# TEACHER NOTES MATH NSPIRED

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1 Math Objectives Students will understand that normal distributions can be used to approximate binomial distributions whenever both np and n(1 p) are sufficiently large. Students will understand that when either np or n(1 p) is small, the normal distribution probabilities for impossible numbers of successes (less than 0 or greater than n) are unreasonable. Students will formulate guidelines to determine what they mean by sufficiently large for good approximations. Vocabulary binomial random variable mean normal random variable number of trials, n probability distribution function probability of success, p standard deviation About the Lesson This lesson involves examining the general shape of binomial distributions for a variety of values of n and p. As a result, students will: Compare the shapes of binomial distributions to those of related normal distributions, recognizing the distinction between discrete and continuous random variables. Recognize that normal approximations of binomial probabilities become less and less accurate as either np or n(1 p) falls below 5 (or 10 or 15) by examining the probabilities calculated from the normal distribution for having the number of successes be less than 0 or greater than n. Prerequisites Knowledge Familiarity with binomially distributed random variables, including their fundamental definition and formulas for determining exact probabilities, mean, and standard deviation. Familiarity with normally distributed random variables, including the general shape of the normal pdf and how the graph of the pdf depends on the mean and standard deviation. Tech Tips: This activity includes class Nspire CX handheld. It is also appropriate for use with the TI-Nspire family of products including TI-Nspire software and TI-Nspire App. Slight variations to these directions may be required if using other technologies besides the handheld. Watch for additional Tech Tips throughout the activity for the specific technology you are using. Access free tutorials at captures taken from the TI- calculators/pd/us/online- Learning/Tutorials Lesson Files: Student Activity Why_np_min_Student.pdf Why_np_min_Student.doc TI-Nspire document Why_np_min.tns 2014 Texas Instruments Incorporated 1 education.ti.com

2 TI-Nspire Navigator System Send out the Why_np_min.tns file. Monitor student progress using Class Capture. Use Live Presenter to spotlight student answers. Activity Materials Compatible TI Technologies: TI-Nspire CX Handhelds, TI-Nspire Apps for ipad, TI-Nspire Software Discussion Points and Possible Answers Teacher Tip: This lesson uses the binomial random variable count of successes in all calculations and graphs in order that it can be used during or immediately after the study of random variables, before beginning formal inference methods. If you want to wait for the study of inference, you might also want to replace the count of successes, x, with the more familiar proportion of successes, p, making appropriate adjustments in the corresponding means and standard deviations. This lesson refers to (1 p) as the probability of a failure outcome. Some texts use the notation q for that quantity. Tech Tip: Be sure students understand how to use the n and p slider controls. For n, select the up or down indicator to step through values of n from 0 to 30 in increments of 5. For p, grab and drag the slider left or right to move through values of p in the interval [0, 1]. 1. You have already studied binomial random variables. This question reviews important facts about the distributions of such random variables. a. Describe exactly what a binomial random variable measures (counts). Answer: A binomial random variable counts the number of successes in exactly n identical and independent random trials, each having probability exactly p of producing a success. b. Write the formula for the binomial probability of exactly x successes. Be sure to state clearly what each letter in your formula represents Texas Instruments Incorporated 2 education.ti.com

3 Answer:, where n is the number of trials, p is the probability of success on each trial, and x is the number of successes for which the probability is being calculated, which is B(x). c. In the context of a binomial scenario, what do the values of np and n(1 p) mean? Answer: np is the average number of successes and n(1 p) is the average number of failures in n trials. Move to page The image on Page 1.2 is the probability distribution function (pdf) for a binomially-distributed random variable defined by n trials and probability p of success on each trial. a. Explain why the graph consists of a set of separated vertical bars. Answer: Every binomial random variable counts the number of successes in some fixed number of trials. A count can only be a positive integer, so only positive integers can have a non-zero probability of occurring and appear as a location for a bar. b. What does the height of each bar represent, and what formula is used to calculate that height? Answer: The height of the bar gives the probability of having x number of successes using the given number n of trials and probability p of success per trial, 2014 Texas Instruments Incorporated 3 education.ti.com

7 Teacher Tip: Note that decisions in question 7 will be subjective. Steer discussion of student answers toward methods that can be replicated and used by others. One such method is defined in question 8 ( restrict the approximating probabilities of impossible events ), but students should be encouraged to come up with ideas of their own. For example, they might suggest that approximations are poor whenever two or three standard deviations away from the mean falls outside the domain of [0, n]. Tech Tip: The unlabeled slider to the left of the vertical axis on Page 3.1 permits the binomial histogram to be displayed or removed for easier visualization of the shaded normal probabilities for P(x < 0) and P(x > n). Move to page a. What are the possible values for a binomial random variable (count of successes)? Answer: Only integers from 0 through n, inclusive. b. Explain why your answer to part a is problematic when using a normal distribution to approximate a binomial distribution. Sample Answers: A normal distribution always allows values from the entire number line, so numbers less than 0 and greater than n will cause conflict between the two models. c. Page 3.1 displays the normal probabilities for x < 0 and x > n. For the actual binomial distribution, what are the exact values of these two probabilities? In general, how do the normal and binomial probabilities compare? Sample Answers: The binomial probabilities are always 0. In general, the two models are pretty close unless the mean of the distributions gets close to 0 or n Texas Instruments Incorporated 7 education.ti.com

8 d. Observe these two probabilities as you vary n and p. Record the combinations of n and p for which the normal approximation gives a value of at least 0.01 for either of these two extreme probabilities. Be sure to check for values of p near 1 as well as for values near 0. (Note: Use the unlabeled slider to the left of the vertical axis to hide/show the binomial histogram bars.) Sample Answers: For n = 30, P(x < 0) first exceeds 0.01 at about p = For n = 20, it happens around p = For n = 10, it s around p = For n = 5, at least one of these probabilities is always greater than For P(x > n), replace each p with 1 p and the results remain valid. TI-Nspire Navigator Opportunity: Quick Poll See Note 3 at the end of this lesson. 9. Many textbooks include guidelines for using normal random variables to approximate binomial random variables. Those guidelines usually are of the form, Check that each of the values of np and n(1 p) is at least. Different books disagree on what they prefer to put in the blank as their magic number. a. What number would you put there, and why? Sample Answers: I would use 5 since it seems to be big enough for the rule we used in question 8. b. How does the quality of the approximation of a binomial distribution by a normal distribution vary as the magic number increases? Explain. Sample Answers: As the products increase the approximations get better, at least in the sense that less and less of the normal distribution corresponds to impossible events. Wrap Up Upon completion of the discussion, the teacher should ensure students are able to understand: Binomial distributions are actually discrete and finite. The mean and standard deviation of a binomial random variable are determined by n and p. Since a normal distribution is determined by its mean and standard deviation, n and p completely determine the appropriate normal distribution to approximate a binomial distribution. Normal distributions are reasonable approximations to binomial distributions whenever both np and n(1 p) are sufficiently large (usually taken to be 5, 10 or 15) Texas Instruments Incorporated 8 education.ti.com

9 Assessment Use Quick Poll or an exit quiz to assess students' understanding of the key concepts in the lesson. T or F. (Students should be asked for reasons for their choices.) 1. If n = 20 and p =.15, the binomial distribution can be fairly well approximated using a normal distribution. Answer: F. According to usual guidelines, np should be at least 5 (or 10 or 15). Here np is only The quantity n(1 p) represents the average number of failures in n trials in the context of a binomial setting. Answer: T. 3. When p is close to 1, the average number of successes is small. Answer: F. np is the average number of successes, so it would be near n, its maximum possible value. 4. When p is close to 1, the binomial distribution is skewed right and has no or a very small left tail. Answer: F. The exact opposite is true. The description in the stem applies to p near If np >10, you do not have to worry about the size of n(1 - p) in order to approximate the binomial with a normal distribution. Answer: F. If the average number of successes is large then the average number of failures can be too small, so it has to be checked as well. 6. An essential difference between a normal distribution and a binomial distribution is that the binomial distribution can be used only for discrete outcomes while the normal distribution can be used for continuous outcomes. Answer: T Texas Instruments Incorporated 9 education.ti.com

10 TI-Nspire Navigator Note 1 Question 3, Class Capture Use Class Capture for the entire class during their exploration of question 2. You might either ask students to produce graphs that they find interesting or assign various combinations of n and p for a more systematic look at the various possible graphs. Note 2 Question 5, Class Capture Use Class Capture for the entire class during their exploration of question 5, especially if different values for n have been assigned. Have students begin to notice how symmetry is related to both n and p. Note 3 Question 9a, Quick Poll Use a quick poll to collect and display students suggested cut-off values. Discuss how various values were selected Texas Instruments Incorporated 10 education.ti.com

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