1) The document provides instructions for students to conduct a coin toss experiment to demonstrate probability and use a chi-square test to analyze the results. It explains that genetics involves random chance processes that can be modeled with coin tosses.
2) Students will toss two coins 100 times in groups and record head/head, head/tail, and tail/tail outcomes. They will then use a chi-square test to compare observed results to expected results based on probability laws.
3) The chi-square test allows students to determine if any differences between observed and expected results are statistically significant, which could mean factors other than chance are influencing the outcomes. This analysis method is important for studying inheritance patterns in genetics.
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Chapter 4: Probability
4.3: Complements and Conditional Probability, and Bayes' Theorem
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Chapter 5: Discrete Probability Distribution
5.1: Probability Distribution
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Chapter 4: Probability
4.3: Complements and Conditional Probability, and Bayes' Theorem
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Chapter 5: Discrete Probability Distribution
5.1: Probability Distribution
Answer questions Minimum 100 words each and reference (questions.docxnolanalgernon
Answer questions Minimum 100 words each and reference (questions #1-3) KEEP questions WITH ANSWER
1) If we had a multiple number of coin tosses and considered this an experiment, what distribution would this experiment follow and why?
2) Virtually all experiments and studies deal with mutually exclusive outcomes. Why is this important?
3) Random variables are part of probability and statistics! Mutual exclusiveness applies to the definition of this. How?
A minimum of 75 words each question and References (IF NEEDED)(Response #1 – 6) KEEP RESPONSE WITH ANSWER
Make sure the Responses includes the Following: (a) an understanding of the weekly content as supported by a scholarly resource, (b) the provision of a probing question. (c) stay on topic
1) I think my friend would have the wrong idea in my opinion. A coin has two sides and if it is a fair coin, then when it is tossed it will have a 50-50 chance of either being heads or tails. There is nothing that would make it tails more than heads. The odds or probability of it landing on tails over heads is 50-50. There is no way of specifically knowing how many times it would be heads or tails an infinite number of times. It will not always land on heads half of the time nor will it always land on tails half of the time, but there is always the probability that it could.
2) No, she is not correct in her theory on the probability of getting heads in a coin toss. The only two outcomes possible are heads or tails. According to the textbook, “the formula for probability then is the frequency of times an outcome occurs, f(x), divided by the sample space or the total number of possible outcomes” (Privitera, 2018). The frequency (f(x)) divided by the sample size is ½. In other words, there is a probability of getting heads one out of two times. The coins could be flipped multiple times and the chances are still 50/50 of getting heads or tails.
3) Considering that tossing a coin can be considered a random event, fixed event, or possibly have a sample space the outcome may vary. “Probability is the frequency of times the outcome occurs divided by the total number of possible outcomes.” (Privitera, G. J., 2018, p.139) If a friend and I had a single coin toss, I would have to disagree on the likeliness of landing on heads having the advantage. The coin toss is a fixed event and there are only 2 options in a single toss. Heads or tails both have a 50 % chance of being the outcome.
Tossing the coin an infinite number of times would be consist of different variations on probability. The outcome could vary amongst every individual. For instance, my father, my son, and I, all just tossed a quarter 10 times each. I landed heads twice, my father landed heads 6 times and my son landed heads 4 times. Therefore, no outcome was the same. The probability of landing heads in 30 tosses was 12, there for two times out of every 6 tosses. However that is if we added up all 3 sets of 10 tosses otherwise with my tosses the.
Review: experimental vrs. theoretical probability, designing simulations, using the randBin() command on the TI-83, calculating theoretical probabilities using "and" and "or".
Tom Selleck Net Worth: A Comprehensive Analysisgreendigital
Over several decades, Tom Selleck, a name synonymous with charisma. From his iconic role as Thomas Magnum in the television series "Magnum, P.I." to his enduring presence in "Blue Bloods," Selleck has captivated audiences with his versatility and charm. As a result, "Tom Selleck net worth" has become a topic of great interest among fans. and financial enthusiasts alike. This article delves deep into Tom Selleck's wealth, exploring his career, assets, endorsements. and business ventures that contribute to his impressive economic standing.
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Early Life and Career Beginnings
The Foundation of Tom Selleck's Wealth
Born on January 29, 1945, in Detroit, Michigan, Tom Selleck grew up in Sherman Oaks, California. His journey towards building a large net worth began with humble origins. , Selleck pursued a business administration degree at the University of Southern California (USC) on a basketball scholarship. But, his interest shifted towards acting. leading him to study at the Hills Playhouse under Milton Katselas.
Minor roles in television and films marked Selleck's early career. He appeared in commercials and took on small parts in T.V. series such as "The Dating Game" and "Lancer." These initial steps, although modest. laid the groundwork for his future success and the growth of Tom Selleck net worth. Breakthrough with "Magnum, P.I."
The Role that Defined Tom Selleck's Career
Tom Selleck's breakthrough came with the role of Thomas Magnum in the CBS television series "Magnum, P.I." (1980-1988). This role made him a household name and boosted his net worth. The series' popularity resulted in Selleck earning large salaries. leading to financial stability and increased recognition in Hollywood.
"Magnum P.I." garnered high ratings and critical acclaim during its run. Selleck's portrayal of the charming and resourceful private investigator resonated with audiences. making him one of the most beloved television actors of the 1980s. The success of "Magnum P.I." played a pivotal role in shaping Tom Selleck net worth, establishing him as a major star.
Film Career and Diversification
Expanding Tom Selleck's Financial Portfolio
While "Magnum, P.I." was a cornerstone of Selleck's career, he did not limit himself to television. He ventured into films, further enhancing Tom Selleck net worth. His filmography includes notable movies such as "Three Men and a Baby" (1987). which became the highest-grossing film of the year, and its sequel, "Three Men and a Little Lady" (1990). These box office successes contributed to his wealth.
Selleck's versatility allowed him to transition between genres. from comedies like "Mr. Baseball" (1992) to westerns such as "Quigley Down Under" (1990). This diversification showcased his acting range. and provided many income streams, reinforcing Tom Selleck net worth.
Television Resurgence with "Blue Bloods"
Sustaining Wealth through Consistent Success
In 2010, Tom Selleck began starring as Frank Reagan i
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Scandal! Teasers June 2024 on etv Forum.co.zaIsaac More
Monday, 3 June 2024
Episode 47
A friend is compelled to expose a manipulative scheme to prevent another from making a grave mistake. In a frantic bid to save Jojo, Phakamile agrees to a meeting that unbeknownst to her, will seal her fate.
Tuesday, 4 June 2024
Episode 48
A mother, with her son's best interests at heart, finds him unready to heed her advice. Motshabi finds herself in an unmanageable situation, sinking fast like in quicksand.
Wednesday, 5 June 2024
Episode 49
A woman fabricates a diabolical lie to cover up an indiscretion. Overwhelmed by guilt, she makes a spontaneous confession that could be devastating to another heart.
Thursday, 6 June 2024
Episode 50
Linda unwittingly discloses damning information. Nhlamulo and Vuvu try to guide their friend towards the right decision.
Friday, 7 June 2024
Episode 51
Jojo's life continues to spiral out of control. Dintle weaves a web of lies to conceal that she is not as successful as everyone believes.
Monday, 10 June 2024
Episode 52
A heated confrontation between lovers leads to a devastating admission of guilt. Dintle's desperation takes a new turn, leaving her with dwindling options.
Tuesday, 11 June 2024
Episode 53
Unable to resort to violence, Taps issues a verbal threat, leaving Mdala unsettled. A sister must explain her life choices to regain her brother's trust.
Wednesday, 12 June 2024
Episode 54
Winnie makes a very troubling discovery. Taps follows through on his threat, leaving a woman reeling. Layla, oblivious to the truth, offers an incentive.
Thursday, 13 June 2024
Episode 55
A nosy relative arrives just in time to thwart a man's fatal decision. Dintle manipulates Khanyi to tug at Mo's heartstrings and get what she wants.
Friday, 14 June 2024
Episode 56
Tlhogi is shocked by Mdala's reaction following the revelation of their indiscretion. Jojo is in disbelief when the punishment for his crime is revealed.
Monday, 17 June 2024
Episode 57
A woman reprimands another to stay in her lane, leading to a damning revelation. A man decides to leave his broken life behind.
Tuesday, 18 June 2024
Episode 58
Nhlamulo learns that due to his actions, his worst fears have come true. Caiphus' extravagant promises to suppliers get him into trouble with Ndu.
Wednesday, 19 June 2024
Episode 59
A woman manages to kill two birds with one stone. Business doom looms over Chillax. A sobering incident makes a woman realize how far she's fallen.
Thursday, 20 June 2024
Episode 60
Taps' offer to help Nhlamulo comes with hidden motives. Caiphus' new ideas for Chillax have MaHilda excited. A blast from the past recognizes Dintle, not for her newfound fame.
Friday, 21 June 2024
Episode 61
Taps is hungry for revenge and finds a rope to hang Mdala with. Chillax's new job opportunity elicits mixed reactions from the public. Roommates' initial meeting starts off on the wrong foot.
Monday, 24 June 2024
Episode 62
Taps seizes new information and recruits someone on the inside. Mary's new job
Meet Crazyjamjam - A TikTok Sensation | Blog EternalBlog Eternal
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Skeem Saam in June 2024 available on ForumIsaac More
Monday, June 3, 2024 - Episode 241: Sergeant Rathebe nabs a top scammer in Turfloop. Meikie is furious at her uncle's reaction to the truth about Ntswaki.
Tuesday, June 4, 2024 - Episode 242: Babeile uncovers the truth behind Rathebe’s latest actions. Leeto's announcement shocks his employees, and Ntswaki’s ordeal haunts her family.
Wednesday, June 5, 2024 - Episode 243: Rathebe blocks Babeile from investigating further. Melita warns Eunice to stay clear of Mr. Kgomo.
Thursday, June 6, 2024 - Episode 244: Tbose surrenders to the police while an intruder meddles in his affairs. Rathebe's secret mission faces a setback.
Friday, June 7, 2024 - Episode 245: Rathebe’s antics reach Kganyago. Tbose dodges a bullet, but a nightmare looms. Mr. Kgomo accuses Melita of witchcraft.
Monday, June 10, 2024 - Episode 246: Ntswaki struggles on her first day back at school. Babeile is stunned by Rathebe’s romance with Bullet Mabuza.
Tuesday, June 11, 2024 - Episode 247: An unexpected turn halts Rathebe’s investigation. The press discovers Mr. Kgomo’s affair with a young employee.
Wednesday, June 12, 2024 - Episode 248: Rathebe chases a criminal, resorting to gunfire. Turf High is rife with tension and transfer threats.
Thursday, June 13, 2024 - Episode 249: Rathebe traps Kganyago. John warns Toby to stop harassing Ntswaki.
Friday, June 14, 2024 - Episode 250: Babeile is cleared to investigate Rathebe. Melita gains Mr. Kgomo’s trust, and Jacobeth devises a financial solution.
Monday, June 17, 2024 - Episode 251: Rathebe feels the pressure as Babeile closes in. Mr. Kgomo and Eunice clash. Jacobeth risks her safety in pursuit of Kganyago.
Tuesday, June 18, 2024 - Episode 252: Bullet Mabuza retaliates against Jacobeth. Pitsi inadvertently reveals his parents’ plans. Nkosi is shocked by Khwezi’s decision on LJ’s future.
Wednesday, June 19, 2024 - Episode 253: Jacobeth is ensnared in deceit. Evelyn is stressed over Toby’s case, and Letetswe reveals shocking academic results.
Thursday, June 20, 2024 - Episode 254: Elizabeth learns Jacobeth is in Mpumalanga. Kganyago's past is exposed, and Lehasa discovers his son is in KZN.
Friday, June 21, 2024 - Episode 255: Elizabeth confirms Jacobeth’s dubious activities in Mpumalanga. Rathebe lies about her relationship with Bullet, and Jacobeth faces theft accusations.
Monday, June 24, 2024 - Episode 256: Rathebe spies on Kganyago. Lehasa plans to retrieve his son from KZN, fearing what awaits.
Tuesday, June 25, 2024 - Episode 257: MaNtuli fears for Kwaito’s safety in Mpumalanga. Mr. Kgomo and Melita reconcile.
Wednesday, June 26, 2024 - Episode 258: Kganyago makes a bold escape. Elizabeth receives a shocking message from Kwaito. Mrs. Khoza defends her husband against scam accusations.
Thursday, June 27, 2024 - Episode 259: Babeile's skillful arrest changes the game. Tbose and Kwaito face a hostage crisis.
Friday, June 28, 2024 - Episode 260: Two women face the reality of being scammed. Turf is rocked by breaking
Maximizing Your Streaming Experience with XCIPTV- Tips for 2024.pdfXtreame HDTV
In today’s digital age, streaming services have become an integral part of our entertainment lives. Among the myriad of options available, XCIPTV stands out as a premier choice for those seeking seamless, high-quality streaming. This comprehensive guide will delve into the features, benefits, and user experience of XCIPTV, illustrating why it is a top contender in the IPTV industry.
In the vast landscape of cinema, stories have been told, retold, and reimagined in countless ways. At the heart of this narrative evolution lies the concept of a "remake". A successful remake allows us to revisit cherished tales through a fresh lens, often reflecting a different era's perspective or harnessing the power of advanced technology. Yet, the question remains, what makes a remake successful? Today, we will delve deeper into this subject, identifying the key ingredients that contribute to the success of a remake.
Young Tom Selleck: A Journey Through His Early Years and Rise to Stardomgreendigital
Introduction
When one thinks of Hollywood legends, Tom Selleck is a name that comes to mind. Known for his charming smile, rugged good looks. and the iconic mustache that has become synonymous with his persona. Tom Selleck has had a prolific career spanning decades. But, the journey of young Tom Selleck, from his early years to becoming a household name. is a story filled with determination, talent, and a touch of luck. This article delves into young Tom Selleck's life, background, early struggles. and pivotal moments that led to his rise in Hollywood.
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Early Life and Background
Family Roots and Childhood
Thomas William Selleck was born in Detroit, Michigan, on January 29, 1945. He was the second of four children in a close-knit family. His father, Robert Dean Selleck, was a real estate investor and executive. while his mother, Martha Selleck, was a homemaker. The Selleck family relocated to Sherman Oaks, California. when Tom was a child, setting the stage for his future in the entertainment industry.
Education and Early Interests
Growing up, young Tom Selleck was an active and athletic child. He attended Grant High School in Van Nuys, California. where he excelled in sports, particularly basketball. His tall and athletic build made him a standout player, and he earned a basketball scholarship to the University of Southern California (U.S.C.). While at U.S.C., Selleck studied business administration. but his interests shifted toward acting.
Discovery of Acting Passion
Tom Selleck's journey into acting was serendipitous. During his time at U.S.C., a drama coach encouraged him to try acting. This nudge led him to join the Hills Playhouse, where he began honing his craft. Transitioning from an aspiring athlete to an actor took time. but young Tom Selleck became drawn to the performance world.
Early Career Struggles
Breaking Into the Industry
The path to stardom was a challenging one for young Tom Selleck. Like many aspiring actors, he faced many rejections and struggled to find steady work. A series of minor roles and guest appearances on television shows marked his early career. In 1965, he debuted on the syndicated show "The Dating Game." which gave him some exposure but did not lead to immediate success.
The Commercial Breakthrough
During the late 1960s and early 1970s, Selleck began appearing in television commercials. His rugged good looks and charismatic presence made him a popular brand choice. He starred in advertisements for Pepsi-Cola, Revlon, and Close-Up toothpaste. These commercials provided financial stability and helped him gain visibility in the industry.
Struggling Actor in Hollywood
Despite his success in commercials. breaking into large acting roles remained a challenge for young Tom Selleck. He auditioned and took on small parts in T.V. shows and movies. Some of his early television appearances included roles in popular series like Lancer, The F.B.I., and Bracken's World. But, it would take a
Meet Dinah Mattingly – Larry Bird’s Partner in Life and Loveget joys
Get an intimate look at Dinah Mattingly’s life alongside NBA icon Larry Bird. From their humble beginnings to their life today, discover the love and partnership that have defined their relationship.
240529_Teleprotection Global Market Report 2024.pdfMadhura TBRC
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Panchayat Season 3 - Official Trailer.pdfSuleman Rana
The dearest series "Panchayat" is set to make a victorious return with its third season, and the fervor is discernible. The authority trailer, delivered on May 28, guarantees one more enamoring venture through the country heartland of India.
Jitendra Kumar keeps on sparkling as Abhishek Tripathi, the city-reared engineer who ends up functioning as the secretary of the Panchayat office in the curious town of Phulera. His nuanced depiction of a young fellow exploring the difficulties of country life while endeavoring to adjust to his new environmental factors has earned far and wide recognition.
Neena Gupta and Raghubir Yadav return as Manju Devi and Brij Bhushan Dubey, separately. Their dynamic science and immaculate acting rejuvenate the hardships of town administration. Gupta's depiction of the town Pradhan with an ever-evolving outlook, matched with Yadav's carefully prepared exhibition, adds profundity and credibility to the story.
New Difficulties and Experiences
The trailer indicates new difficulties anticipating the characters, as Abhishek keeps on wrestling with his part in the town and his yearnings for a superior future. The series has reliably offset humor with social editorial, and Season 3 looks ready to dig much more profound into the intricacies of rustic organization and self-awareness.
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As the delivery date draws near, expectation for "Panchayat" Season 3 is at a record-breaking high. The authority trailer has previously created critical buzz, with fans enthusiastically anticipating the continuation of Abhishek Tripathi's excursion and the new undertakings that lie ahead in Phulera.
All in all, the authority trailer for "Panchayat" Season 3 recommends that watchers are in for another drawing in and engaging ride. Yet again with its charming characters, convincing story, and ideal mix of humor and show, the new season is set to enamor crowds. Write in your schedules and prepare to get back to the endearing universe of "Panchayat."
As a film director, I have always been awestruck by the magic of animation. Animation, a medium once considered solely for the amusement of children, has undergone a significant transformation over the years. Its evolution from a rudimentary form of entertainment to a sophisticated form of storytelling has stirred my creativity and expanded my vision, offering limitless possibilities in the realm of cinematic storytelling.
From Slave to Scourge: The Existential Choice of Django Unchained. The Philos...Rodney Thomas Jr
#SSAPhilosophy #DjangoUnchained #DjangoFreeman #ExistentialPhilosophy #Freedom #Identity #Justice #Courage #Rebellion #Transformation
Welcome to SSA Philosophy, your ultimate destination for diving deep into the profound philosophies of iconic characters from video games, movies, and TV shows. In this episode, we explore the powerful journey and existential philosophy of Django Freeman from Quentin Tarantino’s masterful film, "Django Unchained," in our video titled, "From Slave to Scourge: The Existential Choice of Django Unchained. The Philosophy of Django Freeman!"
From Slave to Scourge: The Existential Choice of Django Unchained – The Philosophy of Django Freeman!
Join me as we delve into the existential philosophy of Django Freeman, uncovering the profound lessons and timeless wisdom his character offers. Through his story, we find inspiration in the power of choice, the quest for justice, and the courage to defy oppression. Django Freeman’s philosophy is a testament to the human spirit’s unyielding drive for freedom and justice.
Don’t forget to like, comment, and subscribe to SSA Philosophy for more in-depth explorations of the philosophies behind your favorite characters. Hit the notification bell to stay updated on our latest videos. Let’s discover the principles that shape these icons and the profound lessons they offer.
Django Freeman’s story is one of the most compelling narratives of transformation and empowerment in cinema. A former slave turned relentless bounty hunter, Django’s journey is not just a physical liberation but an existential quest for identity, justice, and retribution. This video delves into the core philosophical elements that define Django’s character and the profound choices he makes throughout his journey.
Link to video: https://youtu.be/GszqrXk38qk
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From the Editor's Desk: 115th Father's day Celebration - When we see Father's day in Hindu context, Nanda Baba is the most vivid figure which comes to the mind. Nanda Baba who was the foster father of Lord Krishna is known to provide love, care and affection to Lord Krishna and Balarama along with his wife Yashoda; Letter’s to the Editor: Mother's Day - Mother is a precious life for their children. Mother is life breath for her children. Mother's lap is the world happiness whose debt can never be paid.
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Lab23 chisquare2007
1. Name __________________________________
AP Biology
Period _________
Date ______________________
LAB ____: THE CHI-SQUARE TEST
Probability, Random Chance, and Genetics
Why do we study random chance and probability at the beginning of a unit on genetics?
Genetics is the study of inheritance, but it is also a study of probability. Most eukaryotic
organisms are diploid, meaning that each cell contains two copies of every chromosome, so
there are two copies of each gene that controls a trait (alleles). In sexual reproduction, these
two copies of each chromosome separate, and are randomly sorted into the reproductive cells
(gametes). When gametes from two different parents combine in fertilization, new combinations
of alleles are created. Thus chance plays a major role in determining which alleles, and
therefore which combinations of traits end up in each new individual. The important point is that
the inheritance of characteristics is the result of random chance. Therefore, it is important to
understand the nature of chance and probability and the resulting implications for the science of
genetics. In short, the genes that an individual organism inherits depends on the “luck of the
draw,” and the luck of the draw is dependent on the laws of probability.
The Laws of Probability
There are three Laws of Probability that are important in genetics and they can be easily
demonstrated using simple models like flipping a coin or choosing cards from a deck:
•
The Rule of Independent Events: Past events have no influence on future
events.
Question: If a coin is tossed 5 times, and each time a head appears, then what is
the chance that the next toss will be heads?
Answer: 1/2 (1 chance in 2), because coins have 2 sides.
•
The Rule of Multiplication: The chance that two or more independent events
will occur together is equal to the product of the probabilities of each individual
event.
Question: What are the chances of drawing a red nine from a standard deck of
cards?
Answer: 1/26 (1 chance in 26), because there is 1/2 chance of drawing a red
card and 1 chance in 13 of drawing a nine. Therefore, 1/2 x 1/13 = 1/26 or 1
chance in 26 of drawing a red nine.
•
The Rule of Addition: The chance of an event occurring when that event can
occur two or more different ways is equal to the sum of the probabilities of each
individual event
Question: If 2 coins are tossed, what is the chance that the toss will yield 2
unmatched coins (1 head & 1 tail)?
Answer: 1/2 (1 chance in 2) because the combination of 2 unmatched coins can
come about in 2 ways: Result A (coin #1 heads, coin #2 tails) as well as Result B
(coin #1 tails, coin #2 heads). Therefore (1/2 x 1/2) + (1/2 x 1/2) = 1/2, or the
chance of Result A plus the chance of Result B.
1 of 13
Adapted by Kim B. Foglia • www.ExploreBiology.com • 2005-2006
2. Name __________________________________
AP Biology
Paired Coins and Genetics
Using paired coins, in fact, mimics genetics closely. Each coin can serve as the model for a
gamete during fertilization, because it’s the ”luck of the draw” governing which sperm fertilizes
which egg.
When you toss two coins, there are three possible outcomes:
•
2 heads
•
2 tails
•
1 head, 1 tail
The probability of each of these outcomes is based on the 3 Laws of Probability we just
discussed:
•
2 heads: 1/4 chance
1/2 heads on coin #1 x 1/2 heads on coin #2 = 1/4,
which is generalized as p2 because [p x p = p2]
•
2 tails: 1/4 chance
1/2 tails on coin #1 x 1/2 tails on coin #2 = 1/4,
which is generalized as q2 because [q x q = q2]
•
1 head, 1 tail: 1/2 chance
(1/2 heads on coin #1 x 1/2 tails on coin #2) + (1/2 tails on coin #1 x 1/2 heads on coin #2),
which is generalized as 2pq because [(p x q) + (q x p) = 2pq]
Therefore, all the expected results from tossing two coins can be summarized as follows:
2
2
p + 2pq + q = 1
(double heads) + (heads/tails) + (double tails) = 100%
You will see this formula again when we learn about genetics of populations, so it would be
good to become familiar with it now.
Lab Activity
1. Divide the class into 10 teams.
2. Each team of students will toss a pair of coins exactly 100 times and record the results on
the data table labeled “Team Data.” Each team must check their results to be certain that
they have exactly 100 tosses.
3. Record your team results on the chalkboard and then record the summarized results on
the data table labeled “Class Data.”
4. Analyze both the team data and the class data separately using the Chi-square analysis
explained below.
2 of 13
Adapted by Kim B. Foglia • www.ExploreBiology.com • 2005-2006
3. Name __________________________________
AP Biology
Chi-square Analysis
The Chi-square is a statistical test that makes a comparison between the data collected in an
experiment versus the data you expected to find. It is used beyond genetics studies and can be
used whenever you want to compare the differences between expected results and
experimental data.
Variability is always present in the real world. If you toss a coin 10 times, you will often get a
result different than 5 heads and 5 tails. The Chi-square test is a way to evaluate this variability
to get an idea if the difference between real and expected results are due to normal random
chance, or if there is some other factor involved (like an unbalanced coin).
Genetics uses the Chi-square to evaluate data from experimental crosses to determine if the
assumed genetic explanation is supported by the data. In the case of genetics (and coin tosses)
the expected results can be calculated using the Laws of Probability (and possibly the help of a
Punnett square). The Chi-square test helps you to decide if the difference between your
observed results and your expected results is probably due to random chance alone, or if there
is some other factor influencing the results.
•
Is the variance in your data probably due to random chance alone and therefore
your hypothesis about the genetics of a trait is supported by the data?
•
Are the differences between the observed and expected results probably not due
to random chance alone, and your hypothesis about the genetics of a trait is
thereby not supported by the data?
•
Should you consider an alternative inheritance mechanism to explain the results?
The Chi-square test will not, in fact, prove or disprove if random chance is the only thing causing
observed differences, but it will give an estimate of the likelihood that chance alone is at work.
Determining the Chi-square Value
Chi-square is calculated based on the formula below:
X
2
=
∑
(observed – expected) 2
expected
A. For your individual team results, complete column A of the Chi-square Analysis Data
Table by entering your observed results in the coin toss exercise.
B. For your individual team results, complete column B of the Chi-square Analysis Data
Table by entering your expected results in the coin toss exercise.
C. For your individual team results, complete column C of the Chi-square Analysis Data
Table by calculating the difference between your observed and expected results.
D. For your individual team results, complete column D of the Chi-square Analysis Data
Table by calculating the square of the difference between your observed and expected
results. (This is done to force the result to be a positive number.)
E. For your individual team results, complete column E of the Chi-square Analysis Data
Table by dividing the square in column D by the expected results.
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4. Name __________________________________
AP Biology
F. Calculate the X2 value by summing each of the answers in column E. The ∑ symbol
means summation.
G. Repeat these calculations for the full class data and complete the Class Data Chi-square
Analysis Table.
H. Enter the “Degrees of Freedom” based on the explanation below.
Interpreting the Chi Square Value
With the Chi-square calculation table completed, you would look up your Chi-square value on
the Chi-square Distribution table at the back of this lab. But to know which column and row to
use on that chart, you must now determine the degrees of freedom to be used and the
acceptable probability that the Chi-square you obtained is caused by chance alone or by other
factors. The following two steps will help you to determine the degrees of freedom and the
probability.
Degrees of Freedom
Which row do we use in the Chi-square Distribution table?
The rows in the Chi-square Distribution table refer to degrees of freedom. The degrees
of freedom are calculated as the one less than the number of possible results in your
experiment.
In the double coin toss exercise, you have 3 possible results: two heads, two tails, or
one of each. Therefore, there are two degrees of freedom for this experiment.
In a sense degrees of freedom is measuring how many classes of results can “freely”
vary their numbers. In other words, if you have an accurate count of how many 2-heads,
and 2-tails tosses were observed, then you already know how many of the 100 tosses
ended up as mixed head-tails, so the third measurement provides no additional
information.
Probability = p
Which column do we use in the Chi-square Distribution table?
The columns in the Chi-square Distribution table with the decimals from .99 through .50
to .01 refer to probability levels of the Chi-square.
For instance, 3 events were observed in our coin toss exercise, so we already calculated
we would use 2 degrees of freedom. If we calculate a Chi-square value of 1.386 from the
experiment, then when we look this up on the Chi-square Distribution chart, we find that
our Chi-square value places us in the “p=.50” column. This means that the variance
between our observed results and our expected results would occur from random
chance alone about 50% of the time. Therefore, we could conclude that chance alone
could cause such a variance often enough that the data still supported our hypothesis,
and probably another factor is not influencing our coin toss results.
However, if our calculated Chi-square value, yielded a sum of 5.991 or higher, then
when we look this up on the Chi-square Distribution chart, we find that our Chi-square
value places us in the “p=.05” column. This means that the variance between our
observed results and our expected results would occur from random chance alone only
about 5% of the time (only 1 out of every 20 times). Therefore, we would conclude that
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5. Name __________________________________
AP Biology
chance factors alone are not likely to be the cause of this variance. Some other factor is
causing some coin combinations to come up more than would be expected. Maybe our
coins are not balanced and are weighted to one side more than another.
So what value of Probability (p) is acceptable in scientific research?
Biologists generally accept p=.05 as the cutoff for accepting or rejecting a hypothesis. If the
difference between your observed data and your expected data would occur due to chance
alone fewer than 1 time in 20 (p = 0.05, or 5%) then the acceptability of your hypothesis may be
questioned. In other words, there’s a 95% that the differences between your observed and your
expected data are due to some other factor beyond chance. Biologists consider a p value of .05
or less to be a “statistically significant” difference.
A probability of more than 0.05 by no means proves that the hypothesis from which you worked
is correct but merely tells you that from a statistical standpoint that it could be correct, and that
the variation from your expected results is probably due to random chance alone. Furthermore,
a probability of less than 0.05 does not prove that a hypothesis is incorrect; it merely suggests
that you have reason to doubt the correctness or completeness of one or more of the
assumptions on which your hypothesis is based. At that point, it would be wise as a researcher
to explore alternative hypotheses.
Null hypothesis
So how is this directly applied to genetics research?
In classical genetics research where you are trying to determine the inheritance pattern of a
phenotype, you establish your predicted genetic explanation and the expected phenotype ratios
in the offspring as your hypothesis. For example, you think a mutant trait in fruit flies is a simple
dominant inheritance. To test this you would set up a cross between 2 true-breeding flies:
mutant female x wild type male
You would then predict the ratios of phenotypes you would expect from this cross. This then
establishes an hypothesis that any difference from these results will not be significant and will
be due to random chance alone. This is referred to as your “null hypothesis”. It, in essence,
says that you propose that nothing else — no other factors — are creating the variation in your
results except for random chance differences.
After the cross, you would then compare your observed results against your expected results
and complete a Chi-square analysis. If the p value is determined to be greater than .05 then you
would accept your null hypothesis (differences are due to random chance alone) and your
genetic explanation for this trait is supported. If the p value is determined to be .05 or less then
you would reject your null hypothesis — random chance alone can only explain this level of
difference fewer than 1 time out of every 20 times — and your genetic explanation for this trait is
unsupported. You therefore have to consider alternative factors influencing the inheritance of
the mutant trait.
You would repeat this cycle of prediction-hypothesis-analysis for each of your crosses in your
genetic research.
Stating conclusions
Once you have collected your data and analyzed them using the Chi-square test, you are ready
to determine whether your original hypothesis is supported or not. If the p value in your Chisquare test is .05 or less (.05, .01, etc.) then the data do not support your null hypothesis that
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6. Name __________________________________
AP Biology
nothing else but random chance is at work here. So, as a scientist, you would state your
"acceptable" results from the Chi-square analysis in this way:
"The differences observed in the data were not statistically significant at the .05
level." You could then add a statement like, "Therefore the data support the
hypothesis that..."
And you will see that over and over again in the conclusions of research papers.
This is how a scientist would state "unacceptable" results from the Chi-square analysis:
"The differences observed in the data were statistically significant at the .05
level." You could then add a statement like, "Therefore the data do not support
the hypothesis that..."
N.B.: Do not forget in your writings that the word "data" is plural (datum is singular & rarely
used).
"Data are" is correct.
"Data is" is not correct.
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7. Name __________________________________
AP Biology
TEAM DATA
Toss
H/H
H/T
T/T
Toss
H/H
H/T
T/T
Toss
1
35
36
37
38
39
40
74
7
41
75
8
42
76
9
43
77
10
44
78
11
45
79
12
46
80
13
47
81
14
48
82
15
49
83
16
50
84
17
51
85
18
52
86
19
53
87
20
54
88
21
55
89
22
56
90
23
57
91
24
58
92
25
59
93
26
60
94
27
61
95
28
62
96
29
63
97
30
64
98
31
65
99
32
66
100
33
67
—
73
6
—
72
5
—
71
4
T/T
70
3
H/T
69
2
H/H
34
68
* The sum of the total of each column must equal 100 tosses.
—
Total*
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8. Name __________________________________
AP Biology
CLASS DATA
1
2
3
4
5
6
7
8
9
10
H/H
H/T
T/T
Total 100
100
100
100
100
100
100
100
100
100
Totals
Obs
Exp
1000
1000
TEAM DATA: CHI SQUARE ANALYSIS
A
B
C
D
Obs
Exp
Obs - Exp
(Obs – Exp)2
E
(Obs – Exp)2
Exp
H/H
H/T
T/T
X2 Total
Degrees of Freedom
CLASS DATA: CHI SQUARE ANALYSIS
A
Obs
B
Exp
C
D
Obs - Exp
(Obs – Exp)
2
H/H
H/T
T/T
X2 Total
Degrees of Freedom
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E
(Obs – Exp)2
Exp
9. Name __________________________________
AP Biology
CHI-SQUARE DISTRIBUTION TABLE
Degrees
of
freedom
Probability (p) value
0.95
0.80
0.70
ACCEPT NULL HYPOTHESIS REJECT
0.50
0.30
0.20
0.10
0.05
0.01
1
0.004
0.06
0.15
0.46
1.07
1.64
2.71
3.84
6.64
7.88
2
0.10
0.45
0.71
1.30
2.41
3.22
4.60
5.99
9.21
10.59
3
0.35
1.00
1.42
2.37
3.67
4.64
6.25
7.82
11.34
12.38
4
0.71
1.65
2.20
3.36
4.88
5.99
7.78
9.49
13.28
14.86
5
1.14
2.34
3.00
4.35
6.06
7.29
9.24
11.07
15.09
16.75
6
1.64
3.07
3.38
5.35
7.23
8.56
10.65
12.59
16.81
18.55
7
2.17
3.84
4.67
6.35
8.38
9.80
12.02
14.07
18.48
20.28
0.005
In scientific research, the probability value of 0.05 is taken as the common cut off level of
significance. A probability value (p-value) of .05 means that there is a 5% chance that the
difference between the observed and the expected data is a random difference, and a 95%
chance that the difference is real and repeatable — in other words, a significant difference.
Therefore, if your p-value is greater than .05, you would accept the null hypothesis: “The
difference between my observed results and my expected results are due to random chance
alone and are not significant.”
In genetics experiments (like your upcoming Fly Lab), accepting the null hypothesis would mean
that your data are supporting your proposal for the genetics and inheritance scheme for the flies
that you were breeding.
In medical research, the chi-square test is used in a similar — but interestingly different — way.
When a scientist is testing a new drug, the experiment is set up so that the control group
receives a placebo and the experimental group receives the new drug. Analysis of the data is
trying to see if there is a difference between the two groups. The expected values would be that
the same number of people get better in the two groups — which would mean that the drug has
no effect. If the chi-square test yields a p-value greater than .05, then the scientist would accept
the null hypothesis which would mean the drug has no significant effect. The differences
between the expected and the observed data could be due to random chance alone. If the chisquare test yields a p-value ≤ .05, then the scientist would reject the null hypothesis which
would mean the drug has a significant effect. The differences between the expected and the
observed data could not be due to random chance alone and can be assumed to have come
from the drug treatment.
In fact, chi-square analysis tables can go to much lower p-values than the one above — they
could have p-values of .001 (1 in 1000 chance), .0001 (1 in 10,000 chance), and so forth. For
example, a p-value of .0001 would mean that there would only be a 1 in 10,000 chance that the
differences between the expected and the observed data were due to random chance alone,
whereas there is a 99.99% chance that the difference is really caused by the treatment. These
results would be considered highly significant.
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10. Name __________________________________
AP Biology
QUESTIONS
1. What is the Chi-square test used for? ____________________________________________
___________________________________________________________________________
___________________________________________________________________________
2. Why is probability important in genetics? _________________________________________
___________________________________________________________________________
___________________________________________________________________________
3. Briefly describe how the Chi-square analysis may be used in genetics.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
4. Suppose you were to obtain a Chi-square value of 7.82 or greater in your data analysis (with
2 degrees of freedom). What would this indicate?
___________________________________________________________________________
___________________________________________________________________________
5. Suppose you were to obtain a Chi-square value of 4.60 or lower in your data analysis (with 2
degrees of freedom). What would this indicate?
___________________________________________________________________________
___________________________________________________________________________
6. A heterozygous white-fruited squash plant is crossed with a yellow-fruited plant, yielding 200
seeds. Of these, 110 produce white-fruited plants while only 90 produce yellow-fruited
plants. Are these results statistically significant? Explain using Chi-square analysis.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
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11. Name __________________________________
AP Biology
7. What if there were 2000 seeds and 1100 produced white-fruited plants & 900 yellow-fruited?
Are these results statistically significant? Explain using Chi-square analysis.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
8. TEAM DATA: What was your hypothesis (expected values) for your individual team coin
toss?
___________________________________________________________________________
___________________________________________________________________________
What was your calculated Chi-square value for your individual team data? ______________
What p value does this Chi-square correspond to? _________________________________
Was your hypothesis supported by your results? Explain using your Chi-square analysis.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
9. CLASS DATA: What was your hypothesis (expected values) for the class coin toss?
___________________________________________________________________________
___________________________________________________________________________
What was your calculated Chi-square value for the class data? _______________________
What p value does this Chi-square correspond to? _________________________________
Was your hypothesis supported by the results? Explain using your Chi-square analysis.
___________________________________________________________________________
___________________________________________________________________________
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12. Name __________________________________
AP Biology
10. When scientists design research studies they purposely choose large sample sizes. Work
through these scenarios to see why:
a. Just as in your experiment, you flipped 2 coins, but you only did it 10 times. You
collected these data below. Use the chart to calculate the Chi-square value:
Obs
H/H
(Obs – Exp)2
(Obs – Exp)2
Exp
8
T/T
Obs - Exp
1
H/T
Exp
1
X2 Total
Would you accept or reject the null hypothesis? Explain using your Chi-square analysis.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
b. Now you flipped your 2 coins again, but you did it 100 times. You collected these data
below. Use the chart to calculate the Chi-square value:
Obs
H/H
(Obs – Exp)
80
T/T
Obs - Exp
(Obs – Exp)2
Exp
10
H/T
Exp
2
10
X2 Total
Would you accept or reject the null hypothesis? Explain using your Chi-square analysis.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
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13. Name __________________________________
AP Biology
c. Now you flipped your 2 coins again, but you did it 1000 times. You collected these data
below. Use the chart to calculate the Chi-square value:
Obs
H/H
(Obs – Exp)
800
T/T
Obs - Exp
(Obs – Exp)2
Exp
100
H/T
Exp
2
100
X2 Total
Would you accept or reject the null hypothesis? Explain using your Chi-square analysis.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
d. Now, using your understanding of the Chi-square test, explain why scientists purposely
choose large sample sizes when they design research studies.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
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