Applet Exercise The population corresponding to the upper face observed on a single toss

Chapter 7, Problem 7.4

(choose chapter or problem)

Applet Exercise The population corresponding to the upper face observed on a single toss of a balanced die is such that all six possible values are equally likely. Would the results analogous to those obtained in Exercises 7.1 and 7.2 be observed if the die was not balanced? Access the applet DiceSample and scroll down to the part of the screen dealing with Loaded Die. a If the die is loaded, the six possible outcomes are not equally likely. What are the probabilities associated with each outcome? Click on the buttons 1 roll, 10 rolls, and/or 1000 rolls until you have a good idea of the probabilities associated with the values 1, 2, 3, 4, 5, and 6. What is the general shape of the histogram that you obtained? b Click the button Show Stats to see the true values of the probabilities of the six possible values. If Y is the random variable denoting the number of spots on the uppermost face, what is the value for = E(Y )? What is the value of , the standard deviation of Y ? [Hint: These values appear on the Stat Report screen.] c How many times did you simulate rolling the die in part (a)? How do the mean and standard deviation of the values that you simulated compare to the true values = E(Y ) and ? Simulate 2000 more rolls and answer the same question. d Scroll down to the portion of the screen labeled Rolling 3 Loaded Dice. Click the button Roll 1000 Sets until you have generated 3000 observed values for the random variable Y .i What is the general shape of the simulated sampling distribution that you obtained? ii How does the mean of the 3000 values y1, y2,..., y3000 compare to the value of = E(Y ) computed in part (a)? How does the standard deviation of the 3000 values compare to /3? e Scroll down to the portion of the screen labeled Rolling 12 Loaded Dice. i In part (ii), you will use the applet to generate 3000 samples of size 12, compute the mean of each observed sample, and plot these means on a histogram. Before using the applet, predict the approximate value that you will obtain for the mean and standard deviation of the 3000 values of y that you are about to generate. ii Use the applet to generate 3000 samples of size 12 and obtain the histogram associated with the respective sample means, yi,i = 1, 2,..., 3000. What is the general shape of the simulated sampling distribution that you obtained? Compare the shape of this simulated sampling distribution with the one you obtained in part (d). iii Click the button Show Stats to observe the mean and standard deviation of the 3000 values y1, y2,..., y3000. How do these values compare to those you predicted in part (i)?