.  Suppose the points of intersection are  x=1 and x=3.

If f(x) is the upper curve then the integral looks like

Integral (1 to 3) [f(x) - g(x)] dx

If g(x) is the upper curve the the integral looks like

Integral (1 to 3) [g(x) - f(x)] dx

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Suppose you wish to obtain the exact value of sin(165).  We know exact values of the following angles on the unit circle : 0, 30, 45, 60, 90, 120, 135, 150,. 180, 210, 225, 240, 270, 300, 315, 330, 360. 

If we pick two angles from the unit circle that add to 165, we can obtain the exact value for sin(165). The reason is we can substitute those values in for this formula.

sin(a + b) = sin(a)cos(b)+cos(a)sin(b)

Notice that 120 + 45 = 165 and both of those values are on the unit circle

sin(120 + 45)= sin(120)cos(45) + cos(120)sin(45)

                     = (sqrt(3)/2)(sqrt(2)/2) + (-1/2)(sqrt(2)/2)

                     = sqrt(6)/4  - sqrt(2)/4

                    =  (sqrt(6) - sqrt(2))/4

Here's a great little video on hypothesis testing. It really explains the concept simply, in easy to understand terminology.

The Most Simple Introduction to Hypothesis Testing

Remember there is a difference between finding that Z value for a confidence interval and the Z (critical) value for tests and probability.

For example, suppose you want a 95 percent confidence interval for the sample mean.  The formula is mean +/- standard error which is  Z(alpha/2)standard deviation/square root(n).  alpha is 1-.95 = .05. So the Z value is 1.96.


Now if we want an x value that 95% of the data falls below, then we need Z(.05) which is 1.645.

Be careful to understand the difference between the two.

Notice the graph of a quadratic function f(x) = x^2 + 3x  + 8 is a parabola that opens up.  What if you know how to find the vertex but forget how to determine whether the vertex is the maximum or minimum point, therefore not knowing that is opens up or down.  You can take the second derivative and set equal to zero to determine whether the function is concave up or concave down.

The first derivative f'(x) = 2x + 3

Second derivative f"(x) = 2

Notice the second derivative is positive for all values of x therefore the function is concave up, therefore opening up.

When finding the area of a sector of a circle take the angle of the sector in degrees and divide by 360. Take the result and multiply by  Pi times r^2.

For example, suppose the radius of a circle is 6 and the angle of the sector is 45 degrees. Therefore the area of the sector is

(45/360)Pi(6)(6) = (1/9)(36)Pi = 14.14

Suppose you want to know how long it will take for an investment to double if it is compounded quarterly at 7%.

Use the formula A + P(1+ r/n)^(nt) where A is the amount, P is the initial investment, r is the interest rate, n is the number of times compounded annually and t is the time in years.

2A = A(1 + .07/4)^(4t)

2 = (1.0175)^4t

log 2 + log (1.0175)^(4t)

log 2 = 4t log(1.0175)

log2/log(1.0175) = 4t

t = 10 years

Another way to represent an exponential equation is by using logarithms. 

Suppose you have 5^x = 33 and you want to solve for x. That is difficult as it is set up. It can be made easier by using logarithms.

We rewrite the exponential equation above in logarithmic form as follow:

log (base b) y = x is the same ax b^x = y.

In our example, the logarthmic equivalent is log (base 5) 33 = x.

Using the change of base formula you get log 33/log 5 = x.  This can be solve on a calculator to get 2.1725. We can check this by substituting for x in the equation to get 5^(2.1725) which is approximately 30.

What is the difference between "highest 89 percent" and the "89th percentile" ?

There is a huge difference and one that is often confused.

The "highest 89 percent" is from the 11th percent on upward. In other words, if there are 100 people taking a test and you tare in the highest 89 percent, you have scored better than only 11 people.

The "89th percentile" would be that only 11 people scored higher. So it's the exact opposite.

"89th percentile" = "highest 11 percent"

"highest 89 percent" = "11th percentile"

Remember the key to consistency with math is to work problems daily. Master the technique of solving problems, know the steps to the correct solution. Do not just simply memorize steps, though, because it's important to understand what is going on in the problem.

Suppose we know that f"(x) = 5x^4 + 2x^3 + x + 3 and f'(0) = 4 and f(1) = 9.883333.  What is f(x)?

We have to integrate f''(x) first to get f'(x).

f'(x) = x^5 + (1/4)x^4 + (1/2)x^2 + 3x + c, since f'(0) = 4, we get

4 = 0^5 + (1/4)(0)^4 +(1/2)(0)^2 + 3(0) = c, therefore c = 4

f'(x) = x^5 + (1/4)x^4 + (1/2)x^2 + 3x + 4.  We get f(x) by integrating f'(x).

Therefore f(x) = (1/6)x^6 + (1/20)x^5 + (1/6)x^3 + (3/2)x^2 + 3x + c. Since we know f(1) = 9.883333, we get

9.883333 = 1/6 + 1/20 + 1/6 + 3/2 + 3  + c.   Therefore, c =5 and f(x) is

f(x) = (1/6)x^6 + (1/20)x^5 + (1/6)x^3 + (3/2)x^2 + 3x + 5.

Suppose we want to estimate the area under the curve f(x) = x^2 + 3x - 4 over the interval [1, 4].  We can do this be creating rectangles of equal width (noted as w). Suppose we want to have 4 rectangles, each would be of width (4-1)/4 = 3/4. Using the width of 3/4 we can get the left endpoints of each triangle.

x1 = 1,   x2= 1.75, x3 = 2.5, x4 = 3.25 and x5 = 4

f(x1) is the height of the first rectangle using the left endpoint.
f(x2) is the height of the second rectangle using the left endpoint and so on.

Now we take f(x1) = 1 + 3 - 4 = 0
                     f(x2) =  4.3125
                     f(x3) = 9.75
                     f(x4) = 16.3125
                   

To get the area we take w*f(x1) + w*f(x2) + w*f(x3) + w*f(x4). Doing so, we get 22.7815.

Now we can estimate the area again using the right endpoint of each triangle. Those values are

f(x2), f(x3), f(x4) and f(x5)

f(x5) = 24

The estimated area is 22.7815 + 24 = 46.7815.

Using integration on the function f(x), we get (1/3)x^3 + (3/2)x^2 - 4x. Substituting 4 for x, we get 29.333333 Substituting 1 for x, we get -2.166666 Subtracting the values we get the area to be 31.5

We can use the anti-derivative or integral to find the area under a curve between two points.  For example, suppose we want to find the area under the curve defined by the function f(x) = x^3 + 2x^2 - 4x + 6 between x = 1 and x = 4.

First we integrate the function. 

(1/4)x^4 + (2/3)x^3 - 2x^2 + 6x..

Now we evaluate the integral at 4 and then at 1 and subtract.

(1/4)(4)^4 + (2/3)(4)^3 - 2(4)^2 + 6(4) = 98.6666666666

(1/4)(1)^4 + (2/3)(1)^3 - 2(1)^2 + 6(1) = 4.91666666666

Subtract the two to get the area of 93.75

There is often confusion between a statistic and a parameter. A statistic is a measure of some attribute of a sample from a population. Examples include mean, standard deviation, variance, quartiles, median, test statistics.

A parameter is a numerical characteristic from a population.  According to Wikipedia, a parameter is  sometimes taken to be unobservable, and in this case the statistician's task is to infer what they can about the parameter based on observations of random variables distributed according to the probability distribution in question, or, more concretely stated, based on a random sample taken from the population of interest. In other situations, parameters may be fixed by the nature of the sampling procedure used or the kind of statistical procedure being carried out

Source of above paragraph is  http://en.wikipedia.org/wiki/Statistical_parameter

So the easy way to distinguish between the two is  S-S and P-P. Statistic and sample go together as does parameter and population.

Suppose a conical tank with a height of 20 feet and a radius of 10 is filled with water at the rate of 5 cubic feet per minute.  What will the rate of change of the height be when the level of the tank is 8 feet.

Volume of a cone is 1/3(Pi)r^2(h)

We know dv/dt = 5 and we know that r = (1/2)h. Substitute (1/2)h in for r to get

V = (1/3)Pi(1/2 h)^2(h)

V= (1/3)Pi(1/4)(h^3)

V = (1/12)(Pi)h^3

Now take the derivative to get the rate of change of the volume.

V' = (3/12)(Pi)h^2(dh/dt)

V' = (1/4)Pi(h^2)dh/dt

Substitute 5 for V' and 8 for h to get

5 = (1/4)Pi(64)dh/dt

5 = 50.3dh/dt

.1= dh/dt

When evaluating limits at infinity or negative infinity, there are a couple methods you can use.  Look at these three examples.


Lim (x --> infinity)  (x^3 + 4x^2 - 3)/(4x^3 + 6x)

You can divide each term by x to the highest power in the problem, which is 3. So divide each term by x^3 to get

Lim (x --> infinity) (1 + 4/x - 3/x^3)/(4 + 6/x^2)

as x tends to infinity, the numerator is just 1 because 4/x and 3/x^3 approach zero. Similarly 6/x^2 approaches zero so the denominator is just 4.  Therefore the limit is 1/4.

Another way to do this problem is that since the highest power of x is the same in both numerator and denominator, divide the coefficients of those two terms which is just 1/4.


If the highest power in the numerator is less than the highest power in the denominator, the lim as x tends to infinity or negative infinity is 0. If the highest power in the numerator is greater than the highest power in the denominator, the lim as x tends to infinity or negative infinity is undefined

The easy way to tell the nature and number of roots of a quadratic equation is by evaluating its discriminate. The discriminate of a quadratic equation in the form ax^2 + bx + c = 0 is b^2 - 4ac.

If b^2 - 4ac = 0 there is 1 real root

If b^2 - 4ac > 0 there are 2 distinct real roots

If b^2 - 4ac < 0 there are 2 imaginary roots.