tag:blogger.com,1999:blog-38329222291954512912024-03-13T18:50:24.396-07:00Math Easy As πMath Tips, Tricks and MoreKKauffman1969http://www.blogger.com/profile/10725188874999582883noreply@blogger.comBlogger445125tag:blogger.com,1999:blog-3832922229195451291.post-67283372856920531932017-03-31T12:27:00.000-07:002017-03-31T12:27:15.361-07:00Probability of A or B<ul class="conversation" id="lesson-chat-messages" style="box-sizing: border-box; color: #333333; font-family: 'Aspira W01', HelveticaNeue, 'Helvetica Neue', Helvetica, Arial, sans-serif; font-size: 13px; height: 579px; line-height: 18.2px; margin: 0px; overflow-x: hidden; overflow-y: auto; padding: 10px;">
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<li class="message sent" style="font-size: 12px; line-height: 15px; list-style: none; margin-bottom: 4px;"> if A and B ARE mutually exclusive then the events cannot occur at the same time</li>
<li class="message sent" style="font-size: 12px; line-height: 15px; list-style: none; margin-bottom: 4px;">Then there is no intersection</li>
<li class="message " style="font-size: 12px; line-height: 15px; list-style: none; margin-bottom: 4px;"><br /></li>
<li class="message sent" style="font-size: 12px; line-height: 15px; list-style: none; margin-bottom: 4px;">Since they cannot both occur so in that case it's just P(A) + P(B)</li>
<li class="message sent" style="font-size: 12px; line-height: 15px; list-style: none; margin-bottom: 4px;"><br /></li>
<li class="message sent" style="font-size: 12px; line-height: 15px; list-style: none; margin-bottom: 4px;"><b>For</b> NON mutually exclusive events they can both occur at the same time</li>
<li class="message sent" style="font-size: 12px; line-height: 15px; list-style: none; margin-bottom: 4px;"><b>s</b>o we have an intersection of the events A and B, so we still have P(A) + P(B) but now see if we add A and B we are also adding the intersected part, so really adding a part of A and a Part of B twice so to just get A or B we have to remove that part, so subtract off P(A and B)</li>
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KKauffman1969http://www.blogger.com/profile/10725188874999582883noreply@blogger.com0tag:blogger.com,1999:blog-3832922229195451291.post-7916831659732948622017-03-20T07:50:00.001-07:002017-03-20T07:50:24.783-07:00hypothesis test for proportions, things to rememberHere's a few things to remember<br />
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Z scores for proportion<br />
(p^ - p)/square root(p*(1-p)/n)<br />
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Z for difference of two proportion<br />
(p1^ - p2^)/square root(p-bar(1-p-bar)/n1 + (p-bar(1-p-bar)/n2))<br />
p-bar = (x1 +x2)/(n1 + n2)<br />
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Note that p-bar might also be noted at p-pooled<br />
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For hypotheses, remember that Ho always contains = and Ha contains <, > or "does not equal"<br />
Confidence intervals for single proportion<br />
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p^ +/- Z*square root(p^(1-p^)/n)<br />
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For difference of two propotions<br />
p1^ - p2^ +/- Z*square root(p1^(1-p1^)/n1 + p2^(1-p2^)/n2)<br />
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Z values for confidence intervals<br />
90% = 1.645<br />95% = 1.96<br />98% = 2.33<br />99% = 2.575<br />
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You can also get these from Z chart<br />
<br />P-values are the value from
the Z chart for corresponding Z score if Ha contains < and 1- value
from the Z chart for corresponding Z score if Ha contains >. If Ha is
"does not equal" you take 1 - value from the Z chart for corresponding Z
score then multiply the result by 2.<br />
You can also get p-values from Z scores using the link below.<br />
<br />
<br />
http://www.socscistatistics.com/pvalues/normaldistribution.aspxKKauffman1969http://www.blogger.com/profile/10725188874999582883noreply@blogger.com0tag:blogger.com,1999:blog-3832922229195451291.post-82395596850864206092017-03-06T18:30:00.001-08:002017-03-06T18:30:17.332-08:00Finding half lifeSolving for the half life is easy.<br />
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Suppose A(t) = Ao*e^(-4t)<br />
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To find the half life, let A(t) = (1/2)Ao<br />
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(1/2)Ao = Ao*e^(-4t)<br />
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1/2 = e^(-4t)<br />
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ln (1/2) = ln(e^(-4t))<br />
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ln (1/2) = -4t<br />
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t = (-1/4)ln(1/2)<br />
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<br />KKauffman1969http://www.blogger.com/profile/10725188874999582883noreply@blogger.com0tag:blogger.com,1999:blog-3832922229195451291.post-77022283322687013382017-02-27T10:10:00.002-08:002017-02-27T10:10:57.237-08:00Bayes Theorem<div style="background-color: white; color: #333333; font-family: 'Aspira W01', HelveticaNeue, 'Helvetica Neue', Helvetica, Arial, sans-serif; font-size: 13px; line-height: 21px; margin-bottom: 12px;">
Bayes Theorem</div>
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P(B/A)=P(A and B)/P(A) but from Bayes Theorem we have<br />P(B/A) = P(A/B)*P(B)/P(A)</div>
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in our problem let A = identify correctly and B = cat person<br />therefore A' = identify incorrectly and B' = dog person</div>
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P(A) = P(A/B)*P(B) + P(A/B')P(B')</div>
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Note the tree diagram in the written work.</div>
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The values used and obtained are as follows</div>
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P(B) = .33<br />P(B') = .67<br />P(A/B) = .96<br />P(A'/B) = .04<br />P(A/B') = .71<br />P(A'/B') = .29<br />P(A and B) = .33(.76) = .3168<br />P(A' and B) = .33(.04) = .0132<br />P(A and B') = .67(.71) = .4757<br />P(A' and B') = .67(.29) = .1943</div>
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Notice that all the joint probabilities add to 1</div>
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Now put those values into the formula and you'll get P(B/A) = .3997</div>
KKauffman1969http://www.blogger.com/profile/10725188874999582883noreply@blogger.com0tag:blogger.com,1999:blog-3832922229195451291.post-53508496940168079552017-02-20T13:38:00.001-08:002017-02-20T13:38:30.170-08:00Assumptions for hypothesis test with proportions<div style="background-color: white; color: #333333; font-family: 'Aspira W01', HelveticaNeue, 'Helvetica Neue', Helvetica, Arial, sans-serif; font-size: 13px; line-height: 21px; margin-bottom: 12px;">
First we must see if np >10 and n(1-p)> 10</div>
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Also the sampling method must be a simple random sample, with only two possible outcomes, p (success) and 1-p (failure). </div>
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The sample must include at least 10 successes and 10 failures and the population size is at least 20 times larger than the sample size.</div>
KKauffman1969http://www.blogger.com/profile/10725188874999582883noreply@blogger.com0tag:blogger.com,1999:blog-3832922229195451291.post-57346750026188651982017-02-15T17:50:00.001-08:002017-02-15T17:50:07.933-08:00<div style="background-color: white; color: #333333; font-family: 'Aspira W01', HelveticaNeue, 'Helvetica Neue', Helvetica, Arial, sans-serif; font-size: 13px; line-height: 21px; margin-bottom: 12px;">
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.</div>
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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.</div>
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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.</div>
KKauffman1969http://www.blogger.com/profile/10725188874999582883noreply@blogger.com0tag:blogger.com,1999:blog-3832922229195451291.post-42489629835884752532017-02-09T12:15:00.001-08:002017-02-09T12:15:24.466-08:00<div style="background-color: white; color: #333333; font-family: 'Aspira W01', HelveticaNeue, 'Helvetica Neue', Helvetica, Arial, sans-serif; font-size: 13px; line-height: 21px; margin-bottom: 12px;">
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.</div>
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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.</div>
KKauffman1969http://www.blogger.com/profile/10725188874999582883noreply@blogger.com0tag:blogger.com,1999:blog-3832922229195451291.post-21736367091105380252017-02-02T11:45:00.001-08:002017-02-02T11:45:56.443-08:00<div style="color: #333333; font-family: 'Aspira W01', HelveticaNeue, 'Helvetica Neue', Helvetica, Arial, sans-serif; font-size: 13px; line-height: 21px; margin-bottom: 10px;">
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%</div>
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(X > 280) , we need to get the Z score, which is Z =(x -mean)/standard deviation</div>
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Z = (280 - 258)/6<br />Z = 3.67<br />1-Z(3.67) ,.... approximately 0, less then .0001</div>
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Z for upper 25th percent is .67</div>
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Mean + Z*standard deviation<br />258 +.67(6) = 262.02</div>
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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.</div>
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mean = 212.3, standard deviation is 20<br />n = 50</div>
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P(x-bar < 210)<br />Z = (x-bar -mean)/(standard deviation/square root(n))<br />Z = (210 - 212.3)/(20/square root(50))<br />Z = -0.81</div>
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Z(-0.81) = .2090</div>
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answer is .2090</div>
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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?</div>
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p = .51<br />1-p = .49</div>
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Z = (p^ - p)/(sqrt(p(1-p)/n))</div>
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Z = (.49 - .51)/(sqrt(.51*.49)/100)<br />Z = -0.4</div>
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Z(-0.4) = .3446</div>
KKauffman1969http://www.blogger.com/profile/10725188874999582883noreply@blogger.com0tag:blogger.com,1999:blog-3832922229195451291.post-89701471533242372392017-01-24T16:06:00.001-08:002017-01-24T16:06:43.853-08:00When solving an SSA triangle, you could possible have 0,1 or 2 triangles<div>
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Suppose in triangle ABC, you know side a = 10, b = 16 and angle A is 30 degrees</div>
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<br /></div>
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By the law of sines, </div>
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SinB/16 = Sin30/10</div>
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10*sinB = (1/2)!6</div>
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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</div>
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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.</div>
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If you try to solve for an angle an get Sin > 1 or < -1 then there are no solutions.</div>
KKauffman1969http://www.blogger.com/profile/10725188874999582883noreply@blogger.com0tag:blogger.com,1999:blog-3832922229195451291.post-44432361444331144372017-01-13T15:13:00.000-08:002017-01-13T15:13:33.589-08:00suppose 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.<br />
<br />
Now use the formula L(x) (linear approximation as function of x) = f(a) + f'(a)(x-a)<br />
<br />
f(0) = 0<br />
f'(x) = cosx, so f'(a) = f'(0) = 1<br />
<br />
L(x) = 0 + 1(0.3 - 0)<br />
<br />
= 0 + 0.3<br />
<br />
=0.3<br />
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The value of sin(0.3) = .295. so the approximation is close. Note that the value of x = 0.3 is in radiansKKauffman1969http://www.blogger.com/profile/10725188874999582883noreply@blogger.com0tag:blogger.com,1999:blog-3832922229195451291.post-71776985533549611452017-01-01T20:17:00.001-08:002017-01-01T20:17:47.451-08:00<div style="background-color: white; color: #333333; font-family: 'Aspira W01', HelveticaNeue, 'Helvetica Neue', Helvetica, Arial, sans-serif; font-size: 13px; line-height: 21px; margin-bottom: 12px;">
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.</div>
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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.</div>
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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.</div>
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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.</div>
KKauffman1969http://www.blogger.com/profile/10725188874999582883noreply@blogger.com0tag:blogger.com,1999:blog-3832922229195451291.post-49258154386249507122016-11-21T15:54:00.001-08:002016-11-21T15:54:30.644-08:00With 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.KKauffman1969http://www.blogger.com/profile/10725188874999582883noreply@blogger.com1tag:blogger.com,1999:blog-3832922229195451291.post-84224368042793055432016-09-29T14:01:00.001-07:002016-09-29T14:01:24.680-07:00<div style="background-color: white; color: #333333; font-family: HelveticaNeue, 'Helvetica Neue', Helvetica, Arial, sans-serif; font-size: 13px; line-height: 21px; margin-bottom: 12px;">
For the hypothesis test, we need to find the test statistics</div>
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t = (x-bar - Mu)/standard error</div>
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standard error = sample standard deviation/square root of n</div>
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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</div>
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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.</div>
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For the confidence interval it's mean +/- tcritical*standard error.</div>
KKauffman1969http://www.blogger.com/profile/10725188874999582883noreply@blogger.com0tag:blogger.com,1999:blog-3832922229195451291.post-39561378439936011382016-09-23T11:17:00.001-07:002016-09-23T11:17:46.928-07:00<span style="background-color: white; color: #333333; font-family: HelveticaNeue, 'Helvetica Neue', Helvetica, Arial, sans-serif; font-size: 13px; line-height: 21px;">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</span>KKauffman1969http://www.blogger.com/profile/10725188874999582883noreply@blogger.com0tag:blogger.com,1999:blog-3832922229195451291.post-77210966980494210802016-09-14T14:34:00.000-07:002016-09-14T14:34:30.269-07:00<div style="background-color: white; color: #333333; font-family: HelveticaNeue, 'Helvetica Neue', Helvetica, Arial, sans-serif; font-size: 13px; line-height: 21px; margin-bottom: 12px;">
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</div>
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The Var(x + y) = Var(x) + Var(y) if x and y are independent.<br />Var(x) = 6^2 = 36<br />Var(y) = 8^2 + 64<br />Var(x+y) = 100. The standard deviation is the square root of the variance, so standard deviation = 10</div>
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The mean of the difference is the difference of the means E(X - Y) = mean (x - y) = meanx - meany</div>
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So the difference of the means is 75-70 = 5</div>
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The standard deviation of the difference is the square root of the Var(x-y)</div>
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Var(x - y) = Var(x) + Var(y) if x and y are independent, same as for Var(x +y) . Therefore the answer is 10</div>
KKauffman1969http://www.blogger.com/profile/10725188874999582883noreply@blogger.com0tag:blogger.com,1999:blog-3832922229195451291.post-51195239426695012442016-08-30T11:17:00.001-07:002016-08-30T11:17:14.889-07:00Just 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.<br />
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http://www.lulu.com/spotlight/KKauffman1969KKauffman1969http://www.blogger.com/profile/10725188874999582883noreply@blogger.com0tag:blogger.com,1999:blog-3832922229195451291.post-31242156798226333012016-08-20T12:49:00.000-07:002016-08-20T12:49:05.175-07:00As 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!<br />
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http://www.lulu.com/spotlight/KKauffman1969KKauffman1969http://www.blogger.com/profile/10725188874999582883noreply@blogger.com0tag:blogger.com,1999:blog-3832922229195451291.post-6248437657594818712016-08-09T19:10:00.000-07:002016-08-09T19:10:04.871-07:00Remember 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.<br />
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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 ZKKauffman1969http://www.blogger.com/profile/10725188874999582883noreply@blogger.com0tag:blogger.com,1999:blog-3832922229195451291.post-6477182671585085282016-07-31T16:31:00.000-07:002016-07-31T16:31:44.153-07:00Note that if you want to figure out if a function has a slant asymptote, you have to realize that the equation of a slant asymptote is linear, so the exponent of the leading coefficient of the numerator of the function must be one greater than that of the denominator.<br />
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When doing long division (denominator into numerator), it might not divide evenly. Do not concern yourself with the remainder. That slant or oblique asymptote is just the linear portion.KKauffman1969http://www.blogger.com/profile/10725188874999582883noreply@blogger.com0tag:blogger.com,1999:blog-3832922229195451291.post-12224196116535952852016-07-10T12:22:00.000-07:002016-07-10T12:22:35.265-07:00<div class="wu-sendmoney-pickup-instructions-title" style="background-color: white; color: #caa13c; font-family: sans-serif; font-size: 15px; font-weight: bold; line-height: 17px; margin-bottom: 7px;">
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Suppose you roll two 6 sides dice and we want to see the outcomes and probability distribution for the difference between the two dice. 3 on the first and 2 on the second would be 3-2=1 and 2 on the first and 3 on the second would be 2-3 = -1</div>
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To get a 0 when subtracting the numbers on the dice, the numbers must be the same. This happens 6 ways.</div>
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(1,1), (2,2), (3,3), (4,4), (5,5), (6,6).. since there are 36 possibilities, we have 6/36 = 1/6</div>
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For problem 6, to subtract numbers to get -2, the second number must be two larger than the first number.</div>
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(1,3), (2,4), (3,5), (4,6) . So we have 4 out of 36 outcomes that will subtract to -2. The probability is 4/36 = 1/9</div>
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For problem 7, It's basically the same as problem 6, except now the first number is two larger than the second number.</div>
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This gives us (3,1), (4,2), (5,3), (6,4). Again this is 4/36, simplified to 1/9</div>
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For the last problem, we want the possible outcomes with corresponding probabilities</div>
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We can get 0 if the numbers are the same, which I showed in problem 5, that is probability 1/6</div>
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We can get 1 when subtracting the numbers if we have (2,1), (3,2), (4,3), (5,4), (6,5) that is 5/36</div>
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We can get -1 when subtracting if we have (1,2), (2,3), (3,4), (4,5), (5,6), again with probability 5/36</div>
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We can get 2 as shown in problem 7 with probability of 1/9<br />We can get -2 as shown in problem 6 with probability of 1/9<br />We can get 3 with rolls of (4,1), (5,2), (6,3) with probability of 3/36 = 1/12<br />We can get -3 with rolls (1,4), (2,5), (3,6) with probability of 3/36 = 1/12<br />We can get 4 with rolls (5,1), (6,2) with probability of 2/36 = 1/18<br />We can get -4 with rolls (1,5), (2,6) with probability of 2/36 = 1/18<br />We can get 5 with a roll of (6,1) with probability 1/36<br />We can get -5 with a roll of (1,6) with probability of 1/36</div>
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Putting it all together we get this model</div>
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outcome Probability<br />5 1/36<br />4 1/18<br />3 1/12<br />2 1/9<br />1 5/36<br />0 1/6<br />-1 5/36<br />-2 1/9<br />-3 1/12<br />-4 1/18<br />-5 1/36</div>
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Notice that the probabilities will add to 1. That must always be the case for a legitimate probability model and the probabilities for each event must be between 0 and 1 inclusive</div>
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KKauffman1969http://www.blogger.com/profile/10725188874999582883noreply@blogger.com0tag:blogger.com,1999:blog-3832922229195451291.post-73402439016588817682016-07-04T18:28:00.003-07:002016-07-04T18:28:52.225-07:00<div style="background-color: white; color: #333333; font-family: HelveticaNeue, 'Helvetica Neue', Helvetica, Arial, sans-serif; font-size: 13px; line-height: 21px; margin-bottom: 12px;">
Suppose you have the following distribution</div>
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x = 0, 2, 5</div>
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p(x) = 1/4, 1/4, 1/2</div>
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find the mean and variance.</div>
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To get the mean you take the sum of x(P(x))</div>
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so for x = 0, 1,5 with P(x) = 1/4, 1/4, and 1/2</div>
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You get 0(1/4) + 1(1/4) + 5(1/2) = 0 + 1/4 + 5/2 = 2.75</div>
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To calculate sigma squared (variance)</div>
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It's the [sum (x- mean)^2P(x)]/n</div>
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So we have (0 - 2.75)^2 + (2-2.75)^2 + (5-2.75)^2</div>
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The equals 7.5625 + 0.5625 + 5.0625 = 13.1875<br />now take 13.1875/3 = 4.396</div>
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For a sample size of two you can have these possibilities (0,0), (0,2), (0,5), (2,0), (2,2), (2,5), (5,0), (5,2), (5,5)</div>
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The means are the two numbers added and divided by two. That gives us<br />(0 + 0)/2 = 0<br />(0 + 2)/2 = 1<br />(0 + 5)/2 = 2.5<br />(2 + 0)/2 = 1<br />(2 + 2)/2 = 2<br />(2 + 5)/2 = 3.5<br />(5 + 0)/2 = 2.5<br />(5 +2)/2 = 3.5<br />(5 + 5)/2 + 5</div>
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So you can have mean of 0 with (0, 0) with probability (1/4)(1/4) = 1/16<br />mean of 1 with (0, 2) and (2,0) with probability 2(1/4)(1/4) = 1/8<br />mean of 2 with (2, 2) with probability of (1/4)(1/4) = 1/16<br />mean of 2.5 with (0,5) and (5,0) with probability of 2(1/4)(1/2) = 1/4<br />mean of 3.5 with (2,5) and (5,2) with probability of 2(1/4)(1/2) = 1/4<br />mean of 5 with (5,5) with probability of (1/2)(1/2) = 1/4</div>
KKauffman1969http://www.blogger.com/profile/10725188874999582883noreply@blogger.com0tag:blogger.com,1999:blog-3832922229195451291.post-6112382488407308012016-06-25T17:40:00.000-07:002016-06-25T17:40:41.539-07:00When conducting an ANOVA, it's important to realize that the test in and of itself will not tell which of the means in the null hypothesis are different from each other. The only thing we will know is whether or not all the means are equal or if one or more of the means are different. If you want to know which mean or means are different from the others you must perform one of the post hoc tests. You can find more information about these with a Google search on the topic.KKauffman1969http://www.blogger.com/profile/10725188874999582883noreply@blogger.com0tag:blogger.com,1999:blog-3832922229195451291.post-43376625081470436312016-06-11T21:45:00.004-07:002016-06-11T21:45:48.086-07:00<br />
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Suppose you need to take a log transformation on set of data that is non-linear in nature. You can tell this from a scatterplot. Take ln (natural logarithm) of each x
value and y value to complete the transformation. In doing so, you will get some
errors in x for ln(0) and negative values for ln of values between 0
and 1. So to compensate for that add 1 to each x value before taking
natural log. Then you will eliminate such problems and can get the regression equation and scatterplot accordinglyKKauffman1969http://www.blogger.com/profile/10725188874999582883noreply@blogger.com0tag:blogger.com,1999:blog-3832922229195451291.post-7975347134259849972016-06-05T07:36:00.001-07:002016-06-05T07:36:56.628-07:00If we are now testing to see if two variances are equal, as compared to where we wanted to see if the variance equaled a certain
value, we use an F test instead of a Chi-square test. So the test statistic is an F statistic equal to s1^2/s2^2 and we compare to an F
critical value found in any F chart. The decision rule is reject if F falls in the rejection region and do not reject if F is not in the rejection region. The hypotheses for this test would be Ho: variance1 = variance2, Ha: variance1 does not equal variance2KKauffman1969http://www.blogger.com/profile/10725188874999582883noreply@blogger.com0tag:blogger.com,1999:blog-3832922229195451291.post-35100617825111426042016-05-27T23:18:00.001-07:002016-05-27T23:18:43.394-07:00Remember when solving a system of equations, you can either work using substitution or elimination. When dealing with a 3 variable, 3 equation system, generally you want to eliminate a variable from two equations, then eliminate the same variable from another set of two equations. From there solve for one of the two variables in the newly formed system of 2 equations with two variables. From here you can do back substitution to find the values of the other variables. Be sure to check all answers with each equation to make sure they satisfy each.KKauffman1969http://www.blogger.com/profile/10725188874999582883noreply@blogger.com0