(Chapter 8) How does a complex survey design affect confidence intervals?

Confidence interval provides the possibility of various proportions of the sample or the sample mean of the proportion of true / it was found in the population. This allows us to estimate the accuracy of the results obtained from our sample, compared with the actual population. It usually appears as estimate plus or minus(+/-) of margin of error. Margin of error only includes random sampling error where non-responses survey are not included. Also, Outliers can have a big impact on the confidence interval. This is not surprising because we are using the mean and standard deviation to calculate the confidence interval and outliers are definitely can skewed the data and in the end give inaccurate confidence intervals. A confidence interval is a measure of the uncertainty around the sample mean, giving us the range of values ​​in which the population mean lies, with a given confidence level. In other words, confidence intervals show us the range in which 95% or 99% or 99.9% of the sample mean can be expected to lie if repeated surveys occurred. This helps us to decide whether the sample mean is reliable enough for different purposes.

For example:
The Canada government used a clustered, stratified multi-stage sample design to survey about their health care facility wellness in 2012. In addition, weights were applied when obtaining survey estimates. One of the effects of using the complex design and weighting is that standard errors for survey estimates are higher than the standard errors that would be caused from an unweighted simple random sample of the same size. Hence, the true standard errors of the complex design are calculated by multiplying the standard error (of an estimate from a simple random sample) by the design factor.

The ratio of the standard error of the complex sample to that of a simple random sample of the same size is known as the design factor.

The 95% confidence interval of a complex survey design is equal to:

p +/- (1.96 x true standard error)

where:

-true standard error = design factor x standard error of a simple random sample; and

-p = the point estimate, which is the percentage or proportion estimated from our sample (or sample mean)

Canadian government, thus, can quickly produce the confidence intervals of an estimate from a complex survey design with a big sample of Canadian citizens and they can measure how satisfied Canadian citizens with the health care that the government provided.

(Chapter 7) The Role of Probability Distribution in Business Management

A probability distribution is a statistical model that shows the possible outcomes of a particular event or course of action as well as the statistical likelihood of each event. For example, a company might have a probability distribution for the change in sales given a particular marketing campaign. The values on the “tails” or the left and right end of the distribution are much less likely to occur than those in the middle of the curve.Future events are unpredictable in the business world. For this reason, probability distributions can be a great tool for estimating future returns and profitability for businesses

-Scenario Analysis
Probability distributions can be used to create scenario analysis. A scenario analysis uses probability distributions to create several distinct possibilities for the outcome of a particular action or future event that businesses take. For example, a business might create three scenarios: worst-case, likely and best-case. The worst-case scenario would contain some value from the lower end of the probability distribution; the likely scenario would contain a value towards the middle of the distribution; and the best-case scenario would contain a value in the upper end of the scenario.

-Sales Forecasting
One use for probability distributions and scenario analysis in business is to predict future levels of sales. It is essentially impossible to predict the exact value of a future sales level; however, businesses still need to be able to thoroughly plan for future events. Using a scenario analysis based on probability distribution can help a company draw its possible future values in terms of a likely sales level and a worst-case and best-case scenario. By doing probability distribution, the company can base its business plans on the likely scenario but still be aware of the alternative possibilities.

-Risk Evaluation
In addition to predicting future sales levels, probability distribution can be a useful tool for evaluating risk. Consider, for example, a company considering entering a new business line. If the company needs to generate $500,000 in revenue in order to break even and their probability distribution tells them that there is a 10 percent chance that revenues will be less than $500,000, the company knows roughly what level of risk it is facing if it decides to pursue that new business line.

(Chapter 3) How Does Regression Analysis Can be Applied to Help Businesses

On Chapter 3, we learned various type of regressions that we used to find the model that fit perfectly to the data. Linear, Quadratic, Cubic, Power, and Exponential regressions are some example of regressions that could explain if the trends within the data have whether perfect, strong, moderate, or weak correlation between Independent and Dependent Variable. From this explanation that had been discussed in the class, it makes me wonder of how regression analysis can be applied to help businesses develop and thrive. Apparently, through some research i conducted, Regression analysis is play a vital role in business as it helps managers to make vital business decision for their companies, for example. Businesses use regression to predict such things as future sales, stock prices, currency exchange rates, and productivity gains resulting from training program. Regression models that can be used for businesses can be either linear or nonlinear. A linear model assumes the relationships between variables are straight-line relationships, while a nonlinear model assumes the relationships between variables are represented by curved lines. In business, you will often see the relationship between the return of an individual stock and the returns of the market modeled as a linear relationship, while the relationship between the price of an item and the demand for it is often modeled as a nonlinear relationship. In finance, linear regressions are commonly used to describe the returns of an individual security (dependent variable) compared to the returns of the market in general (independent variable). The equation for the simple linear regressions used to describe security movements is also a straight line and is expressed in a format, which, while similar, does contain a couple of twists.

(Chapter 2) How do surveyors planning the sampling techniques and conclude in 1 in……survey?

Throughout the course of learning Chapter 2 about statistics, chapter 2.3 which about sampling techniques was interested me the most. After learning what this chapter about, i found interesting topic to write in this chapter thoughts about how surveyors planning the sampling techniques in 1…surveys and conclude the results of their surveys. We may ever heard about studies result that state; “1 in 4 smokers are alcoholic”, “1 in 5 people in USA have mental problem” or “1 in 7 people in the world have cancer”. To explain more in depth about how the surveyors do and conclude the results of the survey, i will use the actual study about this kind of survey; “1 in 4 Users in USA lie on Facebook”. This survey is conducted by Consumer Report Investigation specifically investigate what kind of lie that people put in their facebook whether in their personal information or when they introduce themselves to a stranger in Facebook. The first step for this survey, surveyors chose a random provinces and from the selected provinces they chose cities within the province and surveyed about 20,000 people within the city that have a facebook account to be asked series of questions regarding the studies topic. Surveyors in other words used Multi-stage sampling technique to surveyed whether people who have facebook account lied or not. Furthermore, after they finished all the surveys and get the results from the surveys, they found that 25% or 5000 of the 20,000 people surveyed were lied on the facebook. Lastly, the most important part of this study is to conclude the study results using 1 in…style. The following calculation is how the surveyors conclude the results:

20,000 * 0.25= 5000 people lied
Then, 20,000/5,000= 4 represents people who did not lied

After the calculation above, Consumer Report Investigation can conclude that 1 in 4 users in USA lie on Facebook. However, this study report conclusion obviously have bias in it. Sample size in this study is too small to represents the whole population of USA. With bigger sample size, the people who lie on facebook might be bigger than the Consumer Report Investigation conclude.

(Chapter 1) Is there any relation between the Movie “Matrix” and Mathematics concept of Matrix?

Things that come to mind if we heard about matrix certainly would be the movie with all of those slow motion act to avoid bullets that come to Neo, the leading character. After we finished learning about the concept of Matrix in Chapter 1, it leaves me with a curiosity if there is any relation between the movie “Matrix” and Mathematics concept of Matrix. In the Movie “Matrix”, Thomas Anderson, played by Keanu Reeves, is a man with two lives. By day, he is an ordinary computer programmer and by night he is a hacker known as Neo. In one of the scenes in the movie, Neo was awaken to the world called Matrix where most of humanity have been captured by a race of machines that live off of the humans’ body heat and electrochemical energy and who imprison their minds within an artificial reality. This where the math taking place in the movie. In math, Matrix is an array of organized arrangement of number and we can apply operators to it for calculations. If we apply the matrix in a computer, a single operation in a split of seconds, can alter data across the board in a specified way or in other word a matrix-wide manipulation with a specific intention. In the movie The Matrix, the entire Matrix world was a digital world, an enormous database, which rather relate to Matrix of Mathematics, of information that was self-interactive and under the control of the machines and DeJaVu is a manipulation of the matrix by the machines.

(Chapter 6) How to calculate the Probability of Precipitation?

Probability of precipitation is a formal measure of likelihood of precipitation that is often published from weather forecasting model, which is often expressed as the “chance of rain” or “chance of precipitation”. Mathematically, Probability of Precipitation(Pop) is defined as:
Pop= CxA
C= the confidence that precipitation will occur somewhere in the forecast area
A= the percent of the area that will receive measurable precipitation

For example, if a forecaster says:
“THIS AFTERNOON…MOSTLY CLOUDY WITH A 40 PERCENT CHANCE OF
SHOWERS AND THUNDERSTORMS. WINDY. HIGHS IN THE LOWER 80S. NEAR
STEADY TEMPERATURE IN THE LOWER 80S. SOUTH WINDS 15 TO 25 MPH.
.TONIGHT…MOSTLY CLOUDY WITH A CHANCE OF SHOWERS AND
THUNDERSTORMS IN THE EVENING…THEN A SLIGHT CHANCE OF SHOWERS
AND THUNDERSTORMS AFTER MIDNIGHT. LOWS IN THE MID 60S. SOUTHWEST
WINDS 5 TO 15 MPH. CHANCE OF RAIN 40 PERCENT”

What does this “40 percent” mean? Will it rain 40 percent of of the time? Or Will it rain over 40 percent of the area?
The “Probability of Precipitation” (PoP) describes the chance of precipitation occurring at any point you select in the area

In this case, if the forecaster knows the precipitation will surely occur(C=100%) The forecaster is expressing how much of the area will receive measurable rain. ( PoP = “C” x “A” or “1” times “.4” which equals .4 or 40%.)

However, also, most of the time, the forecaster is expressing a combination of degree of confidence and areal coverage. If the forecaster is only 50% sure that precipitation will occur, and expects that it will produce measurable rain over about 80 percent of the area, the PoP (chance of rain) is 40%. ( PoP = .5 x .8 which equals .4 or 40%. )

Either way, the correct way to interpret the forecast is: there is a 40 percent chance that rain will occur at any given point in the area.

(Chapter 5) Real Life Application of Binomial Theorem

As we learned in Chapter 5.4, Binomial theorem is an useful method to expand the power (a+b)^n into the sum involving terms of the form nCr*a^n-r*b^r. However, we are not quite learned about what are some real life contributions that used Binomial theorem as the main tool. In fact, there are a lot of fields where the application of binomial theorem can be applied in.

For instance, there are a lot of areas where the application of binomial theorem is inevitable, even in the modern world areas such as computing. In computing areas, binomial theorem has been very useful such a in distribution of IP addresses. With binomial theorem, the automatic distribution of IP addresses is not only possible but also the distribution of virtual IP addresses.

Another field that used Binomial Theorem as the important tools is the nation’s economic prediction. Economists used binomial theorem to count probabilities that depend on numerous and very distributed variables to predict the way the economy will behave in the next few years. To be able to come up with realistic predictions, binomial theorem is used in this field.

Binomial Theorem has also been a great use in the architecture industry in design of infrastructure. It allows engineers, to calculate the magnitudes of the projects and thus delivering accurate estimates of not only the costs but also time required to construct them. For contractors, it is a very important tool to help ensuring the costing projects is competent enough to deliver profits.

(Chapter 4) Why Does 0! is equal to 1?

Why 0 Factorial is 1?

The factorial is a short way to write the multiplication of successive decreasing integer numbers. For example, 4 factorial can be written in the form of 4!, and the definition of 4! Is 4 x 3 x 2 x1 which gives the answer of 24. It is pretty clear from this definition how to calculate the factorial for any number greater than or equal to one. But why does 0 factorial is 1? With the basic knowledge of factorial, we might think 0! would be equal to 0 as well because we think that after there are no positive integer that come up after 0. In fact, if we go back to the definition of permutation, we can explain why does 0! Factorial is 1. A permutation is where we order elements in set. For example, there are six permutations of the set {1,2,3}. There are three elements in the set and there are six ways to arrange the set of {1,2,3}. We could also state this fact through the form of 3!=6. The idea of the factorial is used to calculate the number of permutations of arranging a set of n numbers. This idea can be explain in this following table:

n Number of permuations (n!) Number of Ways to arrange
1 1 {1}
2 2 {1,2},{2,1}
3 6 {1,2,3}, {1,3,2}, {2,1,3}, {2,3,1}, {3,1,2}, {3,2,1}

The set with zero elements is called empty set. Even though there is nothing to put in an order, but still there is one way to arrange that empty set. Therefore, we have 0!=1

n Number of permutations(n!) Number of ways to arrange
0 1 {}

Sources:
http://statistics.about.com/od/ProbHelpandTutorials/a/Why-Does-Zero-Factorial-Equal-One.htm

http://www.zero-factorial.com/whatis.html