Applet Exercises

Applet Exercises

399

To do these exercises, go to www.macmillanhighered.com/fapp10e.

Question 8.114

1. When we toss a coin, experience shows that the probability (long-term proportion) of a head is close to . Suppose now that we toss the coin repeatedly until we get a head. What is the probability that the first head comes up in an odd number of tosses (1, 3, 5, and so on)? Use the Probability applet to estimate this probability. Set the probability of heads to 0.5. Toss coins one at a time until the first head appears. Do this 50 times (click “Reset” after each trial). What is your estimate of the probability that the first head appears on an odd toss?

Question 8.115

2. The table of random digits (Table 7.1, page 298) was produced by a random mechanism that gives each digit probability 0.1 of being a 0.

  1. What proportion of the digits in the first row of Table 7.1 are 0s? This proportion is an estimate, based on 40 repetitions, of the true probability, which in this case is known to be 0.1.
  2. The Probability applet can imitate random digits. Set the probability of heads in the applet to 0.1. Check “Show true probability” to show this value on the graph. A head stands for a 0 in the random digit table and a tail stands for any other digit. Simulate 200 digits (50 at a time—don’t click “Reset"). If you kept going forever, presumably you would get 10% heads. What was the percentage of heads in your 200 tosses?

Question 8.116

3. One of the few players to have a better field goal percentage than free throw percentage, basketball star Shaquille O’Neal made about half (53%) of his free throws in his 21-year NBA career. Use the Probability applet to simulate 100 free throws shot independently by a player who has probability 0.53 of making each shot. (Toss 50, 50, without clicking “Reset.”)

  1. What percentage of the 100 shots were made?
  2. Start the process again, this time repeating 10 tosses 10 times so that you can keep track of the individual hits and misses. Examine the sequence of hits and misses after each click on “Toss” and keep track of the longest run of shots made and the longest run of shots missed. How long were the longest runs in the 100 shots taken? (Sequences of random outcomes often show longer runs than our intuition expects.)

Question 8.117

4. The central limit theorem is the basis for the confidence intervals that have been discussed in this chapter and in Chapter 7 (page 321). Next, you will use the Central Limit Theorem applet to generate individual data values from two different continuous probability models: the uniform probability model and the exponential probability model. You will find data from these distributions don’t look very normal, and then you will take samples of size 30 and generate means from the samples.

  1. Go to the Central Limit Theorem applet. Choose “Uniform” for the distribution. Set the sample size to 1, so that you can see the results of individual data values drawn from this distribution. (The “Show normal curve” should be unchecked.) Click the “Generate samples” button. Do data from the uniform distribution appear to have the characteristic normal shape?
  2. Now change the sample size to 10. Instead of generating individual outcomes from a uniform distribution, the applet will draw many samples of size 10 and then make a histogram of the sample means, . Click the “Generate Samples” button. Do these data appear to be from a normal distribution? Check the box for “Show normal curve.”
  3. This time, choose “Exponential” for the distribution. Set the sample size to 1 as you did in part (a). Click the “Generate samples” button. Describe the shape of exponential data.
  4. Now, continue with the exponential distribution but change the sample size to 10. Instead of generating individual outcomes from an exponential distribution, the applet will draw many samples of size 10 and then make a histogram of the sample means, . Click the “Generate samples” button. Do these data appear to be from a normal distribution? Check the box for “Show normal curve.”
  5. Repeat part (d), but this time change the sample size to 30.
  6. Summarize the patterns you have observed in parts (a) through (e). How do these patterns relate to the central limit theorem?