177

*For Exercise 4.1, see page 176.*

4.2 Are these phenomena random?

Identify each of the following phenomena as random or not. Give reasons for your answers.

- The outside temperature in Chicago at noon on New Year's Day.
- The first character to the right of the “@” symbol in an employee's company email address.
- You draw an ace from a well-shuffled deck of 52 cards.

4.3 Interpret the probabilities.

Refer to the previous exercise. In each case, interpret the term “probability” for the phenomena that are random. For those that are not random, explain why the term “probability” does not apply.

4.4 Are the trials independent?

For each of the following situations, identify the trials as independent or not. Explain your answers.

- The outside temperature in Chicago at noon on New Year's Day, each year for the next five years.
- The number of tweets that you receive on the next 10 Mondays.
- Your grades in the five courses that you are taking this semester.

4.5 Financial fraud.

It has been estimated that around one in six fraud victims knew the perpetrator as a friend or acquaintance. Financial fraud includes crimes such as unauthorized credit card charges, withdrawal of money from a savings or checking account, and opening an account in someone else's name. Suppose you want to use a physical device to simulate the outcome that a fraud victim knew the perpetrator versus the outcome that the fraud victim does not know the perpetrator. What device would you use to conduct a simulation experiment? Explain how you would match the outcomes of the device with the fraud scenario.

4.6 Credit monitoring.

In a recent study of consumers, 25% reported purchasing a credit-monitoring product that alerts them to any activity on their credit report. Suppose you want to use a physical device to simulate the outcome of a consumer purchasing the credit-monitoring product versus the outcome of the consumer not purchasing the product. Describe how you could use two fair coins to conduct a simulation experiment to mimic consumer behavior. In particular, what outcomes of the two flipped coins would you associate with purchasing the product versus what outcomes would you associate with not purchasing the product?

4.7 Random digits.

As discussed in Chapter 3, generation of random numbers is one approach for obtaining a simple random sample (SRS). If we were to look at the random generation of digits, the mechanism should give each digit probability 0.1. Consider the digit “0” in particular.

- The table of random digits (Table B) was produced by a random mechanism that gives each digit probability 0.1 of being a 0. What proportion of the first 200 digits in the table are 0s? This proportion is an estimate, based on 200 repetitions, of the true probability, which in this case is known to be 0.1.
- Now use software assigned by your instructor:
*Excel users:*Enter the formula**RANDBETWEEN(0,9)**in cell A1. Now, drag and copy the contents of cell A1 into cells A2:A1000. You will find 1000 random digits appear. Any attempt to copy these digits for sorting purposes will result in the digits changing. You will need to “freeze” the generated values. To do so, highlight column 1 and copy the contents and then**Paste Special as Values**the contents into the same or any other column. The values will now not change. Finally, use Excel to sort the values in ascending order.*JMP users:*With a new data table, right-click on the header of Column 1 and choose**Column Info**. In the drag-down dialog box named**Initialize Data**, pick**Random**option. Choose the bullet option of**Random Integer**, and set**Minimum/Maximum**to 0 and 9. Input the value of 1000 into the**Number of rows**box, and then click**OK**. The values can then be sorted in ascending order using the**Sort**option found under**Tables**.*Minitab users:*Do the following pull-down sequence:**Calc→ Random Data→ Integer**. Enter “1000” in the**Number of rows of data to generate**box, type “c1” in the**Store in column(s)**box, enter “0” in the**Minimum value**box, and enter “9” in the**Maximum**box. Click**OK**to find 1000 realizations of outputted in the worksheet. The values can then be sorted in ascending order using the**Sort**option found under**Data**.

Based on the software you used, what proportion of the 1000 randomly generated digits are 0s? Is this proportion close to 0.1?

4.8 Are McDonald's prices independent?

Over time, stock prices are always on the move. Consider a time series of 1126 consecutive daily prices of McDonald's stock from the beginning of January 2010 to the near the end of June 2014.^{1}

178

- Using software, plot the prices over time. Are the prices constant over time? Describe the nature of the price movement over time.
- Now consider the relationship between price on any given day with the price on the prior day. The previous day's price is sometimes referred to as the
*lag*price. You will want to get the lagged prices in another column of your software:*Excel users:*Highlight and copy the price values, and paste them in a new column shifted down by one row.*JMP users:*Click on the price column header name to highlight the column of price values. Copy the highlighted values. Now click anywhere on the nearest empty column, resulting in the column being filled with missing values. Double-click on the cell in row 2 of the newly formed column. With row 2 cell open, paste the price values to create a column of lagged prices. (Note: A column of lagged values can also be created with JMP's**Lag**function found in the**Formula**option of the column.)*Minitab users:***Stat→ Time Series→ Lag**.

Refering back to Chapter 2 and scatterplots, create a scatterplot of McDonald's price on a given day versus the price on the previous day. Does the scatterplot suggest that the price series behaves as a series of independent trials? Explain why or why not.

4.9 Are McDonald's price changes independent?

Refer to the daily price series of McDonald's stock in Exercise 4.8. Instead of looking at the prices themselves, consider now the daily *changes* in prices found in the provided data file.

- Using software, plot the price changes over time. Describe the nature of the price changes over time.
- Now consider the relationship between a given price change and the previous price change. Create a lag of price changes by following the steps of Exercise 4.8(b). Create a scatterplot of price change versus the previous price change. Does the scatterplot seem to suggest that the price-change series behaves essentially as a series of independent trials? Explain why or why not.
- This exercise only explored the relationship or lack of it between price changes of successive days. If you want to feel more confident about a conclusion of independence of price changes over time, what additional scatterplots might you consider creating?

4.10 Use the Probability applet.

The idea of probability is that the *proportion* of heads in many tosses of a balanced coin eventually gets close to 0.5. But does the actual *count* of heads get close to one-half the number of tosses? Let's find out. Set the “Probability of Heads” in the *Probability* applet to 0.5 and the number of tosses to 50. You can extend the number of tosses by clicking “Toss” again to get 50 more. Don't click “Reset” during this exercise.

- After 50 tosses, what is the proportion of heads? What is the count of heads? What is the difference between the count of heads and 25 (one-half the number of tosses)?
- Keep going to 150 tosses. Again record the proportion and count of heads and the difference between the count and 75 (half the number of tosses).
- Keep going. Stop at 300 tosses and again at 600 tosses to record the same facts. Although it may take a long time, the laws of probability say that the proportion of heads will always get close to 0.5 and also that the difference between the count of heads and half the number of tosses will always grow without limit.

4.11 A question about dice.

Here is a question that a French gambler asked the mathematicians Fermat and Pascal at the very beginning of probability theory: what is the probability of getting at least one 6 in rolling four dice? The *Law of Large Numbers* applet allows you to roll several dice and watch the outcomes. (Ignore the title of the applet for now.) Because simulation—just like real random phenomena—often takes very many trials to estimate a probability accurately, let's simplify the question: is this probability clearly greater than 0.5, clearly less than 0.5, or quite close to 0.5? Use the applet to roll four dice until you can confidently answer this question. You will have to set “Rolls” to 1 so that you have time to look at the four up-faces. Keep clicking “Roll dice” to roll again and again. How many times did you roll four dice? What percent of your rolls produced at least one 6?

4.12 Proportions of McDonald's price changes.

Continue the study of daily price changes of McDonald's stock from the Exercise 4.9. Consider three possible outcomes: (1) positive price change, (2) no price change, and (3) negative price change.

- Find the proportions of each of these outcomes. This is most easily done by sorting the price change data into another column of the software and then counting the number of negative, zero, and positive values.
- Explain why the proportions found in part (a) are reasonable estimates for the true probabilities.

179

4.13 Thinking about probability statements.

Probability is a measure of how likely an event is to occur. Match one of the probabilities that follow with each statement of likelihood given. (The probability is usually a more exact measure of likelihood than is the verbal statement.)

0 | 0.01 | 0.3 | 0.6 | 0.99 | 1 |

- This event is impossible. It can never occur.
- This event is certain. It will occur on every trial.
- This event is very unlikely, but it will occur once in a while in a long sequence of trials.
- This event will occur more often than not.