5.84 The cost of Internet access. In Canada, households spent an average of $80.63 CDN monthly for high-speed broadband access.²⁴ Assume that the standard deviation is $27.32. If you ask an SRS of 500 Canadian households with high-speed broadband access how much they pay, what is the probability that the average amount will exceed $85?

5.85 Dust in coal mines. A laboratory weighs filters from a coal mine to measure the amount of dust in the mine atmosphere. Repeated measurements of the weight of dust on the same filter vary Normally with standard deviation σ = 0.09 milligram (mg) because the weighing is not perfectly precise. The dust on a particular filter actually weighs 137 mg.

5.86 The effect of sample size on the standard deviation. Assume that the standard deviation in a very large population is 100.

5.87 Marks per round in cricket. Cricket is a dart game that uses the numbers 15 to 20 and the bull’s-eye. Each time you hit one of these regions, you score either 0, 1, 2 or 3 marks. Thus, in a round of three throws, a person can score 0 to 9 marks. Lex plans to play 20 games. Her distribution of marks per round is discrete and strongly skewed. A majority of her rounds result in 0, 1, or 2 marks and only a few are more than 4 marks. Assume that her mean is 2.07 marks per round with a standard deviation of 2.11.

5.88 Common last names. The U.S. Census Bureau says that the 10 most common names in the United States are (in order) Smith, Johnson, Williams, Brown, Jones, Miller, Davis, Garcia, Rodriguez, and Wilson.²⁵ These names account for 4.9% of all U.S. residents. Out of curiosity, you look at the authors of the textbooks for your current courses. There are 12 authors in all. Would you be surprised if none of the names of these authors were among the 10 most common? Give a probability to support your answer and explain the reasoning behind your calculation.

5.89 Benford’s law. It is a striking fact that the first digits of numbers in legitimate records often follow a distribution known as Benford’s law (see Example 4.12, page 226). Here it is:

Fake records usually have fewer first digits 1, 2, and 3. What is the approximate probability, if Benford’s law holds, that among 1000 randomly chosen invoices there are 575 or fewer in amounts with first digit 1, 2, or 3?

5.90 Genetics of peas. According to genetic theory, the blossom color in the second generation of a certain cross of sweet peas should be red or white in a 3:1 ratio. That is, each plant has probability 3/4 of having red blossoms, and the blossom colors of separate plants are independent.

5.91 Leaking gas tanks. Leakage from underground gasoline tanks at service stations can damage the environment. It is estimated that 25% of these tanks leak. You examine 15 tanks chosen at random, independently of each other.

5.92 A roulette payoff. A $1 bet on a single number on a casino’s roulette wheel pays $35 if the ball ends up in the number slot you choose. Here is the distribution of the payoff X:

339

Each spin of the roulette wheel is independent of other spins.

5.93 A roulette payoff revisited. Refer to the previous exercise. In part (d), the central limit theorem was used to approximate the probability that Sam ends the year ahead. The estimate was about 0.10 too large. Let’s see if we can get closer using the Normal approximation to the binomial with the continuity correction.

5.94 Learning a foreign language. Does delaying oral practice hinder learning a foreign language? Researchers randomly assigned 25 beginning students of Russian to begin speaking practice immediately and another 25 to delay speaking for four weeks. At the end of the semester both groups took a standard test of comprehension of spoken Russian. Suppose that in the population of all beginning students, the test scores for early speaking vary according to the N(32, 6) distribution and scores for delayed speaking have the N(29, 5) distribution.

5.95 Summer employment of college students. Suppose (as is roughly true) that 88% of college men and 82% of college women were employed last summer. A sample survey interviews SRSs of 400 college men and 400 college women. The two samples are of course independent.

5.96 Income of working couples. A study of working couples measures the income X of the husband and the income Y of the wife in a large number of couples in which both partners are employed. Suppose that you knew the means and and the variances and of both variables in the population.

5.97 A random walk. A particle moves along the line in a random walk. That is, the particle starts at the origin (position 0) and moves either right or left in independent steps of length 1. If the particle moves to the right with probability 0.6, its movement at the ith step is a random variable X_i with distribution

P(X_i = 1) = 0.6

P(X_i = −1) = 0.4

The position of the particle after k steps is the sum of these random movements,

Y = X₁ + X₂ + · · · + X_k

Use the central limit theorem to find the approximate probability that the position of the particle after 500 steps is at least 200 to the right.

5.98 A lottery payoff. A $1 bet in a state lottery’s Pick 3 game pays $500 if the three-digit number you choose exactly matches the winning number, which is drawn at random. Here is the distribution of the payoff X:

340

Each day’s drawing is independent of other drawings.

5.99 Poisson distribution? Suppose you find in your spam folder an average of two spam emails every 10 minutes. Furthermore, you find that the rate of spam mail from midnight to 6 A.M. is twice the rate during other parts of the day. Explain whether or not the Poisson distribution is an appropriate model for the spam process.

5.100 Tossing a die. You are tossing a balanced die that has probability 1/6 of coming up 1 on each toss. Tosses are independent. We are interested in how long we must wait to get the first 1.

5.101 The geometric distribution. Generalize your work in Exercise 5.100. You have independent trials, each resulting in a success or a failure. The probability of a success is p on each trial. The binomial distribution describes the count of successes in a fixed number of trials. Now the number of trials is not fixed; instead, continue until you get a success. The random variable Y is the number of the trial on which the first success occurs. What are the possible values of Y? What is the probability P(Y = k) for any of these values? (Comment: The distribution of the number of trials to the first success is called a geometric distribution.)

5.102 Wi-fi interruptions. Suppose that the number of wi-fi interruptions on your home network follows the Poisson distribution with an average of 0.9 wi-fi interruptions per day.