The Central Limit Theorem

Wheelan, Chapter 8, 9 and 11
➔ Chapter 8 : The Central Limit Theorem
◆ “The central limit theorem is the ‘power source’ for many of the statistical activities that involve using a
sample to make inferences about a large population.”
◆ “The core principle underlying the central limit theorem is that a large, properly drawn sample will resemble
the population from which it is drawn.”
◆ “If a sample usually looks like the population from which it’s drawn, it must also be true that a population
will usually look like a sample drawn from that population.”
◆ Per the central limit theorem, sample means (averages) for any population will be distributed roughly as a
normal distribution around the population mean.
◆ This means we can use the same standard deviation probabilities from Chapter 2 when analyzing data and
making statistical inferences.
◆ Standard error measures dispersion of sample means. It is the standard deviation of a sample’s distribution (or
an estimate of that standard deviation).
● Standard deviation measures dispersion in the underlying population.
● Standard error measures dispersion of sample means.
● “A large standard error means that the sample means are spread out widely around the population
mean; a small standard error means that they are clustered relatively tightly.”
● “Sample means will cluster more tightly around the population mean as the size of each sample
gets larger.”
● Standard error = standard deviation of population from which the population is drawn divided by
the square of the sample size.
◆ “The less likely it is that an outcome has been observed by chance, the more confident we can be in
surmising that some other factor is in play.”
➔ Chapter 9 : Inference
◆ “Statistics cannot prove anything with certainty. Instead, the power of statistical inference derives from
observing some pattern or outcome and then using probability to determine the most likely explanation for
that outcome.”
◆ Remember: The most likely explanation isn’t always the correct one. Improbable things do happen
◆ Statistical inference: The use of statistical analysis to answer important questions. For example: Is a new drug
effective at treating heart disease? Does increased school spending result in better student outcomes?
◆ Hypothesis testing:
● Statistical inference begins with an implicit or explicit null hypothesis (starting assumption).
● If we reject the null hypothesis, then we must accept an alternative hypothesis consistent with the
data observed.
● Example: In a court of law, the null hypothesis is that the defendant is innocent. The prosecution
must convince the judge or jury to reject the null hypothesis and accept the alternative hypothesis
(that the defendant is guilty).
◆ Significance level: The arbitrary probability threshold at which a null hypothesis can be rejected.
● A common threshold is 5% (typically represented in decimal form as .05).
● P-value is the specific probability. In statistics, the p-value is the probability of obtaining results at
least as extreme as the observed results of a statistical hypothesis test. A smaller p-value means that
there is evidence favoring the alternative hypothesis.
● “When we can reject a null hypothesis at some reasonable significance level, the results are said to
be statistically significant.”
● “Statistical significance says nothing about the size of the association.”
◆ Confidence interval: a range associated with a given probability level. Example: we can say with 95%
confidence (95 times out of 100) that the interval between 1202.8 and 1274.8 cubic cm includes the average
brain volume for children in the general population without autism.
◆ Choosing an appropriate significance level involves a trade-off:
● A low burden of proof (e.g. 0.1 or 10%) means we are more likely to reject the null hypothesis
when it is true. Alternatively we may erroneously select a false alternative (false positive). This is
called a Type I error.
○ Example: An innocent person is convicted.
● A high burden of proof (e.g. .01 or 1%) means we are more likely to reject an alternative
hypothesis when it is true (false negative). This is called a Type II error.
○ Example: A guilty person is not convicted.
● Consider some real world situations involving the trade-off:
○ Spam filters: Null hypothesis is that a message is not spam.
◆ Type I error means screening emails that are not spam (false positive).
◆ Type II error means allowing spam into your inbox (false negative).
● Cancer screening: Null hypothesis is that there is no cancer.
○ Type I error means identifying cancer in healthy patients (false positive).
○ Type II error means failing to identify cancer in sick patients (false negative).
● Capturing terrorists: Null hypothesis is that a person is not a terrorist.
○ Type I error identifies innocent people as terrorists (false positive).
○ Type II error means failing to identify a real terrorist (false negative).
➔ Chapter 11
◆ Regression analysis : A tool that quantifies the relationship between variables while controlling for other
◆ Regression analysis finds the “best fit” linear relationship between two variables.
◆ Ordinary least squares (OLS): A methodology that fits the line based on minimizing the sum of squared
residuals (vertical distance from the regression line).
◆ Linear regression equation: Y = a + bX
● Y is the y-axis variable (dependent variable)
● a is the y-intercept (value for Y when X = 0)
● X is the x-axis variable (explanatory variable)
● b is the slope of the line
◆ Basic terminology (using an example that considers the relationship between height and weight).
● Dependent variable is the variable being explained (it depends on other factors). Example: weight
● Explanatory variable or i ndependent variable is the variable that explains our dependent
variable. Example: height (x-axis).
● Weight = -135 + (4.5) * Height in Inches
○ Y = a + bX
○ 4.5 is the c oefficient of height. Every one-unit increase in the independent variable is
associated with a 4.5 unit increase in the dependent variable (weight).
○ In other words: without any other information, we could use the height of a person from
the studied population to generate a good guess about their weight.
● Regression coefficient has three interesting characteristics:
○ Sign (positive or negative): Shows the direction of the association with the dependent
variable. In the above example, the coefficient on height is positive.
○ Size: How big is the observed effect between independent and dependent variables? In
the above example, the size is 4.5 pounds which is significant.
○ Significance: Is the association between variables meaningful? One way to confirm is to
determine if other samples express this association.
○ A standard error can be calculated for the coefficient. This measures dispersion of the
value in repeated analyses on further samples.
○ “One rough rule of thumb is that the coefficient is likely to be statistically significant
when the coefficient is at least twice the size of the standard error.”
● R2 (R-squared): The measure of the total amount of variation explained by the regression equation.
○ In the example above, R2 tells us how much of the variation around the mean is
associated with differences in height alone.
○ For the weight-height analysis, the R2 is .25 (25%). Which is to say that 75% of the
variance is unexplained by the analysis (perhaps factors such as sex, diet, age, genetics,
◆ Regression analysis can include multiple variables . A m ultivariate regression or m ultiple regression
analysis estimates the linear relationship between each explanatory variable and the dependent variable
(while holding other dependent variables constant).
Eisenhower, “A Chance for Peace”
➔ When Stalin died in 1953, Eisenhower saw a chance to change the tone with the new Soviet leadership and bring the
Cold War to a close.
➔ This speech concentrated on the future and sent a warning that the enemy on the outside could ruin the country from
◆ The United States, like the rest of the Western world, desired peace with the Soviet Union, as shown by this
➔ Clear precepts that governs the US’s conduct in world affairs
◆ No people on earth can be held, as a people, to be an enemy, for all humanity shares the common hunger for
peace and fellowship and justice
◆ No nation’s security and well-being can be lastly achieved in isolation but only in e ffective cooperation with
fellow nations
◆ Any nation’s right to a form of government and an economic system of its own choosing is inalienable
◆ Any nation’s attempt to dictate to other nations their form of government is indefensible
◆ A nation’s hope of lasting peace cannot be firmly based upon any race in armaments but rather upon just
relations and honest understanding with all other nations
➔ Reminders in the Speech
◆ As long as freedom has a threat, free nations must remain armed and strong for the risk of war
◆ The best would be this: a life of perpetual fear and tension; a burden of arms draining the wealth
and the labor of all peoples; a wasting of strength that defies the American system or the Soviet
system or any system to achieve true abundance and happiness for the peoples of this earth.
● Trade-offs → “ this world in arms is not spending money alone”
○ power plants, wheat, etc.
◆ Leader turnover as a political opportunity ⇒ “i ts future is, in great part, its own to make.”
◆ Strength to serving the needs, rather than the fears, of the world
◆ Ready, by these and all such actions, to make of the United Nations an institution that can effectively guard
the peace and security of all peoples
➔ The Plan
◆ The limitation, by absolute numbers or by an agreed international ratio, of the sizes of the military and
security forces of all nations
◆ A commitment by all nations to set an agreed limit upon that proportion of total production of certain
strategic materials to be devoted to military purposes
◆ International control of atomic energy to promote its use for peaceful purposes only and to insure the
prohibition of atomic weapons
◆ A limitation or prohibition of other categories of weapons of great destructiveness
◆ The enforcement of all these agreed limitations and prohibitions by adequate safeguards, including a practical
system of inspection under the United Nations
Ismay, John. 2013, “What Would a Fighter Jet Buy 60 Years After Eisenhower’s Speech?”
➔ President Dwight D. Eisenhower
◆ He believed that Stalin’s death presented an opening to end the accelerating arms race
● Speechwriters were told to convey the desire to Russia
● Tactics
○ Idea of future peace that unified Germany → removing occupying forces in Austria and
human terms of what both sides lost when spending their wealth on armaments
◆ More Tactics
● Talk about money that could go elsewhere, other than “jet plane”(s) and other armed weapons
● Avoided addressing the UN → did not want delegates to immediately make his words lose meaning
● Delivering his message to the American Society of Newspaper Editors, Mr. Hughes and Paul Nitze
○ The numbers provided in his address were based off of no evidence, BUT commodity
does give the idea that these numbers were not completely random
➔ What Things Could Buy 60 Years Later (PRICES IN TODAY’S WORLD)
◆ Modern Bomber
● The B-47 Stratojet cost $2,440,000 in 1951 → compared to the most expensive bomber today, B-2
Spirit, Stealth Bomber c ost $1,461,500,000
● “Modern brick school” was the first & now costs $14,800,000
○ One B-2 would buy 99 schools priced at this cost
● Electric power plants → costs 1 megawatt of electricity powers 749 houses, 80 megawatts would
be needed to power 60,000 homes.. A power plant → costs $78,800,000
○ One B-2 would by 19 plants
● Constructing a hospital without equipment and such would cost $231,000,000
○ One B-2 would buy six of these hospitals
● 50 miles of roadway → costs $222,400,000
○ One B-2 would buy 328 miles
◆ Fighter Jet
● F-22 Raptor is worth about $214,000,000 per plane
● Today, a wheat bushel is $7 per bushel → half-million costs $3,500,000
○ One F-22 Raptor would buy 29,500,000 bushels of wheat today
◆ Destroyer
● Single family house ( 4 people) → costs $173,600 in Feb. 2013,
○ Building enough for 8,000 people would cost $347,200,000
● Flight IIA DDG-51 Arleigh Burke – class destroyer costs about $1.5 billion
○ Would put d urable roofs o ver a little more than 34,000 Americans
● “Flight III” → costs $3 billion to $4 billion apiece
○ Enough to rebuild every home destroyed by Hurricane Sandy in NJ
➔ Conclusion
◆ Current total defense spending represents 4.3 percent of GDP
● The defense spending decreased since Eisenhower, the cost of each item Eisenhower listed has
grown tremendously since then
◆ Eisenhower was a five-star general, BUT he did decline requests made, such as proposals for new weapons
◆ Impact
● He believed in the safety of the US military, but he could not assure that it would be given
● We cannot completely assure safety given any amount of spending, we can definitively show what
that spending could otherwise be used for.
➔ Theories (n ot from document)
◆ Sees an alliance between the warrior class and the corporate interests, or the “military-industrial complex”
● Came from Eisenhower
Heo, Uk and Min Ye. 2016, “Defense Spending and Economic Growth around the Globe: The Direct and Indirect Link.
International Interactions”
➔ National Security
◆ Security concerns dominate domestic politics
● Military receives a large share of budget spending in MOST countries
● Some politicians believe that expanding (b y increasing military budget spending) is the state’s
ability to accomplish other goals while giving the country strength
➔ Three Groups of Research
◆ First one
● Argues that defense spending stimulates the state’s economy by increasing purchase power and
aggregate demand and by producing positive externalities → accelerating economic growth
◆ Second One
● Argues a negative relationship between defense spending and economic growth → they focus on
trade-offs ⇒ the allocation effect (g un v butter tradeoff) & growth effect (g un v growth tradeoff)
◆ Third One
● No meaningful relationship → could positively or negatively affect the economic growth
➔ Demand-Side Model
◆ Notion of crowding out
● One source demands scarce or capital resources with another source
● Dependent variables → include savings, investment, unemployment. Education, public health
● Lacks a dominant theoretical framework for empirical research
➔ Supply-Side Model
◆ Aggregate production function and investigate the direct economic effects of defense spending on growth
◆ Has a strong theoretical framework
◆ “Demand effects operate through the level and composition of expenditure. The most obvious is the
Keynesian multiplier effect: an exogenous rise in military spending increases demand, and if there is spare
capacity, increases utilization and reduces unemployment of resources. . . . Military expenditures have
opportunity costs and may crowd-out other forms of expenditures, such as investment.”

○ The direct and indirect link of the defense growth nexus
➔ Explanation of the Theory and Models
◆ Investment
● Most common argument → indirect effects of defense spending on economic growth is military
spending has an opportunity cost that comes from a trade off between defense spending and civil
resource use
○ Guns v butter tradeoff
◆ Increasing military spending leads to a reduction in social welfare spending
◆ Came from Eisenhower
● Military spending and private investment compete for human and capital resources in similar
○ Ex. capital-goods industries, such as engineering, electronics, aerospace, etc., produce
most military equipment. Thus, changes in defense spending lead to altered demand and
supply for human and capital resources in these industries. Due to the short-term
inelasticity in the supply of human and capital resources, these industries adjust to the
new situation and reduce investment
◆ Testing the Effects of Defense Spending on Investment
● Assumes investment as a function of national income

Get Professional Assignment Help Cheaply

Buy Custom Essay

Don't use plagiarized sources. Get Your Custom Essay on
The Central Limit Theorem
Just from $9/Page
Order Essay

Are you busy and do not have time to handle your assignment? Are you scared that your paper will not make the grade? Do you have responsibilities that may hinder you from turning in your assignment on time? Are you tired and can barely handle your assignment? Are your grades inconsistent?

Whichever your reason is, it is valid! You can get professional academic help from our service at affordable rates. We have a team of professional academic writers who can handle all your assignments.

Why Choose Our Academic Writing Service?

  • Plagiarism free papers
  • Timely delivery
  • Any deadline
  • Skilled, Experienced Native English Writers
  • Subject-relevant academic writer
  • Adherence to paper instructions
  • Ability to tackle bulk assignments
  • Reasonable prices
  • 24/7 Customer Support
  • Get superb grades consistently

Online Academic Help With Different Subjects


Students barely have time to read. We got you! Have your literature essay or book review written without having the hassle of reading the book. You can get your literature paper custom-written for you by our literature specialists.


Do you struggle with finance? No need to torture yourself if finance is not your cup of tea. You can order your finance paper from our academic writing service and get 100% original work from competent finance experts.

Computer science

Computer science is a tough subject. Fortunately, our computer science experts are up to the match. No need to stress and have sleepless nights. Our academic writers will tackle all your computer science assignments and deliver them on time. Let us handle all your python, java, ruby, JavaScript, php , C+ assignments!


While psychology may be an interesting subject, you may lack sufficient time to handle your assignments. Don’t despair; by using our academic writing service, you can be assured of perfect grades. Moreover, your grades will be consistent.


Engineering is quite a demanding subject. Students face a lot of pressure and barely have enough time to do what they love to do. Our academic writing service got you covered! Our engineering specialists follow the paper instructions and ensure timely delivery of the paper.


In the nursing course, you may have difficulties with literature reviews, annotated bibliographies, critical essays, and other assignments. Our nursing assignment writers will offer you professional nursing paper help at low prices.


Truth be told, sociology papers can be quite exhausting. Our academic writing service relieves you of fatigue, pressure, and stress. You can relax and have peace of mind as our academic writers handle your sociology assignment.


We take pride in having some of the best business writers in the industry. Our business writers have a lot of experience in the field. They are reliable, and you can be assured of a high-grade paper. They are able to handle business papers of any subject, length, deadline, and difficulty!


We boast of having some of the most experienced statistics experts in the industry. Our statistics experts have diverse skills, expertise, and knowledge to handle any kind of assignment. They have access to all kinds of software to get your assignment done.


Writing a law essay may prove to be an insurmountable obstacle, especially when you need to know the peculiarities of the legislative framework. Take advantage of our top-notch law specialists and get superb grades and 100% satisfaction.

What discipline/subjects do you deal in?

We have highlighted some of the most popular subjects we handle above. Those are just a tip of the iceberg. We deal in all academic disciplines since our writers are as diverse. They have been drawn from across all disciplines, and orders are assigned to those writers believed to be the best in the field. In a nutshell, there is no task we cannot handle; all you need to do is place your order with us. As long as your instructions are clear, just trust we shall deliver irrespective of the discipline.

Are your writers competent enough to handle my paper?

Our essay writers are graduates with bachelor's, masters, Ph.D., and doctorate degrees in various subjects. The minimum requirement to be an essay writer with our essay writing service is to have a college degree. All our academic writers have a minimum of two years of academic writing. We have a stringent recruitment process to ensure that we get only the most competent essay writers in the industry. We also ensure that the writers are handsomely compensated for their value. The majority of our writers are native English speakers. As such, the fluency of language and grammar is impeccable.

What if I don’t like the paper?

There is a very low likelihood that you won’t like the paper.

Reasons being:

  • When assigning your order, we match the paper’s discipline with the writer’s field/specialization. Since all our writers are graduates, we match the paper’s subject with the field the writer studied. For instance, if it’s a nursing paper, only a nursing graduate and writer will handle it. Furthermore, all our writers have academic writing experience and top-notch research skills.
  • We have a quality assurance that reviews the paper before it gets to you. As such, we ensure that you get a paper that meets the required standard and will most definitely make the grade.

In the event that you don’t like your paper:

  • The writer will revise the paper up to your pleasing. You have unlimited revisions. You simply need to highlight what specifically you don’t like about the paper, and the writer will make the amendments. The paper will be revised until you are satisfied. Revisions are free of charge
  • We will have a different writer write the paper from scratch.
  • Last resort, if the above does not work, we will refund your money.

Will the professor find out I didn’t write the paper myself?

Not at all. All papers are written from scratch. There is no way your tutor or instructor will realize that you did not write the paper yourself. In fact, we recommend using our assignment help services for consistent results.

What if the paper is plagiarized?

We check all papers for plagiarism before we submit them. We use powerful plagiarism checking software such as SafeAssign, LopesWrite, and Turnitin. We also upload the plagiarism report so that you can review it. We understand that plagiarism is academic suicide. We would not take the risk of submitting plagiarized work and jeopardize your academic journey. Furthermore, we do not sell or use prewritten papers, and each paper is written from scratch.

When will I get my paper?

You determine when you get the paper by setting the deadline when placing the order. All papers are delivered within the deadline. We are well aware that we operate in a time-sensitive industry. As such, we have laid out strategies to ensure that the client receives the paper on time and they never miss the deadline. We understand that papers that are submitted late have some points deducted. We do not want you to miss any points due to late submission. We work on beating deadlines by huge margins in order to ensure that you have ample time to review the paper before you submit it.

Will anyone find out that I used your services?

We have a privacy and confidentiality policy that guides our work. We NEVER share any customer information with third parties. Noone will ever know that you used our assignment help services. It’s only between you and us. We are bound by our policies to protect the customer’s identity and information. All your information, such as your names, phone number, email, order information, and so on, are protected. We have robust security systems that ensure that your data is protected. Hacking our systems is close to impossible, and it has never happened.

How our Assignment Help Service Works

1. Place an order

You fill all the paper instructions in the order form. Make sure you include all the helpful materials so that our academic writers can deliver the perfect paper. It will also help to eliminate unnecessary revisions.

2. Pay for the order

Proceed to pay for the paper so that it can be assigned to one of our expert academic writers. The paper subject is matched with the writer’s area of specialization.

3. Track the progress

You communicate with the writer and know about the progress of the paper. The client can ask the writer for drafts of the paper. The client can upload extra material and include additional instructions from the lecturer. Receive a paper.

4. Download the paper

The paper is sent to your email and uploaded to your personal account. You also get a plagiarism report attached to your paper.

smile and order essay GET A PERFECT SCORE!!! smile and order essay Buy Custom Essay

Place your order
(550 words)

Approximate price: $22

Calculate the price of your order

550 words
We'll send you the first draft for approval by September 11, 2018 at 10:52 AM
Total price:
The price is based on these factors:
Academic level
Number of pages
Basic features
  • Free title page and bibliography
  • Unlimited revisions
  • Plagiarism-free guarantee
  • Money-back guarantee
  • 24/7 support
On-demand options
  • Writer’s samples
  • Part-by-part delivery
  • Overnight delivery
  • Copies of used sources
  • Expert Proofreading
Paper format
  • 275 words per page
  • 12 pt Arial/Times New Roman
  • Double line spacing
  • Any citation style (APA, MLA, Chicago/Turabian, Harvard)

Our guarantees

Delivering a high-quality product at a reasonable price is not enough anymore.
That’s why we have developed 5 beneficial guarantees that will make your experience with our service enjoyable, easy, and safe.

Money-back guarantee

You have to be 100% sure of the quality of your product to give a money-back guarantee. This describes us perfectly. Make sure that this guarantee is totally transparent.

Read more

Zero-plagiarism guarantee

Each paper is composed from scratch, according to your instructions. It is then checked by our plagiarism-detection software. There is no gap where plagiarism could squeeze in.

Read more

Free-revision policy

Thanks to our free revisions, there is no way for you to be unsatisfied. We will work on your paper until you are completely happy with the result.

Read more

Privacy policy

Your email is safe, as we store it according to international data protection rules. Your bank details are secure, as we use only reliable payment systems.

Read more

Fair-cooperation guarantee

By sending us your money, you buy the service we provide. Check out our terms and conditions if you prefer business talks to be laid out in official language.

Read more
error: Content is protected !!
Open chat
Need assignment help? You can contact our live agent via WhatsApp using +1 718 717 2861

Feel free to ask questions, clarifications, or discounts available when placing an order.
  +1 718 717 2861           + 44 161 818 7126           [email protected]
  +1 718 717 2861         [email protected]