Statistics: confidence interval for the mean (two sided)

Tutoring statistics, confidence intervals are important.

A two-sided confidence interval for the population mean is given by

sample_mean – (standard_dev/n1/2)*sig_factor, sample_mean + (standard_dev/n1/2)*sig_factor

The sig_factor (significance factor) depends on the certainty (confidence level) with which we want the confidence interval to include the population mean; typically it’s around 2 (aka, 1.96) for 95% confidence.

The standard deviation might be known or might be calculated from the sample itself. If it’s known, the normal distribution is used; if calculated, then technically the t-distribution should be used (see point 3 below).

There are a few points that make the two-sided confidence interval for the population mean an elegant construct:

  1. Its lower and upper boundaries depend on the sample size, but not the population size.
  2. For sample size n≥31, the parent population needn’t be normal for the sample mean to be normally distriubted. This validates the confidence interval even for a non-normal population for n≥31. It’s a consequence of the Central Limit Theorem. (Actually, the rule of thumb is n≥30, but for the purpose of the next point, I like 31.)
  3. For n≥31, the t-distribution approximates the normal to around 4%, so the normal approximation can probably be used even for unknown population standard deviation.


Harnett, Donald L. and James L. Murphy. Statistical Analysis for Business and Economics, first Can. ed. Don Mills: Addison-Wesley, 1993.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Economics: US trade deficit: Is it real?

The tutor probes more deeply into the concept of the US trade deficit.

In my January 1 post I began about the US trade deficit, mentioning, for example, that it’s exceeded $500 billion per year over the last ten years. The numbers might sound menacing, but are they really?

Between 2003, and 2015, the net worth of US households rose from $60 trillion to $84.9 trillion (inflation adjusted). During that time, the US trade deficit summed to around $7.4 trillion. The US federal debt increased, during that interval, by about $11.2 trillion. Subtracting the outflows of $7.4 trillion and $11.2 trillion from the wealth increase of $24.9 trillion still shows net growth of $6.3 trillion.

The clear question: Is money really leaving the US, as many suggest, or is it actually accumulating in the US?

I’ll be following up:)


Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Economics: oligopoly vs monopolistic competition

The tutor compares big business with small business.

In yesterday’s post I talked about firms in monopolistic competition – the situation characterized by

  1. negligible boundary to entry
  2. large number of small firms competing
  3. average long-run profit approx. zero

Oligopoly is rather the opposite case, characterized by

  1. difficult boundary to entry (perhaps legal, but capital as well)
  2. a few large firms competing

An example of oligopoly might be the computer chip industry.

In the case of monopolistic competition, a given firm holds its market share due to some distinction it offers. Perhaps it’s the firm’s location, or the personalities of its staff. That distinction gives the firm an effective monopoly with clientele that prefer it; hence, monopolistic competition.

In the case of oligopoly, each firm typically enjoys a technological advantage that enables it to compete. For example, for anti-virus software, some people prefer Bitdefender, while others prefer McAfee. A firm’s technological advantage needn’t only be scientific; it can be management as well. Hence a given customer’s preference towards a particular airline or grocery chain.

In a coming post I’ll be looking at the profitability challenge faced by large firms.


Parkins, Michael and Robin Bade. Economics: Canada in the Global Environment, 2nd ed. Don Mills: Addison-Wesley, 1994.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Economics: small business: monopolistic competition

The tutor looks into monopolistic competition and the long-run trend faced by its entrants.

Monopolistic competition exists, in a given business, when there is free entry – that is, when starting a new enterprise engaged in that business is easy. An example might be lawn care. Theoretically, anyone with a lawn mower and possibly a few other implements can start a lawn care business and win market share. The result is that, with monopolistic competition, there tend to be numerous, small, local firms engaged in that same business.

A given firm facing monopolistic competition can generate long-term profit. However, the typical trend faced by a firm in monopolistic competition is as follows:

  1. At first, there is much more demand than supply, inducing one or two “startups”. Those early firms enjoy a few years of rewarding profitability.
  2. Noticing the profitability of the original “startups”, other people decide to join that same market: they start firms that engage in the same business. They can do so easily because there is free entry.
  3. Eventually, enough new firms start so that each has a relatively small portion of the market – to the point that each firm’s market share is only enough to cover its overhead. On average, the profitability of the business is zero.

An economist, using more technical language, might explain the above scenario as follows:

As long as demand exceeds supply, the firm can set its price above its cost per unit and enjoy profitability. However, as competition emerges, supply increases, so the firm loses its power to set its own price, yet maintain its market share. With free entry, competition will be plentiful, so supply will surely grow. Eventually, the typical firm will only be able to charge enough to cover its costs. When production is within capacity, the cost per unit decreases as more units are sold. Therefore, if the firm could sell more units for the same price, its unit cost would decrease, allowing profit; however, with so much competition, it cannot expect to increase its market share. In the case of monopolistic competition, the early entrants to the market might likely enjoy short-run profitability, but long-run profitability will typically be zero.


Parkin, Michael and Robin Bade. Economics: Canada in the Global Environment, 2nd ed. Don Mills: Addison-Wesley, 1994.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Business math: a word problem about market share.

The tutor works a word problem to find time until market share equalization.

Imagine a relatively new device, of which there are two competing versions, A and B. (We assume that no-one owns both.) The potential market is 100 million, but at present, 20 million own A, while 12 million own B. Currently, however, B outsells A 2:1. Combined sales total 600 000 per month.

Assuming only new sales (as opposed to replacement), and only one per customer, when will ownership of B equal A? How many of each will have sold by then? What will be the total market penetration?



x=new sales of A (which means 2x=new sales of B)

We want the owned units of A to equal the owned units of B

20 000 000 + x = 12 000 000 +2x

Subtracting x and also 12 000 000 from both sides gives

8 000 000 = x

Right now, 20 000 000 people own A. So when 8 000 000 more units of A have sold, 28 000 000 will own one. During that time, 16 million units of B will sell; 12 million people own B right now. 16 million + 12 million = 28 million, the same as A will be.

At market equalization, 28 000 000 customers will own each. Therefore, 56 000 000 will own one or the other: total market penetration will be 56%.

How long will market equalization take? Of the 600 000 units selling each month, 200 000 are A (while 400 000 are B). At 200 000 units per month, the time for 8 000 000 units to sell is 8 000 000 ÷ 200 000 = 40 months.



Tan, S.T. Applied Finite Mathematics. Boston: PWS-KENT, 1990.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Economics: the US trade deficit

The tutor looks into the export of US dollars.

Since sometime in the late 1970s, US imports have exceeded exports. More specifically, the US trade deficit has exceeded $500 billion per year over the last ten years.

A casual observer might ask, “How long can the US continue running such a trade deficit?” Apparently, as of 2015, US households held net worth of $84.9 trillion: the US could continue a $500 billion trade deficit annually for 169.8 years.


Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Business and Economics: The price of oil, Part II

The tutor looks into why oil is less than half its price two years ago.

Back when oil was over $100, I thought I understood the reason: China, India, and other developing countries have a growing middle class. Those swelling ranks are buying cars: there are likely thousands, if not tens of thousands, of new drivers on the planet each month. They didn’t use to buy gasoline; their sudden demand for it increases world oil consumption.

At the same time, I reasoned, oil is relatively rare compared with the demand for it. Supply cannot necessarily increase as quickly as the developing world’s demand. Therefore, the price should be much higher than it was before the rise of India, China, Brazil, and other transforming economies.

In fact, the supply of oil is more responsive than I (and perhaps some others) realized. American production of oil more than doubled from around 4 million bpd (barrels per day) in 2009 to 9.7 million bpd in 2015. Considering world oil consumption is about 96 million bpd, the Yanks can supply 10 percent of the entire world’s needs.

At the same time, US oil consumption has decreased by around 12% since 2005. It suggests that, perhaps, economies will eventually progress past dependence on oil.

Some oil exporters are now reflecting that oil not sold today, may never be. Therefore, the oil that’s cheap to produce should be sold at a profit now. That idea, almost impossible to imagine fifteen years ago, squares off against the opposite, much older one: that development means oil consumption.

It’s very hard to predict the price of oil:)


Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Business and Economics: ripples from the price of oil

With some reflections, the tutor opens a discussion about the price of oil.

For years, I tutored high school math to young men headed to Alberta or Saskatchewan. Months later, in the oil fields, they made several times what I did. I recall hearing that one of my students, two years after his tutoring with me, was making a quarter million a year.

I never resented what the oil workers made; I celebrated it. Vancouver Island seems not to have a lot of industrial opportunity; optimistically, there was opportunity you could point to, east.

After the closure of the pulp and paper mill here – I believe in 2009 – many of Campbell River’s fathers left for the oil fields as well. Often they made much more there. Oil receipts kept families living here.

Eventually, a high price of oil came to mean prosperity for Campbell River. With so many of our people working in Fort McMurray or other oil towns, the job situation here often wasn’t considered. Rather, the job situation there was foremost. Campbell River remained an industrial town; the industry was just being carried on in the oil fields.

Inevitably, the drop in oil prices has brought job loss. In my own mind, a question has emerged that I haven’t considered since university, over 20 years ago: Is industrial money too fickle to plan a future around?

As my wife points out, when the people involved in primary (production) industries are working, they seem to make more than anyone. Their earnings infuse their surroundings with opportunity. In my experience, there is no better place to live than a blue-collar town where the mill, mine, or plant is hiring.

Yet, primary industry – the oil industry being a perfect example – seems the most vulnerable to economic cycles. Jobs like teaching or medical professions, as well as trades like auto mechanics or plumbing, seem much more secure. Considering the future, a person can’t ignore these considerations – especially in a place like Vancouver Island, where primary industry seems to be on a long-term decline.

I’ll be talking more about the price of oil in coming posts:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Business and Economics: Holding Companies, part 0

The tutor begins exploring the frontier of holding companies.

Years ago, training in financial planning, I read about holding companies – mainly about their tax treatment. What I learned was that holding companies can be complex situations.

A holding company’s purpose is to possess assets – usually, shares in businesses. Essentially, it’s a wealth management tool: the money is made by the businesses whose shares the holding company possesses. That money is then transferred to the holding company.

What are the advantages of a holding company, in Canada? As I understand, there are three main ones:

  1. Tax Advantage: Potentially, the profits from a Canadian corporation can be received as dividends by the holding company without taxation (at the time).
  2. Risk Insulation: If the business that makes the money is sued, the profits it has sent to the holding company are (likely) not available to the plaintiff. Only the business’s own assets are vulnerable to seizure. Since the holding company receives the business’s profits, that business may have little assets to seize, even if it is very profitable.
  3. Wealth Transfer: Net worth can be transferred via shares of the holding company that stores the assets, which is more convenient than changing the ownership of specific assets, especially if they are currently invested.

I’ll be delving deeper into this fascinating world.


Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Accounting and Business: Depreciation: Sum-of-years’-digits method

The tutor shares a really neat find from the world of accounting.

Although I’m an academic, the first post-secondary diploma I received, believe it or not, was in accounting and office management. While I haven’t used it much professionally, what I learned from that program keeps me interested in related topics, especially theoretical ones.

Depreciation – the loss of value an asset experiences over time – is a great example of a concept that can be fascinating. One reason: there are several ways to calculate it. Just today I’ve discovered the sum-of-years’-digits method of calculating depreciation. What an elegant method it is:


Calculate the yearly depreciation of a 60,000 asset that will last four years, then have a salvage value of 8850.


First, we subtract salvage value from the new value, to arrive at the lost value:

60000-8850 = 51150

Next, we add up the years’ digits:

1+2+3+4 = 10

The greatest digit (in this case, 4) is then divided by their sum (in this case, 10); that quotient is multiplied by the value that will be lost (51150) to get the first year’s depreciation amount. The process is repeated with the second greatest digit, and so on, so that the depreciation, year by year, is as follows:

Year Depreciation amount
1 (4/10)*51150 = 20,460
2 (3/10)*51150 = 15345
3 (2/10)*51150 = 10230
4 (1/10)*51150 = 5115

I love the straightforward calculation procedure of the sum-of-years’-digits method, and hope you will too:)


Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.