Question

c-f on

this page

e) Deseribe the probabuiity histogram (symmetry, shape, center And the mean of this discrete random variable, Х. (Reminden This is asking you to find the mean or a random variable. .not a sample mean.) Show your 二つ、 e) Find the standard deviation of the random variable, X. (Reminder: This is asking you to find the standard deviation of a random variable...not a sample standard deviation.) Keep EXACT fractions throughout the calculation. Round to three digits after the decimal for your final answer only. Show your calculation. 2. 15, 17 1.S e PART 2: SIMULATION (a) Now you will roll two dice, so it is as if you are taking theoretical population you showing. List them in the table below. described in RT 1, and let &"the mean of the two numbers We know X is a random variable. Think about the possibie Now you will simulate taking 50 samples of size n actually rolling the two dice 50 times. Every time you frequency distribution to show how often you 2 from the population in PART 1 by and make a t roll the two dice, find observed each X NOTE: own or on the Internet at You can do this dice rolling at home using two dice of your a site that rolls the dice for you. to calculat table. You need to calculate &after each roll tally the results in the 2 TİObservedFrequencyTollies ur 111 45 b) Neatly construct a frequency histogram using the simulation results from part (a). Label the axes. oT 1.5 a 5 3 3.5 1 4.3 5 5.5 6 C.57 (c) Examine the frequency histogram of X you created in PART 2(b). What type of distribution does it appear to be (symmetric or not, normal or not, center)? nor mal Cbel sha pe (d) In PART 2ta) and (b) you used data to explore the distribution of the sample hme tis cetical samples of size 2 taken from the population in PART 1 but now let's construc probability distribution so we can get an accurate mean and distribution of X. To determine the theoretical probabilities, it will be helpfui toossible possible equally likely outcomes in a table. In the following table, 1iW in X for pair of dice. of s0 the theoret for the standard deviation Die 2 42 3 35 45 5 635 4$ 56 Now, use the fact that there are 36 equally likely of the 11 different values of R. Insert all this information in the folowing table. value of X along with the exact probability X takes on a particular value. Probability Distribution of X: outcomes to calculate the probability of each 2. (e) Construct the probability histogram for X. Label the axes what type of distribution is this (symmetric or not, normal or not, center)? be lI s hape 115 2 25 3 3.5 445S 5.5 6 (f) Find the mean of the random variable a. show your (&) Find the standard deviation of the random variable X. Do not round probabilities in the calculation. Keep three digits after the decimal in your final answer. Show your calculation. 4.35- 3.1 PART 3: Putting It All Together a) Compare the mean from PART 10) to the meařrfrom PART 2 what do you noticer The men n fro m Pa thy hagher t h an Par (b) Compare the standard deviation from PART 1(e) and to the standard deviation from PART 2(8). Which standard deviation is smaller? The dde viati onlis arger hen standard devi Stand ar art artds tandard deviationi malle (e)if you take the standard deviation from PART 1e) and divide it by (the square rot of the r to sample size) how does it compare to the standard deviation in PART 218)? it is Sim pot standor So now finish this sentence: If we know the standard deviation of a population and we t of samples of size 2, then we can predict the standard deviation of R by calculating ake lots (d) Carefully read about the Central Limit Theorem. State the Central Limit Theorem in your own words and be sure to talk about the mean of R and standard deviation of (e) Did your results from PART 3(a) and 3(e) support the Central Limit Theorem? (They should. If they did not then go figure out why and fix it.) Be specific involving the means and standard deviations (f) If we would take samples of size n = 30 from the population in PART 1 and finde every time. then we would expect the mean of the random variableX to b deviation of random variable X to be We would expect the distribution of X to be a distribution. e and the standard

Answer 1

(c) Standard deviation from Part 1(e) E(X) Mean 3.5 E(X2)12(1/6)22 (1/6) +32*(1/6)+42(1/6) 5(1/6)+62 (1/6) 15.1667 σ2 E(X)-E2(X) 15.1667-(3.5)2 2.9167 σ-standard deviation-V 2.9167-1.708 Now, -- = 1.2077 Predict the standard deviation by calculating 1.2077 mx-3.5 N(0,1) (d) From CLT,1,708 (e) No, since the sample size 2 is very small, so need to increase sample size (f) X- 3.5 standard deviation of X = 1.708/v/30 0.31 18 Distribution of XNormal distribution with mean-3.5, standard deviation-0.3118