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McGraw-Hill Ryerson

McGraw-Hill Ryerson. Data Management 12. 7.5. Section 5.1. Connections to Discrete Random Variables. Connections to Discrete Random Variables. 5.1. 7.5. I am learning to • make connections between a normal distribution and a binomial distribution

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McGraw-Hill Ryerson

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  1. McGraw-Hill Ryerson Data Management 12 7.5 Section 5.1 Connections to Discrete Random Variables

  2. Connections to Discrete Random Variables 5.1 7.5 I am learning to •make connections between a normal distribution and a binomial distribution •make connections between a normal distribution and a hypergeometric distribution •recognize the role of the number of trials in these connections Success Criteria I will know I am successful when I can • use a normal distribution to approximate probabilities associated with a binomial distribution •use a normal distribution to approximate probabilities associated with a hypergeometric distribution •tell whether it is appropriate to use a normal approximation for a given number of trials •apply a continuity correction to determine the probability associated with a discrete distribution using a normal distribution •describe the connection between the number of trials and the fit of a normal approximation to a binomial or hypergeometric distribution What are some other success criteria?

  3. Connections to Discrete Random Variables 7.5 Recall from Chapter 4 that tossing a coin several times and recording the number of heads obtained is an example of a binomial distribution. How does the shape of the distribution depend on the number of times the experiment is tried? Example: As the number of trials increases, the shape of the distribution becomes closer to the normal distribution. Predict the form of a graph of the number of heads possible when a coin is flipped five times. Make a sketch of your prediction. Example: The graph will have a normal distribution. Click to Reveal

  4. Connections to Discrete Random Variables 5.1 7.5 Investigate 1 Compare the Binomial Distribution to the Normal Distribution 1. Click on the icon to open the Fathom™ file. The file shows the probability distribution for the number of heads flipped on five trials. To approximate the binomial distribution with a normal distribution, you can calculate the mean using the formula , and the standard deviation using the formula . Check some of the probabilities by calculating them yourself. 2. How well does the normal distribution match the binomial distribution? 3. Right click on the collection box, and add 5 new cases. Adjust the scales on the axes if necessary. How well do the distributions match with 10 tosses of the coin? 4. Right click on the collection box, and add 10 new cases. Adjust the scales on the axes if necessary. How well do the distributions match with 20 tosses of the coin? 5. Reflect How does the fit of the normal distribution to the binomial distribution depend on the number of trials? Use your Fathom™ simulation to try a larger number of trials, such as 100 and then 1000.

  5. Connections to Discrete Random Variables 5.1 7.5 Investigate 2 Compare the Hypergeometric Distribution to the Normal Distribution A committee of 4 is chosen from a group of 200 people. How many males are on the committee? How does the distribution change as the size of the committee is increased? Does the population size have any effect? 1. Click on the icon to open the Fathom™ file. The file shows the probability distribution for the number of males selected in a committee of 4 people from a sample of 100 males and 100 females. Ifthe sample size is small compared to the population size, the probability of selecting a male is approximately equal to the number of males divided by the population size. To approximate the hypergeometric distribution with a normal distribution, you can calculate the mean using the formula and the standard deviation using the formula .

  6. Connections to Discrete Random Variables 5.1 7.5 Investigate 2 Compare the Hypergeometric Distribution to the Normal Distribution 2. How well does the normal distribution match the hypergeometric distribution? 3. Change the committee membership from 4 to 10. Right click on the collection box, and add 6 new cases. Adjust the scales on the axes of the graph if necessary. How is the fit with 10 members on the committee? 4. Change the committee membership to 20. Right click on the collection box, and add 10 new cases. How is the fit with 20 members on the committee? 5. Reflect The committee membership must remain a small fraction of the population size, typically less than one-tenth. Why is this necessary? Consider the values of pand q in the above investigation in your response. 6. Extend Your Understanding Suppose that a committee of 4 were chosen in a random selection from a very small population, say a club with 6 male members and 2 female members. Can you see any problems with calculating the number of males on the committee? Consider some extreme cases. Use the Fathom™ simulation from the investigation to explore different scenarios.

  7. Connections to Discrete Random Variables 5.1 7.5 When is a normal approximation reasonable? Click to Reveal Binomial Distribution usually considered reasonable if np> 5 and nq > 5 Hypergeometric Distribution usually considered reasonable if n < 0.1 NP, where n is the sample size and NPis the population size Why is it not considered reasonable to use a normal approximation outside these parameters?

  8. Connections to Discrete Random Variables 5.1 7.5 Continuity Correction Suppose you want to use a normal approximation to determine the probability of flipping a coin 5 times and getting exactly 1 head. You cannot simply determine the area under the normal curve for that particular point, as there is no area under a point. (A point has a width of zero). We apply a continuity correction in cases like this to determine the probability of discrete outcomes using a (continuous) normal distribution. To understand a continuity correction, it might help to envision a histogram overlaid upon the normal approximation—think of the binomial distribution it is approximating. The area we want is the area of the bar centred on the point at (1,f(1)), with a width of 1 unit. Its left bound is x = 0.5 and its right bound is x = 1.5. This rectangular area can be approximated by considering the area under the normal curve over the same interval. So, the approximate probability is given by P(1 head) = P(0.5 < X < 1.5).

  9. Connections to Discrete Random Variables 5.1 7.5 Example 1 Normal Approximation for a Binomial Distribution A data management quiz consists of 25 multiple choice questions with 4 choices per question. Charlie didn’t study for the quiz, so he guesses an answer for each question. a) Is it reasonable to approximate this distribution with a normal distribution? Give a reason. Click for Hints b) What values need to be determined in order to use a normal approximation? Determine these values. To use a normal approximation, you need to determine the mean and standard deviation. c) What is the probability that Charlie will get a passing grade (50% or more) on this quiz? Use a continuity correction for answering part c).

  10. Connections to Discrete Random Variables 5.1 7.5 Example 2 Normal Approximation for a Hypergeometric Distribution Lizzie deals 5 cards from a standard deck of 52 cards. She would like to deal as many face cards as possible. Click for Hints There are 3 face cards (J, Q, K) in each of 4 suits for a total of 12 face cards per deck. a) Is it reasonable to approximate this distribution with a normal distribution? Give a reason. To use a normal approximation, you need to determine the mean and standard deviation. b) What values need to be determined in order to use a normal approximation? Determine these values. c) What is the probability that Lizzie will deal 3 or more face cards? Use a continuity correction when answering part c).

  11. Connections to Discrete Random Variables 7.5 Reflect In what circumstances would you prefer not to use a normal distribution to approximate a binomial or hypergeometric distribution? Click to Reveal Example: When looking for the probability of a single event, it may be faster to calculate the probability using the binomial or hypergeometric formula.

  12. Connections to Discrete Random Variables 7.5 1. True or false? It is always better to approximate binomial and hypergeometric distributions with normal distributions. False Click for Answer

  13. Connections to Discrete Random Variables 7.5 2. True or false? It is appropriate to use a normal approximation for a binomial distribution with p = 0.6 and n = 10. Click for Answer False

  14. Connections to Discrete Random Variables 7.5 3. True or false? A continuity correction is used when using the area under a normal distribution to model the probability of discrete events. Click for Answer True

  15. Connections to Discrete Random Variables 7.5 4. Select the best answer. When approximating a binomial distribution with a normal distribution, increasing the sample size will make the approximation A better B better as long as the sample does not exceed 10% of the population size C perfect D worse Click for Answer A

  16. Connections to Discrete Random Variables 7.5 5. Select the best answer. The graphing calculator entry to approximate the probability of 5 or more discrete events, using a normal approximation with a mean of 6 and a standard deviation of 2.3 is A normalcdf(5, 999999, 6, 2.3) B normalcdf(5, infinity, 6, 2.3) C normalcdf(4.5, 999999, 6, 2.3) D normalcdf (4.5, infinity, 2.3, 6) C Click for Answer

  17. Section 5.1 The following pages contain 

solutions for the previous 

questions.

  18. Solutions Investigate 1 Compare the Binomial Distribution to the Normal Distribution 1. Click on the icon to open the FathomTMfile. To approximate the binomial distribution with a normal distribution, you can calculate  the mean using the formula , and the standard deviation using the formula . Check some of the probabilities by calculating them yourself. Using the normal distribution to approximate the probability of tossing 2 heads 
 [normalcdf(1.5, 2.5, 2.5, 1.118)] gives a value of about 0.3144. Using the normal approximation to find the probability of tossing 4 heads [normalcdf(3.5, 4.5, 2.5, 1.118)] gives a value of about 0.1487. 2. How well does the normal distribution match the binomial distribution? The normal approximation is a smooth, continuous curve, where the binomial distribution has a small number of discrete points, and its graph looks like a jagged curve which is fairly close to the normal approximation, especially at the integer points.

  19. Solutions Investigate 1 Compare the Binomial Distribution to the Normal Distribution 3. Right click on the collection box, and add 5 new cases. Adjust the scales on the axes if necessary. How well do the distributions match with 10 tosses of the coin? The normal approximation fits slightly better for 10 tosses of the coin. 4. Right click on the collection box, and add 10 new cases. Adjust the scales on the axes if necessary. How well do the distributions match with 20 tosses of the coin? The normal approximation fits better still for 20 tosses of the coin.

  20. Solutions Investigate 1 Compare the Binomial Distribution to the Normal Distribution 5. Reflect How does the fit of the normal distribution to the binomial distribution depend on the number of trials? Use your Fathom™simulation to try a larger number of trials, such as 100 and then 1000. The normal approximation fits better as the number of trials increases.

  21. Solutions Investigate 2 Compare the Hypergeometric Distribution to the Normal Distribution A committee of 4 is chosen from a group of 200 people. How many males are on the committee? How does the distribution change as the size of the committee is increased? Does the population size have any effect? 1. Click on the icon to open the Fathom™ file. To approximate the hypergeometric distribution with a normal distribution, you can calculate the mean using the formula and the standard deviation using the formula . 2. How well does the normal distribution match the hypergeometric distribution? The normal distribution matches the hypergeometric distribution reasonably well.

  22. Solutions Investigate 2 Compare the Hypergeometric Distribution to the Normal Distribution 3. Change the committee membership from 4 to 10. Right click on the collection box, and add 6 new cases. Adjust the scales on the axes of the graph if necessary. How is  the fit with 10 members on the committee? The fit is slightly better with 10 members on the committee.

  23. Solutions Investigate 2 Compare the Hypergeometric Distribution to the Normal Distribution 4. Change the committee membership to 20. Right click on the collection box, and add 10 new cases. How is the fit with 20 members on the committee? The fit is even better with 20 members on the committee. 5. Reflect The committee membership must remain a small fraction of the population size, typically less than one-tenth. Why is this necessary? Consider the values of p  and q in the above investigation in your response. We are approximating p by taking the number of males in the population and dividing by the size of the population. Once a sample is taken, for example when the first person is selected for the committee, the value of p will change, since we are not replacing each committee member before picking the next one. If the value of p changes too much, the approximation for the mean and standard deviation will not be very good.

  24. Solutions Investigate 2 Compare the Hypergeometric Distribution to the Normal Distribution 6. Extend Your Understanding Suppose that a committee of 4 were chosen in a random  selection from a very small population, say a club with 6 male members and 2 female  members. Can you see any problems with calculating the number of males on the  committee? Consider some extreme cases. Use the Fathom™simulation from the  investigation to explore different scenarios. If the number of males, or females as in this example, is less than the sample size, it is possible the normal approximation will predict a non-zero probability for a committee with more males or females than exist in the whole population, which is clearly impossible.

  25. Solutions Example 1 Normal Approximation for a Binomial Distribution a) Is it reasonable to approximate this distribution with a normal distribution? Give a reason. Yes. The probability of guessing correctly on each question is 25%, so p = 0.25 and q = 0.75. Since np= (25)(0.25) = 6.25 > 5 and nq = 18.75 > 5, it is reasonable to approximate this binomial distribution with a normal distribution. b) What values need to be determined in order to use a normal approximation? Determine these values. In order to use a normal approximation, we need to determine the mean and standard deviation: c) What is the probability that Charlie will get a passing grade (50% or more) on this quiz? The lowest passing grade would be 13 out of 25. We should use a continuity correction and calculate P(X > 12.5). P(X > 12.5) = 0.00195 The probability of Charlie passing this quiz is less than 0.2%.

  26. Solutions Example 2 Normal Approximation for a Hypergeometric Distribution a) Is it reasonable to approximate this distribution with a normal distribution? Give a reason. Yes. Since n = 5 is slightly less than 0.1NP = 0.1(52), it is considered reasonable to approximate this scenario with a normal distribution. b) What values need to be determined in order to use a normal approximation? Determine these values. In order to use a normal approximation, we need to determine the mean and standard deviation:

  27. Solutions Example 2 Normal Approximation for a Hypergeometric Distribution c) What is the probability that Lizzie will deal 3 or more face cards? We should use a continuity correction. There is about a 5% probability that Lizzie will deal 3 or more face cards.

  28. Attachments 7.5 CoinToss.ftm 7.5 Committee.ftm

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