If X = the different scores you could get on the quiz, what values could X be?

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1 Example 1: Quiz? Take it. o, there are no questions m giving you. You just are giving me answers and m telling you if you got the answer correct. Good luck: hope you studied! Circle the correct answers that were read to you. Give yourself a score out of 5: Did you pass? f X = the different scores you could get on the quiz, what values could X be? X: score Probability Using your method of choice, simulate taking the quiz 10 more times, and add your 10 scores to the table. When all the scores have been totaled, we will complete our probability distribution in the table. According to our distribution, if we were to take the quiz by guessing, what is the most likely score? This assumption is based on our probability distribution of simulations, not a theoretical distribution. Getting a question right or wrong is an example of a binomial random variable For a variable to be considered which is only for variables, it has to meet the following conditions:? Each trial can result in just two possible outcomes. We call one of these outcomes a and the other, a.? The trials are independent; that is, the outcome on one trial does not affect the outcome on other trials.? The number of trials n of the chance process must be in advance.? The probability of success, denoted by, is the same on every trial.

2 Example 2: Determine if the given random variable has a binomial distribution. Explain your response. Consider the following statistical experiments. Using, are they binomial? 1. huffle a deck of cards. Turn over the top card. Put the card back in the deck, and shuffle again. Repeat this process 10 times. Let X = the number of aces you observe. 2. Choose three students at random from your class. Let Y = the number who are over 6 feet tall. 3. Flip a coin. f it s heads, roll a 6-sided die. f it s tails, roll an 8-sided die. Repeat this process 5 times. Let W = the number of 5s you roll. 4. You are taking a 5-question quiz by guessing. Let C = the number of problems you get correct.

3 Example 3: Our guessing quiz Hey, #4 looks familiar. o now that we know that our quiz-taking is an example of a binomial distribution, what does the theoretical probability distribution look like? X: score Probability To find the probabilities, we have to use the multiplication rule, since these events are independent. a. What would the probability be that we guess none of them correctly? P(X=0) Question 1 question 2 question 3 question 4 question 5 b. What would the probability be that we guess all of them correctly? P(X=5) = Question 1 question 2 question 3 question 4 question 5 c. What would the probability be that we guess exactly one correctly? P(X=1) = This one is more complicated because unlike a and b, there are multiple ways to guess only one problem correctly. d. Create the sample space to show how many ways there are to guess one problem correctly on a five-question quiz.

4 e. What is the probability for guessing exactly 2 correctly? Formula for binomial probability n = k = p = 1-p = The n over k in the parenthesis is called the which is the number of ways an event can happen, and is calculated using combinations,. o let s try it: e. What is the probability for guessing exactly 2 correctly? P(X=2): et up your equation using the formula, with n = k = p = 1 p = Value of binomial coefficient: (on your calculator, type in your value of n, 5, then Math, Prb, 3:nCr, and 2, since k = 2): f. Use the formula to complete our probability distribution, table and graph.

5 Example 4: 1 in 6 wins! As a special promotion for its 20-ounce bottles of soda, a soft drink company printed a message on the inside of each cap. ome of the caps said, Please try again, while others said, You re a winner! The company advertised the promotion with the slogan 1 in 6 wins a prize. even friends each buy one 20-ounce bottle at a local convenience store. The store clerk is surprised when three of them win a prize. s this group of friends just lucky, or is the company s 1 in 6 claim inaccurate? inomial setting: How to use your T calculator to calculate binomial probabilities inomial random variable: X = Probability distribution: Feel free to use, but in order to get credit on a test, want to see the set up using the formula nd /VAR 2. croll down a. inompdf(n,p,k): used when finding P(X = k) b. inomcdf(n,p,k): used when finding P(X k) i. Careful using the binomcdf it can be misleading when you want to use it to find instead P(X k). n these cases, make sure to find 1-P(X k-1) instead of 1-P(X k). Make a list of desired outcomes to be sure. f you don t specify X, it will give you the probability for all values of X, from 0 to n as a list Value Probability hape: Center: pread: P(X 3) =

6 Example 5: Thinking again about that quiz a. How many questions of the 5 do you think you would expect to get correct? This is the mean of this particular binomial distribution. Mean and standard deviation of a binomial random variable b. Calculate the standard deviation of this distribution. μx= σx= c. Find the mean and standard deviation for the 1 in 6 wins example. Example 6: Tastes as Good as the Real Thing? The makers of a diet cola claim that its taste is indistinguishable from the full calorie version of the same cola. To investigate, an AP tatistics student named Emily prepared small samples of each type of soda in identical cups. Then, she had volunteers taste each cola in a random order and try to identify which was the diet cola and which was the regular cola. Overall, 23 of the 30 subjects made the correct identification. f we assume that the volunteers really couldn t tell the difference, then each one was guessing with a 1/2 chance of being correct. Let X = the number of volunteers who correctly identify the colas. Problem: (a) Explain why X is a binomial random variable. (b) Find the mean and the standard deviation of X. nterpret each value in context. (c) Of the 30 volunteers, 23 made correct identifications. Does this give convincing evidence that the volunteers can taste the difference between the diet and regular colas?