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M116 – TI 83/84 CALCULATOR – CH 5 Using the TI-83/84 calculator to find the Mean and Standard Deviation of Probability Distributions To evaluate the mean and standard deviation using the calculator Enter x into L1 Enter the probabilities into L2 Press STAT Arrow right to CALC Select 1: 1-Var Stats L1,L2 Press ENTER 1) When randomly selecting jail inmates convicted of DWI (driving while intoxicated), the probability distribution for the number x of prior DWI sentences is as described in the accompanying table (based on data from the U.S. Department of Justice). x 0 1 2 3 P(x) 0.512 0.301 0.138 0.049 a) What is the population? We are selecting our sample out of jail inmates convicted of DWI b) Describe in words the random variable. (What are we counting?) We are counting the number x of prior DWI sentences c) What are the possible values of the random variable? 0, 1, 2, 3 d) Verify that the given table is a probability distribution - All probabilities are positive numbers less than or equal to 1 - The sum of the probabilities is 1 e) Use the calculator to find the mean and standard deviation of this distribution. μ = .7 σ = .9 f) Which values are usual and which are unusual, according to (i) The probability rule? 3 is unusual because its probability is less than 0.05 We observe about 49 out of 1000 (less than 5 in 100) with 3 prior DWI sentences (ii) The range rule of thumb? [μ-2σ, μ + 2σ] = [.7 – 2(.9), .7 + 2(.9)] = [-1.1, 2.5] Since the numbers 0, 1, and 2 are inside this interval, then, those are usual outcomes. Since the number 3 is outside of this interval, then 3 is unusual It’s common for jail inmates convicted of DWI to have anywhere from 0 to 2 prior DWI sentences. It’s unusual for them to have 3 prior DWI sentences. (Page 3 of notes) 1 M116 – NOTES – CH 5 Section 5.2 & 5.3 – Binomial Experiments Features of a binomial experiment (5.2) 1) 2) 3) 4) The experiment has a fixed number of trials (n) The trials must be independent Each trial has 2 possible outcomes: success (S) and failure (F) Probabilities remain constant for each trial. p is the probability of success, and q is the probability of failure When sampling without replacement, the events can be treated as if they were independent if the sample size is no more than 5% of the population size. (That is, n 0.05 N ) Binomial probability formula (5.2) (not done in the Spring 08 – used calculator) P(r ) n! p r q nr = Cn,r p r q nr (n r )!r ! where x is the number of successes in n trials, p is the probability of success in any one trial, and q is the probability of failure in any one trial. (q = 1 – p) Find binomial probabilities with a shortcut feature of the calculator To find individual probabilities: Use binompdf(n,p,x) Press 2nd VARS Select 0:binompdf( Type n,p,x) Press ENTER To calculate cumulative probabilities from 0 to x, use binomcdf(n,p,x) Mean, Variance, and Standard Deviation for the Binomial Distribution (5.3) In Section 5.1, we used the general formulas for any discrete probability distribution. But the special qualities of binomials distributions, lead to specialized formulas for binomials npq np Both sets of formulas form sections 5.1 and 5.3 are given on the table below Any Discrete Probability Distribution Section 5.1 Mean [ x P( x)] Binomial Distributions Section 5.3 np Did not use, used calculator Standard Deviation [ x 2 P( x)] 2 npq Did not use, used calculator (Page 4 of notes) 2 DID NOT DO THIS ON THE SPRING 2008 M116 – TI 83/84 CALCULATOR – CH 5 2) Count the number of male and female students in class, and complete the table. frequency Probability Male Females Total a) EXPERIMENT: We are going to select 4 students at random from the class, and count the number of females in the selected group. This is an experiment with 4 trials. Then n=4 Since we are counting females, our success probability p, is the probability obtained above for the females category. Then p= (use 3 decimal places) b) On the table given below, indicate the possibilities for the random variable, the number of female students in a group of 4. x Tally…………………... c) frequency Experimental probability Theoretical probability –from part (e) Let’s use the calculator to do the random selection. MATH Arrow right to PRB Select 5:randInt(1,….,4) ENTER d) Use the class roster to identify the gender of the selected students. Count the number of female students in the group of 4. Tally the results on the table. Do the process 5 times with your calculator. Come and tally your results on the board. After calculating the frequencies, determine the experimental probabilities. e) Now let’s use the calculator to obtain the theoretical probabilities STAT 1:Edit Enter the values of the random variable 0-4 into L1 Go to list 2. Arrow up to “sit on the name of the list” 2nd VARS[DISTR] Select 0:binompdf(n,p) (use the values of n and p from part (a)) Press ENTER Compare the experimental and theoretical results. How can you obtain “closer results”? f) Observe the table. What values of x are more likely? g) Use the calculator to find the mean and standard deviation of the probability distribution that you have in lists 1 and 2 of the calculator. (Page 5 of notes) 3 M116 – TI 83/84 CALCULATOR – CH 5 h) Identify usual and unusual results, according to (i) The probability rule? DID NOT DO THIS ON SPRING 2008 (ii) The range rule of thumb? 3) Suppose we select 5 students at random from our class. Use the calculator to construct the theoretical probability distribution for the number of male students in a group of 5. Use the probability of male obtained on the first table of the preceding page. Place this distribution into lists 3 and 4 of your calculator. In this problem n = …………….., and p = ……………….. x P(x)……………… a) Use the calculator to find the mean and standard deviation of the probability distribution that you have in lists 3 and 4 of the calculator. b) Identify usual and unusual results, according to (i) The probability rule? (ii) The range rule of thumb? (Page 6 of notes) 4 DID NOT DO THIS ON THE SPRING 2008 M116 – TI 83/84 CALCULATOR – CH 5 Using the TI-83/84 calculator to find Factorials, Binomial Coefficients and Binomial Probabilities 4) Evaluate factorials with the calculator: Type number Press MATH Arrow left to PRB Select 4:! Press ENTER Examples: a) Find 10! 10! = 10 * 9 * 8 *… * 3 * 2 * 1 = b) Find 6! 5) Evaluate binomial coefficients with the calculator: Type n (number of trials) Press MATH Arrow left to PRB Select 3:nCr Type x Press ENTER Examples: a) Find 10 C3 = b) Find 8 C5 = 6) Evaluate binomial probabilities with the formula: For a binomial experiment with n = 7 and p = 0.8, a) Find P(r = 3) b) Find P(r = 6) (Page 7 of notes) 5 M116 – TI 83/84 CALCULATOR – CH 5 7) Find binomial probabilities with a shortcut feature of the calculator A) To find individual probabilities: Use binompdf(n,p,x) Press 2nd VARS Select 0:binompdf( Type n,p,x) Press ENTER Partial solutions given - Please perform the final calculations Examples: a) For a binomial experiment with n = 7 and p = 0.8, find P(r = 3). Binompdf(7,0.8,3) = b) For a binomial experiment with n = 4 and p = 1/3, find P(r = 2). Binompdf(4,1/3,2) = 5-B) To get a list of all the probabilities corresponding to x = 0, 1, 2, …., n: Use binompdf(n,p) and scroll to the right to read the probabilities Examples: a) For a binomial experiment with n = 4 and p = 1/6, find the probability distribution. Find the mean and standard deviation for the distribution. Identify usual and unusual values with the range rule of thumb and with the probability rule. X P(X=r)=binompdf(4,1/6) 0 .48225 1 .3858 2 .11574 3 .01543 4 .00077 Did you notice that this distribution is right skewed?????(High probabilities on the low end) The lower numbers of successes are MORE COMMON. Because of the small probability of success (1/4) it’s more common to have a low number of successes. And this is what the probability rule tells you. 0, 1, or 2 successes are usual, while 3 or 4 successes are unusual Now let’s use the range rule of thumb: 1 n * p 4* .666667 .7 6 [ 1 5 n * p * q 4* * .745 .7 6 6 Usual range: [.7-2*.7, .7+2*.7] = [-.7, 2.1] Usual outcomes: 0, 1, 2 Unusual outcomes: 3, 4 Think of this example as rolling a fair die 4 times and counting the number of ones obtained. If you have a die, try it. Do the experiment (of rolling a die 4 times and counting the number of ones obtained in the 4 rolls) many times. You will see that it’s “common” to obtain 0, 1, or 2 ones. It’s not so common to obtain 3 or 4 ones. (Page 8 of notes) 6 b) For a binomial experiment with n = 5 and p = 1/2, find the probability distribution. Find the mean and standard deviation for the distribution. Identify usual and unusual values with the range rule of thumb and with the probability rule. X 0 1 2 3 4 5 P(X=r)=binompdf(5,1/2) .03125 .15625 .3125 .3125 .15625 .03125 Did you notice the bell shape in this distribution? According to the probability rule x = 0 and x = 5 are unusual outcomes Now let’s use the range rule of thumb: 1 n * p 5* 2.5 2 [ 1 1 n * p * q 5* * 1.1 2 2 Usual range: [2.5-2*1.1, 2.5 + 2*1.1] = [.3, 4.7] Usual outcomes: 1, 2, 3, 4 Unusual outcomes: 0, 5 (it agrees with the probability rule) Think of this example as tossing a fair coin 5 times and counting the number of heads obtained. If you have a coin, try it. Do the experiment (tossing the coin 5 times and counting the number of heads obtained) many times. You will see that it’s “common” to obtain 1, 2, 3, or 4 heads in 5 tosses, and not so common to obtain 0 heads or 5 heads. (Page 8 of notes-continued) 7 M116 – TI 83/84 CALCULATOR – CH 5 5-C) To calculate cumulative probabilities from 0 to x, use binomcdf(n,p,x) Press 2nd VARS Select A:binomcdf( Type n,p,x) Press ENTER Partial solutions given - Please perform the final calculations Examples: a) For a binomial experiment with n = 7 and p = 0.2, find the probability of at most 3 successes. P(x ≤ 3) = P(x=0) + P(x=1) + P(x=2) + P(x=3) = binomcdf (7, .2, 3) = b) For a binomial experiment with n = 6 and p = 0.46, find the probability of at most 4 successes. P(x ≤ 4) = binomcdf (6, 0.46, 4) = c) For a binomial experiment with n = 4 and p = 0.3, find the probability of at least 2 successes. (This is the complement of at most 1) P(x ≥ 2) = from the WHOLE probability distribution, subtract the sum of the probabilities from 0 through 1) = = 1 - binomcdf (4, 0.3, 1)) = d) For a binomial experiment with n = 8 and p = 0.85, find the probability of at least 5 successes. (This is the complement of at most 4) P(x ≥ 5) = from the WHOLE probability distribution, subtract the sum of the probabilities from 0 through 4) = 1 - binomcdf (8, 0.85, 4) = e) For a binomial experiment with n = 9 and p = 0.35, find the P (2 < x <6) P(2 < x < 6) = P(x = 3) + P(x = 4) + P(x = 5) = (Add probabilities from 0 to 5, and subtract the sum of the probabilities from 0 through 2) = = binomcdf (n,p,5) – binomcdf (n,p,2) = f) For a binomial experiment with n = 10 and p = 0.73, find the probability that x is between 4 and 9 inclusive. P(4 ≤ x ≤ 9) = P(x = 4) + P(x = 5) + ......+ P(x = 9) = (Add probabilities from 0 to 9, and subtract the sum of the probabilities from 0 through 3) = = binomcdf (n,p,9) – binomcdf (n,p,3) = (Page 9 of notes) 8 M116 – TI 83/84 CALCULATOR – CH 5 OPTIONAL (ITP) - Using the TI-83/84 calculator to graph Binomial Probability Distributions 6) To graph a probability distribution follow the steps outlined below: a) For a binomial experiment with n = 4 and p = ¼ Get into the editor of the calculator and clear two lists Place the possible values of the random variable into one of the lists, let’s say L1 (In this case the possible values of x are from 0 to 4) Go to L2, and arrow up until you “sit” on the name of the list Press 2nd VARS[DISTR] Arrow down to select 0:binompdf( Indicate choices of n,p (It should read: binompdf(4,1/4)) Press ENTER The probabilities should show in L2 Sketch a HISTOGRAM for L1, L2 (GO to STAT PLOT) Select appropriate WINDOW values x-min = 0 x-max = 5 (n + 1) x-scale = 1 y-min = - 0.2 y-max = 0.6 Press GRAPH The graph of the distribution should show. Press TRACE and arrow right to see the values of the random variable along with the probabilities. Comment on the shape of this distribution. b) For a binomial experiment with n = 6 and p = ½ (x-max should be 7 here) Sketch the graph of the distribution and comment on its shape. Work on L3, L4. Remember to have only one plot ON. c) For a binomial experiment with n = 10 and p = 4/5 (x-max should be 11 here) Sketch the graph of the distribution and comment on its shape. Work on L5, L6. d) Relationship between the value of p and the shape of the binomial distribution If p < 0.5 the shape is right skewed If p = 0.5 the shape is bell shaped If p > 0.5 the shape is left skewed (Page 10 of notes) 9 M116 – TI 83/84 CALCULATOR – CH 5 Binomial Distributions and Simulations (Chapter 5) 7) – Booking tickets: Air America has a policy of booking as many as 15 persons on an airplane that can seat only 14. Past studies have revealed that only 85% of the booked passengers actually arrive for the flight. Find the probability that if Air America books 15 persons, not enough sits will be available. a) Use the formula. Not done during the fall 07 b) Is this probability low enough so that overbooking is not a real concern for passengers? Is it unusual to find that there are not enough sits available? According to the answer given in part (c), it’s not unusual for 15 people to show up, then overbooking is a real concern. c) Now use a feature in the calculator to calculate the answer. Indicate what you enter in the calculator, and the results. P(not enough seats) = P(x = 15) = binompdf(15, 0.85, 15) = 0.087 (in about 9 out of 100 times) d) OPTIONAL (ITP) Now we are going to simulate this situation by repeating the experiment 20 times. Use MATH PRB 7:randBin(n,p) and press ENTER 20 times. Record results in a table, and then use your table to answer the question to the problem. e) Use class results and answer the question again. f) OPTIONAL (ITP) Here we have another simulation technique. Use the calculator to generate 50 numbers that come from a binomial distribution with n = 15 and p = 0.85 (We’ll clear List 1, generate the numbers and store them into List 1, we’ll sort the list and then explore the editor) STAT 4:ClrList L1 : MATH PRB 7:randBin(n,p,50) STO L1 : STAT 3:SortA(L1) Go to the editor, explore the list and count how many times we had 15 passengers showing up. Then determine the probability, and compare with the theoretical results from part (a). Comment on the law of large numbers. (Page 11 of notes) 10