What's the theoretical maximum score on the TV show Jeopardy? To get the maximum possible score, one has to ring first on all answers (or have a different first ringer answer incorrectly), answer everything correctly, bet the maximum amount on all occasions, and then a luck factor plays in. The Daily Double clues are located at the optimal spot. If all of the non-luck factors are obtained, what's the probability that yields all of the lucks in place to obtain the theoretical maximum score?
In the first round, there are 6 categories, with each with $200, $400, $600, $800, and $1000 clues. Add them up and multiply it by six: 6*(600*5) = 18,000. However, there is one Daily Double clue in this first round. Since the Daily Double clue annuls the monetary value originally associated with the clue, the Daily Double has to be hidden behind a $200 clue to minimize the annulling, and thus maximizing the overall score. Subtract 200 from 18,000 leaves 17,800. If everything is wagered on Daily Double and the answer is correct, that leaves 17,800*2 = $35,600 just from the first round.
In the Double Jeopardy round, everything is doubled: 6 categories, each with $400, $800, $1200, $1600, $2000. Furthermore, there are two Daily Doubles. Start with the $18,000 figure calculated from the previous round. Double that, since it's Double Jeopardy: 18,000*2 = 36,000. Now, we need to backtrack two of the $400 values to account for the Daily Double. That leaves 36,000 - 2*400 = $35,200 gained purely from Double Jeopardy, without the Daily Doubles. Don't forget to add the amount from first round, and that leaves 35,600 + 35,200 = $70,800. Now for the two Daily Doubles, as well as the Final Jeopardy, everything is wagered and answer correctly. That is 3 times of doubling the score: 70,800*2^3 = 70,800*8 = $566,400.
Now the question is, if suppose someone can ring first on all answers, answer everything correctly, and still bet the maximum amount on all
occasions, what's the probability that the Daily Doubles will be located in the correct spot to allow this theoretical maximum score? Well, in the first round, it has to be hidden in one of the $200 clues. That's a 1/5 chance of that happening. Independently from that, the chance of the two Daily Doubles both hidden under $400 in Double Jeopardy is (6/30)*(5/29). The first one can be in any of the six $400 slots, out of the 30 overall clues. After that's taken care of, the second one has to be in any of the five remaining $400 slots, out of the 29 overall remaining clues. Therefore, the overall probability is (1/5)*(6/30)*(5/29) = 0.00689655172, which is exactly 1/145.
So overall, if someone can ring first on all answers, answer everything correctly, still bet the maximum amount on all
occasions, and has the 1/145 luck chance, it is possible to get the theoretically maximum score of $566,400 on Jeopardy. To give a comparison, that value is over 1/5 of the amount Ken Jennings earned during his record-setting 74-win streak.
Sources:
Monday, June 11, 2012
Saturday, June 9, 2012
I-490 Extension Debate in Cleveland
Anyone who lives in the southeastern suburbs of Cleveland driving to Case Western Reserve University or Cleveland Clinic knows this dilemma: there just isn't an effective highway route that connects University Circle to the suburbs. Even for those who don't directly work there, it's an important issue: Cleveland Clinic alone is one of the biggest employers in Northeast Ohio. The following map illustrates the issue:
The closest highway interchange is northward at I-90 and MLK Drive. While this actually is a pretty effective route for commuters coming from the northeastern (via I-90 westbound) and western (via I-90 eastbound) directions, it is not an efficient route for people coming from the southeastern direction. Taking I-271 northbound and transferring to I-90 westbound is an enormous detour. Taking I-480 westbound and transferring to I-77 northbound is still quite a detour. Instead, most commuters know to get off I-271 at Chagrin Blvd and drive another 20+ minutes on the local roads. Another option, getting off I-480 at Broadway, takes a bit less time, but most suburban commuters avoid it due to the long stretch of local roads in the Cleveland city proper. Either way, as shown on the map, there's the enormous area enclosed by I-77, I-271, I-480, and I-90 that lacks a highway.
Try to spot I-490. It's only 2+ miles, running west-east slightly south of downtown. It connects onto I-71 and I-90 on its western terminus, but ends abruptly at East 55th St at a grade crossing. Looking at the map, if I-490 continued eastward, it could easily connect University Circle and join I-271 to the north. That would be the ideal route for commuters coming from Beachwood, Gates Mills, or anything southeast of that, which covers notable communities like Orange, Solon, and Twinsburg. It's quite a portion of the Great Cleveland commuters. Alas, no such highway system exists. However, the idea of it has floated around before.
In fact, in the early 1960s, there was a proposal to construct the Clark Freeway that runs east-west through Shaker Heights and connect onto I-271. However immediately, criticism came in about the elimination of homes and commercial properties that would result. In fact, this reaction was hardly confined to Cleveland. During the 60s and 70s, freeway and expressway revolts spread throughout the nation, as potentially affected neighborhoods voiced their concern of disruption. The unpopularity eventually resulted in the scrap of the proposed Clark Freeway. Instead, the Clark Freeway (I-490) merely stretched for 2+ miles it encompasses today.
Recently there has been rejuvenated efforts to stretch I-490 at its eastern terminus to University Circle, denoted the Opportunity Corridor. While there would be no eastern connection to I-271 and thus would not be the panacea for southeastern commuters, it would bring some relief to all. However, there first is the budget issue. With ODOT putting numerous projects on hold, the fiscal responsibility of the $200+ million project demands an answer. Furthermore, locals living in the path of the route would still be displaced. The area between University Circle and I-490 terminus is quite an economically lackluster region, and leaders hope that an infrastructure upgrade would also stir commerce in the region.
If Great Cleveland can learn something about infrastructure renovation, the recently completed Euclid Corridor has been quite a success. The $200 million development of rapid-bus route and transformation of Euclid Avenue, connecting Downtown Cleveland and East Cleveland while passing University Circle and Cleveland Clinic, has attributed to several billion dollars of new development. One can only hope that the Opportunity Corridor would one day bring similar results. University Circle would also be more easily accessible from highway, the airport, and commuters. Yet despite such, for the commuters coming from the southeastern direction, it doesn't look like they'll get their wish of connection from I-271. Those disgruntled can attribute the demise of a direct connection route to the nationwide phenomenon of freeway protests that rocked the country in the 1960s. That affected area is still heavily populated, and unlike the area affected by Opportunity Corridor that could benefit from revitalization,would likely still face great friction in pushing through any idea of freeway.
Sources:
Try to spot I-490. It's only 2+ miles, running west-east slightly south of downtown. It connects onto I-71 and I-90 on its western terminus, but ends abruptly at East 55th St at a grade crossing. Looking at the map, if I-490 continued eastward, it could easily connect University Circle and join I-271 to the north. That would be the ideal route for commuters coming from Beachwood, Gates Mills, or anything southeast of that, which covers notable communities like Orange, Solon, and Twinsburg. It's quite a portion of the Great Cleveland commuters. Alas, no such highway system exists. However, the idea of it has floated around before.
In fact, in the early 1960s, there was a proposal to construct the Clark Freeway that runs east-west through Shaker Heights and connect onto I-271. However immediately, criticism came in about the elimination of homes and commercial properties that would result. In fact, this reaction was hardly confined to Cleveland. During the 60s and 70s, freeway and expressway revolts spread throughout the nation, as potentially affected neighborhoods voiced their concern of disruption. The unpopularity eventually resulted in the scrap of the proposed Clark Freeway. Instead, the Clark Freeway (I-490) merely stretched for 2+ miles it encompasses today.
Recently there has been rejuvenated efforts to stretch I-490 at its eastern terminus to University Circle, denoted the Opportunity Corridor. While there would be no eastern connection to I-271 and thus would not be the panacea for southeastern commuters, it would bring some relief to all. However, there first is the budget issue. With ODOT putting numerous projects on hold, the fiscal responsibility of the $200+ million project demands an answer. Furthermore, locals living in the path of the route would still be displaced. The area between University Circle and I-490 terminus is quite an economically lackluster region, and leaders hope that an infrastructure upgrade would also stir commerce in the region.
If Great Cleveland can learn something about infrastructure renovation, the recently completed Euclid Corridor has been quite a success. The $200 million development of rapid-bus route and transformation of Euclid Avenue, connecting Downtown Cleveland and East Cleveland while passing University Circle and Cleveland Clinic, has attributed to several billion dollars of new development. One can only hope that the Opportunity Corridor would one day bring similar results. University Circle would also be more easily accessible from highway, the airport, and commuters. Yet despite such, for the commuters coming from the southeastern direction, it doesn't look like they'll get their wish of connection from I-271. Those disgruntled can attribute the demise of a direct connection route to the nationwide phenomenon of freeway protests that rocked the country in the 1960s. That affected area is still heavily populated, and unlike the area affected by Opportunity Corridor that could benefit from revitalization,would likely still face great friction in pushing through any idea of freeway.
Sources:
Wednesday, June 6, 2012
Tempur-Pedic International Inc (NYSE: TPX)
It doesn't look like memory foams will not cushion this latest news for Tempur-Pedic (NYSE: TPX). Shares of the company tumbled over 48% on Wednesday, after it was announced that second-quarter profits will drop 50% from last year. Analysts had expected 86 cents per sharing of net income, but the expectation is now only 34 cents. Much of this is due to increased competition from mattress makers. An analyst from Gilford Securities is quoted in saying that Tempur-Pedic "needs to institute a strategy where they drive more brand loyalty, not only from consumers but also retailers."
The history of Tempur-Pedic goes back to the 1970s and NASA, which was "[developing] pressure-absorbing material to help cushion and support astronauts during lift-off." In 1992, the company was formed in Lexington, Kentucky, where it is headquartered now. From 2008 to 2011, profit had climbed over 270% from $58 million to $219 million. However, as CEO Mark Sarvary claims, the environment in North America has seen "unprecedented number of new competitive product introductions which have been supported by aggressive marketing and promotion." Notably competition has come from Sealy Corporation (NYSE: ZZ) and Select Comfort (NASDAQ: SCSS), which also saw losses of over 5% and 20% on Wednesday, respectively.
Sources:
The history of Tempur-Pedic goes back to the 1970s and NASA, which was "[developing] pressure-absorbing material to help cushion and support astronauts during lift-off." In 1992, the company was formed in Lexington, Kentucky, where it is headquartered now. From 2008 to 2011, profit had climbed over 270% from $58 million to $219 million. However, as CEO Mark Sarvary claims, the environment in North America has seen "unprecedented number of new competitive product introductions which have been supported by aggressive marketing and promotion." Notably competition has come from Sealy Corporation (NYSE: ZZ) and Select Comfort (NASDAQ: SCSS), which also saw losses of over 5% and 20% on Wednesday, respectively.
Sources:
Tuesday, June 5, 2012
Westport Innovations Inc (NASDAQ: WPRT, TSE: WPT)
On Tuesday, the stocks of Westport Innovations (NASDAQ: WPRT, TSE: WPT) climbed over 21% at NASDAQ to close at 27.02. Based in Vancouver, Westport trades not only on the NASDAQ but also on the Toronto-based TSE. According to its website, Westport "is a global leader in alternative fuel, low-emissions technologies that allow engines to operate on clean-burning fuels." Income statements show that Westport has had negative operating income in not only each of the past 5 quarters, but also annually for each of the past 3 years. Profit margin was Q1-2012 was -20% as the company lost $22.63 million USD.
The stocks climbed Tuesday after a deal was announced between Westport, which has market capitalization of slightly above $1 billion and Caterpillar (NYSE: CAT), which has market cap of over $54 billion. The would focus on "[developing] fuel systems for off-road vehicles," including natural gas engines in locomotives. The incentive to focus on natural gas comes from the substantial price difference between natural gas and diesel fuel, which results in "strong financial incentive to enable off-road applications to take advantage of low ... costs without sacrificing operational performance." Shares of Caterpillar climbed a meager 0.47% for the day.
Sources:
The stocks climbed Tuesday after a deal was announced between Westport, which has market capitalization of slightly above $1 billion and Caterpillar (NYSE: CAT), which has market cap of over $54 billion. The would focus on "[developing] fuel systems for off-road vehicles," including natural gas engines in locomotives. The incentive to focus on natural gas comes from the substantial price difference between natural gas and diesel fuel, which results in "strong financial incentive to enable off-road applications to take advantage of low ... costs without sacrificing operational performance." Shares of Caterpillar climbed a meager 0.47% for the day.
Sources:
Thursday, May 24, 2012
Evaluating Methods of Communication: Rules of Phone Calls?
A conversation with a friend led to the question of "are there rules for when one can call someone?" Of course there are not, unless we are dealing with restrictive orders. But in today's living, where the modes of communication range from making phone calls, sending text messages, leaving Facebook posts, to sending instant-messages, is there a respectable convention?
No matter how technology progresses, the modes of communication can be categorized on a spectrum of attention-seeking. Off at the highest end is calling on the phone or video-calling on Skype; it's hard to multi-task while doing that. Off to the other end of the spectrum are methods like sending emails (suppose the recipient checks email consistently), leaving Facebook posts, or even sending text messages. For these, at the moment the information is passed on, the recipient can choose whether or not to put his or her attention onto it. He or she also chooses to reply at a moment of convenience. In-between is sending instant-messages. While the recipient can choose to pay attention and response at his or her discretion, there is a modest amount of attention beheld, for a response is expected soon; otherwise, it turns into relaying messages back-and-forth.
The most crucial component of a piece of information that dictates its communication method is its urgency. If it's an urgent information, it justifies seeking the immediate attention of the other. Here's an illustration. Suppose you're meeting a friend at 12:00. If you realize at that day 11:50 that you can't make it today, the best bet is to call, lest the friend is already on his or her way. This is an urgent message, and that justifies seeking the attention of the friend, regardless what the friend was doing at the time. Now suppose it's one week before the scheduled meeting, and it needs to be rescheduled. All of the forms of communication are acceptable, but methods toward the bottom of the attention-seeking spectrum are most appropriate. These non-attention-seeking methods will be able to convey the necessary information in time to serve its purpose (to schedule a new meeting time in the upcoming week).
Of course, there is nothing inherently "wrong" with calling the friend one week in advance purely to say that the meeting needs to be canceled. Instead, it's a probabilistic inconvenience. Go back to the moment one week before the scheduled meeting. The friend may be idle at the moment you call, in which case there's no real detraction from calling. However, what if the friend is really busy at the time, such that taking the non-urgent call disturbs his or her actions? Let's use the concept of utility, and specifically, the mutual utility of having the information successfully communicated, and the change in personal utility from picking up this information in the particular method. In the disturbed case, the friend's personal utility decreased due to the disturbance, but no mutual utility is gained from having the information successfully conveyed instantly. This wouldn't be true in an urgent situation, where if the information isn't successfully conveyed within time, the mutual utility plummets. Back to the case though, there can be no gain from using the attention-seeking communication method, given that the probability of non-attention-seeking methods getting the information across successfully is almost certain. Instead, regardless of what the probability that the friend is super busy at the time, the total change in combined utility will be non-positive. It could be unchanged, but it can decrease; by no way will it increase.
That's why making a non-urgent phone or video call is inefficient for both parties. If the information to-be-conveyed truly is not urgent, there are no benefits from utilizing the attention-seeking method of communication. Of course, some can argue for the increased heartfelt happiness as a result of those more intimate communication methods. Well, here we are assuming that there are no such effects. Amorous environments are of another realm, because when one tries to quantify love, well, we lose quantifiable results.
No matter how technology progresses, the modes of communication can be categorized on a spectrum of attention-seeking. Off at the highest end is calling on the phone or video-calling on Skype; it's hard to multi-task while doing that. Off to the other end of the spectrum are methods like sending emails (suppose the recipient checks email consistently), leaving Facebook posts, or even sending text messages. For these, at the moment the information is passed on, the recipient can choose whether or not to put his or her attention onto it. He or she also chooses to reply at a moment of convenience. In-between is sending instant-messages. While the recipient can choose to pay attention and response at his or her discretion, there is a modest amount of attention beheld, for a response is expected soon; otherwise, it turns into relaying messages back-and-forth.
The most crucial component of a piece of information that dictates its communication method is its urgency. If it's an urgent information, it justifies seeking the immediate attention of the other. Here's an illustration. Suppose you're meeting a friend at 12:00. If you realize at that day 11:50 that you can't make it today, the best bet is to call, lest the friend is already on his or her way. This is an urgent message, and that justifies seeking the attention of the friend, regardless what the friend was doing at the time. Now suppose it's one week before the scheduled meeting, and it needs to be rescheduled. All of the forms of communication are acceptable, but methods toward the bottom of the attention-seeking spectrum are most appropriate. These non-attention-seeking methods will be able to convey the necessary information in time to serve its purpose (to schedule a new meeting time in the upcoming week).
Of course, there is nothing inherently "wrong" with calling the friend one week in advance purely to say that the meeting needs to be canceled. Instead, it's a probabilistic inconvenience. Go back to the moment one week before the scheduled meeting. The friend may be idle at the moment you call, in which case there's no real detraction from calling. However, what if the friend is really busy at the time, such that taking the non-urgent call disturbs his or her actions? Let's use the concept of utility, and specifically, the mutual utility of having the information successfully communicated, and the change in personal utility from picking up this information in the particular method. In the disturbed case, the friend's personal utility decreased due to the disturbance, but no mutual utility is gained from having the information successfully conveyed instantly. This wouldn't be true in an urgent situation, where if the information isn't successfully conveyed within time, the mutual utility plummets. Back to the case though, there can be no gain from using the attention-seeking communication method, given that the probability of non-attention-seeking methods getting the information across successfully is almost certain. Instead, regardless of what the probability that the friend is super busy at the time, the total change in combined utility will be non-positive. It could be unchanged, but it can decrease; by no way will it increase.
That's why making a non-urgent phone or video call is inefficient for both parties. If the information to-be-conveyed truly is not urgent, there are no benefits from utilizing the attention-seeking method of communication. Of course, some can argue for the increased heartfelt happiness as a result of those more intimate communication methods. Well, here we are assuming that there are no such effects. Amorous environments are of another realm, because when one tries to quantify love, well, we lose quantifiable results.
Tuesday, May 15, 2012
Misrepresentation of GPA
Looking back at this semester for me, there is quite a strong negative correlation between the amount of effort and commitment put into the classes, and the final grades received from the classes. It's a clear demonstration that GPA cannot capture the amount of
take-away for the future, which just happens to be the most crucial element of education.
It's an unfortunate truth that the grades we get are predominantly based on how we perform on the one or two particular exams, relatively to the others in the class. Fine, let that be because there usually is no better way to go about it. But of course in the long run, it's the personally absorbed knowledge and future take-away that actually matter. Cramming all the knowledge the night before, acing the exam, and subsequently forgetting everything afterwards have little value in the future. Knowing actually how to use and apply the learned knowledge in the subject matter will go much further, whether or not we remember how to solve that one particular problem on that particular day on that particular exam.
It's an unfortunate truth that the grades we get are predominantly based on how we perform on the one or two particular exams, relatively to the others in the class. Fine, let that be because there usually is no better way to go about it. But of course in the long run, it's the personally absorbed knowledge and future take-away that actually matter. Cramming all the knowledge the night before, acing the exam, and subsequently forgetting everything afterwards have little value in the future. Knowing actually how to use and apply the learned knowledge in the subject matter will go much further, whether or not we remember how to solve that one particular problem on that particular day on that particular exam.
Sunday, May 13, 2012
Dow Jones and S&P 500 Daily Change Correlations
Do the daily fluctuations in one stock index align well with the change observed in another index? For this exercise, let's take the Dow Jones and S&P 500 index for April 2012. For each index, after extracting the closing numbers for each day, the daily percentage change was computed, summarized below:
When the pairs of daily % changes are plotted, there exists almost an 1:1 relationship, with a strong correlation:
Sources:
| Date | Dow %Δ | S&P %Δ |
| 30-Apr-12 | -0.11% | -0.39% |
| 27-Apr-12 | 0.18% | 0.24% |
| 26-Apr-12 | 0.87% | 0.67% |
| 25-Apr-12 | 0.69% | 1.36% |
| 24-Apr-12 | 0.58% | 0.37% |
| 23-Apr-12 | -0.78% | -0.84% |
| 20-Apr-12 | 0.50% | 0.12% |
| 19-Apr-12 | -0.53% | -0.59% |
| 18-Apr-12 | -0.63% | -0.41% |
| 17-Apr-12 | 1.50% | 1.55% |
| 16-Apr-12 | 0.56% | -0.05% |
| 13-Apr-12 | -1.05% | -1.25% |
| 12-Apr-12 | 1.41% | 1.38% |
| 11-Apr-12 | 0.70% | 0.74% |
| 10-Apr-12 | -1.65% | -1.71% |
| 9-Apr-12 | -1.00% | -1.14% |
| 5-Apr-12 | -0.11% | -0.06% |
| 4-Apr-12 | -0.95% | -1.02% |
| 3-Apr-12 | -0.49% | -0.40% |
| 2-Apr-12 | 0.40% | 0.75% |
| Average | 0.004% | -0.034% |
| Stand Dev | 0.008693038 | 0.009248878 |
When the pairs of daily % changes are plotted, there exists almost an 1:1 relationship, with a strong correlation:
Sources:
Wednesday, May 9, 2012
4 Home-Run Game Probability
On Tuesday night, Texas Rangers outfielder Josh Hamilton hit 4 home-runs in one game. This was only the 16th time in Major League Baseball's history to see the accomplishment of such a feat, and the first since 2003. Hamilton also had a double in the game, going 5-for-5 overall and totaling 18 bases in a single game, which was only one shy of Major League record. The Rangers won the game 10-3 against Baltimore Orioles.
To best estimate the probability of this event, the following data for Hamilton during each of the past four seasons at the Rangers were retrieved:
Since Hamilton hit 99 home-runs in 1965 at-bats during the past four seasons, let's assume that the probability of hitting a home-run is 99/1965 = 5.038%. On Wednesday night, Hamilton had 5 at-bats, and hit 4 home-runs. The probability of hitting 4 home-runs in a game with 5 at-bats will be the product of the following terms:
To put in perspective, each season is 162 games. During each game across the league, there are 30 teams and 9 players who come to bat. That is 30*162*9 = 43,740 total player-performances in a season, assuming uniformity. Already that number is greater than 32,688. This means that if every player in MLB had the HR-hitting probability as Hamilton, the league should expect to see occurrence like Tuesday night's every season. However, clearly that isn't the case. Four-HR games are more rarer than perfect games or no-hitters. In baseball, only 20-K games and unassisted triple players have been less frequent.
Sources:
To best estimate the probability of this event, the following data for Hamilton during each of the past four seasons at the Rangers were retrieved:
| Year | At Bat | Home Run |
| 2008 | 624 | 32 |
| 2009 | 336 | 10 |
| 2010 | 518 | 32 |
| 2011 | 487 | 25 |
| Sum | 1965 | 99 |
Since Hamilton hit 99 home-runs in 1965 at-bats during the past four seasons, let's assume that the probability of hitting a home-run is 99/1965 = 5.038%. On Wednesday night, Hamilton had 5 at-bats, and hit 4 home-runs. The probability of hitting 4 home-runs in a game with 5 at-bats will be the product of the following terms:
- nCr(5,4) to indicate the number of combinations
- (5.038%) ^ 4 to indicate the 4 home-runs
- (1-5.038%) to indicate the 1 non-HR at-bat
To put in perspective, each season is 162 games. During each game across the league, there are 30 teams and 9 players who come to bat. That is 30*162*9 = 43,740 total player-performances in a season, assuming uniformity. Already that number is greater than 32,688. This means that if every player in MLB had the HR-hitting probability as Hamilton, the league should expect to see occurrence like Tuesday night's every season. However, clearly that isn't the case. Four-HR games are more rarer than perfect games or no-hitters. In baseball, only 20-K games and unassisted triple players have been less frequent.
Sources:
Thursday, May 3, 2012
Simple Linear Regression Illustration
Given a set of coordinate points, how
do we find the linear regression using least square method? Imagine the
set of 10 points like this:
| Xi | Yi |
| 215 | 30.8 |
| 201 | 32.5 |
| 196 | 35.4 |
| 226 | 28.1 |
| 226 | 24.4 |
| 348 | 24.1 |
| 226 | 28.5 |
| 348 | 24.2 |
| 148 | 32.8 |
| 226 | 28.0 |
The
regression line will take on the form of y = B0 + B1*x, with variance
σ^2. The equations for the variables B0, B1, and σ^2 are:
- B1 = (Σxi*yi - Σxi*Σyi/n) / (Σxi^2-(Σxi)^2/n)
- B0 = ӯ(n) - B1*x̄(n)
- σ^2 = (Σyi^2-n*ӯ(n)^2-B1*(Σxi*yi-Σxi*Σyi/n)) / (n-2)
| Xi | Yi | Xi^2 | Yi^2 | Xi*Yi | ||
| 215 | 30.8 | 46225 | 948.64 | 6622 | ||
| 201 | 32.5 | 40401 | 1056.25 | 6532.5 | ||
| 196 | 35.4 | 38416 | 1253.16 | 6938.4 | ||
| 226 | 28.1 | 51076 | 789.61 | 6350.6 | ||
| 226 | 24.4 | 51076 | 595.36 | 5514.4 | ||
| 348 | 24.1 | 121104 | 580.81 | 8386.8 | ||
| 226 | 28.5 | 51076 | 812.25 | 6441 | ||
| 348 | 24.2 | 121104 | 585.64 | 8421.6 | ||
| 148 | 32.8 | 21904 | 1075.84 | 4854.4 | ||
| 226 | 28.0 | 51076 | 784 | 6328 | ||
| Sum | 2360 | 288.8 | 593458 | 8481.56 | 66389.7 |
- B1 = (66389.7-2360*288.8/10) / (593458-2360^2/10) = -0.0484
- B0 = (288.8/10) - (-0.0484)(2360/10) = 40.3024
- σ^2 = (8481.56-10*(288.8/10)^2-(-0.0484)*(66389.7-2360*288.8/10)) / 8 = 6.9360
Although the value of the coefficient of determination is quite low, the equation of the least-square best-fit line corresponds with the numbers we obtained. Also, using the function =STEYX(C2:C11,B2:B11) in Excel, the sample standard deviation was calculated to be 2.632951. Squaring that number, we get 6.932, nearly identical as the value we calculated above as well.
Wednesday, May 2, 2012
Calculating Type II (β) Error
In hypothesis test, β is the probability of failing to reject the null hypothesis when it is actually false. Let's first consider the two-sided case to see how it is calculated. Begin with the following given conditions:
What about an one-sided case? It is similarly computed. Consider the following given conditions:
- Let x̄(n) = 174.5, s(n) = 6.9, n = 50, α = 0.05
- H0: μ = 175, H1: μ ≠ 175
- Compare against alternative hypothesis μ = 173
- Recall that for samples (for both two- or one-sided cases), z = (x̄-μ) / (s(n) / sqrt(n))
- 1.96 = (x̄ - 175) / (6.9 / sqrt(50)) --> x̄ = 176.913
- -1.96 = (x̄ - 175) / (6.9 / sqrt(50)) --> x̄ = 173.087
- (173.087 - 173) / (6.9 / sqrt(50)) = 0.0892
- (176.913 - 173) / (6.9 / sqrt(50)) = 4.0100
What about an one-sided case? It is similarly computed. Consider the following given conditions:
- Let p̂ = 0.535, n = 1000, α = 0.05
- H0: p = 0.50, H1: p > 0.50
- Compare against the alternative hypothesis p = 0.52
- Recall that for proportions (for both two- or one-sided cases), z = (p̂-p) / sqrt(p*(1-p)/n)
- 1.645 = (p̂-0.5) / (sqrt(0.5^2/1000)
- p̂ = 0.52601
- Do not forget that p = 0.52 now, instead of 0.50
- (0.52601 - 0.52) / sqrt(0.52*0.48 / 1000)
- = 0.3804
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