Contingency Tables

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1. How does stress affect sleep?  (particularly in college students) Stress can cause insomnia, loss of concentration while in class or doing any of the daily life activities.  It makes the likelihood of deep sleep less favorable and consumes more resources from the individual.  As a result, people and students alike are less focus on everyday task including attention during class. 2. What issues affect sleep with respect to stress? Students often face Family/ personal issues, work-related issues, financial issues, class conflicts, peer pressure, and professors.  Issues can come from every aspect of life that can occupy their mind at all hours of the day and night.  There have also been a positive correlation found between stress in life and sleep-walking, which can leave the student tired in the morning even thought they believe to have slept 8 or 9 hours. 3. What can cause a body to change its circadian rhythm and cause one to have chronic sleep difficulties? Staying up all night for whatever reason, such as studying, stress from school/ work. 4. What are some effects of chronic sleep deprivation? Tired, loss of concentration, poor judgment, loss of productivity 5. How does lack of sleep affect one's performance in academia and day-to-day activities? It can be the reason behind many negative characteristics such as more accident prone, violence, work slower than normal, mood swings, aggravation. 6. What are some health effects of lack of sleep? It can cause mental issues, fatigue, malaise, low immunity system, getting sick all the time, hospitalization 7. What advice do you have for a college student with respect to managing stress and time management? Students should always plan ahead their schedule, Do not wait for the last minute to do their work, spend some time doing appropriate activities to de-stress such as hang out with their friends friends, talk to their friends about their issues he/she might have, balance school/ work/ family life 8. Where can one seek help if necessary? There are many places and people that one can go to in order to get help such as seeing a psychiatrist, school counselor, seek help from friends and family, and talk to other professionals within this field.



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                                                                   Contingency Tables

Probability is usually estimated as a relative frequency, (See Chapter 3 Cumulative and Relative Frequencies) we can use tables of relative frequencies to learn about relationships (e.g. dependent events or conditional probabilities).

A contingency table is a cross-tabulation of frequencies into rows and columns. The intersection of each row and column is a cell that shows a frequency.   The table is like a frequency distribution, except fortwo variables instead of just one.  A table with r rows and c columns is called an r x c table.  They are often used to report the results of a survey.

Ex. Salary Gains for MBA Grads

Tuition	Small (S1) Under $50K	Medium (S2) $50K - $100K	Large (S3) $100K+	Row Total
Low (T1) Under $40K	5	10	1	16
Medium (T2) $40K - $50K	7	11	1	19
Large (T3) $50K+	5	12	15	32
Column Total	17	33	17	67

This table lays out the reported salary gains from 67 top tier graduate business schools and cross tabulates them with their respective tuition levels.  At first glance, there appears to be a high correlation between tuition and salary increase. (In class, he read it as the salary gains reported by an individual MBA grad and the tuition level they paid.  Close examination of the text says differently, but it doesn’t matter in terms of understanding the table.)

Incidentally, each of these examples in the text will cite the source for the information. In this case, this data came from the October 13, 2003 edition of Forbes magazine.  When we’re asked to do similar examples on exams, the data we need will be given to us, even without citation.  

Marginal Probabilities

The marginal probability of an event is found by dividing a row or column by the total sample size.    For example, using the column totals,  we see that 33 out of 67 schools had medium salary gains.   So, the marginal probability of a medium salary gain is P(S2) = 33/67 = 0.4925.   In other words, salary gains at about 49 percent of the top tier schools were between 50 and 100 thousand.

Joint Probabilities

Each of the nine main cells is used to calculate a joint probability representing the intersection of two events.  (See in this Chapter: The Intersection of Two Events)

For example, the upper right hand cell is the joint event that the school has low tuition (T1) and has large salary gains (S3).   We can express this event in this notation: P(T1 ∩ S3).    Only 1 school is in this category, therefore, 1/67.  So this joint probability is 1/67 = 0.0149. In other words, there is a less than 2 percent chance that a top-tier school has both low tuition and a high salary gain.

Conditional Probabilities

Conditional probabilities may be found by restricting ourselves to one row or column (the condition). For example, suppose we know that a school's MBA tuition is high (T3). When restricted to the 32 schools in the third row (those with high tuition) the conditional probabilities of any event may be calculated.

Ex. What is the conditional probability that salary gains will be small, given that the tuition for the MBA was high?

Expressed in the following notation: P(S1|T3) = 5/32 = 0.1563  (Or 15.6%)

So there is about a 16 percent chance that a top-tier school's salary gains will be small despite its high tuition.

Independence

To check whether events in a contingency table are independent, we can look at conditional probabilities. For example, if large salary gains were independent of low tuition, we could expect that the conditional probability of P(S3|T1) would be the same as the marginal probability P(S3).

Conditional: P(S3|T1) = 1/16 = 0.0625

Marginal: P(S3) = 17/67 = 0.2537

It's not the case.  Large salary gains are not independent of low tuition.  (The existence of a low tuition will affect the probability of a large salary gain.)

Taking one of the examples we did in class; We asked “Among medium tuition schools, what percentage of them saw low salary gains?”  This is a conditional probability being asked; P(S1|T2)

“P(S1|T2)” is the same as saying “P(S1 ∩ T2 )”, that is we are being asked for the intersection of a particular cell (Med Tution/Low Salary) and a particular row (Medium Tuition).

7/19 = .3684 or 36.8% of medium tuition top-tier schools saw low salary gains in their MBA grads.

Relative Frequencies

To aid in probability calculations, we can also find the relative frequency for any cell in the table. (f/n) (See Chapter 3: Cumulative and Relative Frequencies)

For example, the upper left-hand cell becomes 5/67 = 0.0746.

The nine joint probabilities in the table sum to 1.0000 since these are all the possible intersections.  Save for rounding, summing the joint probabilities across a row or down a column will produce a marginal probability. The marginal row and column probabilities will sum to 1.0000.

How We Get Contingency Tables

These tables are created from raw data.  In the example of the MBA salary gain/tuition data, the numerical values (40K salary gain, 60K tuition, etc) were mapped out into discrete codes (S1, T3, etc).  

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1. How does stress affect sleep?  (particularly in college students) Stress can cause insomnia, loss of concentration while in class or doing any of the daily life activities.  It makes the likelihood of deep sleep less favorable and consumes more resources from the individual.  As a result, people and students alike are less focus on everyday task including attention during class. 2. What issues affect sleep with respect to stress? Students often face Family/ personal issues, work-related issues, financial issues, class conflicts, peer pressure, and professors.  Issues can come from every aspect of life that can occupy their mind at all hours of the day and night.  There have also been a positive correlation found between stress in life and sleep-walking, which can leave the student tired in the morning even thought they believe to have slept 8 or 9 hours. 3. What can cause a body to change its circadian rhythm and cause one to have chronic sleep difficulties? Staying up all night for whatever reason, such as studying, stress from school/ work. 4. What are some effects of chronic sleep deprivation? Tired, loss of concentration, poor judgment, loss of productivity 5. How does lack of sleep affect one's performance in academia and day-to-day activities? It can be the reason behind many negative characteristics such as more accident prone, violence, work slower than normal, mood swings, aggravation. 6. What are some health effects of lack of sleep? It can cause mental issues, fatigue, malaise, low immunity system, getting sick all the time, hospitalization 7. What advice do you have for a college student with respect to managing stress and time management? Students should always plan ahead their schedule, Do not wait for the last minute to do their work, spend some time doing appropriate activities to de-stress such as hang out with their friends friends, talk to their friends about their issues he/she might have, balance school/ work/ family life 8. Where can one seek help if necessary? There are many places and people that one can go to in order to get help such as seeing a psychiatrist, school counselor, seek help from friends and family, and talk to other professionals within this field.




If you have any test reviews, homeworks, guides, anything school related that you think can be posted on this website, reach out to me at makingschooleasier@gmail.com  a