6.4 Exercises

1. Yoonie is a personnel manager in a large corporation. Each month she must review 16 of the employees. From past experience, she has found that the reviews take her approximately four hours each to do with a population standard deviation of 1.2 hours. Let [latex]Χ[/latex] be the random variable representing the time it takes her to complete one review. Assume [latex]Χ[/latex] is normally distributed.  Suppose 16 review are selected at random.
  1. What is the mean, standard deviation, and sample size?
  2. What is the distribution of the sample means?  Explain.
  3. What is the mean and standard deviation of the sample means?
  4. Find the probability that one review will take Yoonie from 3.5 to 4.25 hours.
  5. Find the probability that the mean of a month’s reviews will take Yoonie from 3.5 to 4.25 hrs.
  6. Why are the probabilities in (d) and (e) different?

2. Suppose that the distance of fly balls hit to the outfield (in baseball) is normally distributed with a mean of 250 feet and a standard deviation of 50 feet. We randomly sample 49 fly balls.

  1. What is the probability that the 49 balls traveled an average of less than 240 feet?
  2. What is the probability that the 49 balls traveled an average of 245 feet to 255 feet?
  3. What is the probability that the 49 balls traveled an average of more than 260 feet?

3. According to the Internal Revenue Service, the average length of time for an individual to complete (keep records for, learn, prepare, copy, assemble, and send) IRS Form 1040 is 10.53 hours (without any attached schedules) with a standard deviation of two hours. Suppose we randomly sample 36 taxpayers.

  1. What is the distribution of the sample means?  Explain.
  2. Would you be surprised if the 36 taxpayers finished their Form 1040s in an average of more than 12 hours? Explain why or why not in complete sentences.
  3. Would you be surprised if one taxpayer finished his or her Form 1040 in more than 12 hours? In a complete sentence, explain why.

4. Suppose that a category of world-class runners are known to run a marathon (26 miles) in an average of 145 minutes with a standard deviation of 14 minutes. Consider 49 of the races. Find the probability that the runner will average between 142 and 146 minutes in these 49 marathons.

 

5. In 1940 the average size of a U.S. farm was 174 acres. Let’s say that the standard deviation was 55 acres. Suppose we randomly survey 38 farmers from 1940.

  1. What is the distribution of the sample means?  Explain.
  2. What is the mean and standard deviation of the sample means?
  3. What is the probability that the sample mean is less than 170 acres?

6. The percent of fat calories that a person in America consumes each day is normally distributed with a mean of about 36 and a standard deviation of about ten. Suppose that 16 individuals are randomly chosen.

  1. What is the distribution of the sample means?
  2. What is the mean and standard deviation of the sample means?
  3. For the group of 16, find the probability that the average percent of fat calories consumed is more than five.

7. The distribution of income in some Third World countries is considered wedge shaped (many very poor people, very few middle income people, and even fewer wealthy people). Suppose we pick a country with a wedge shaped distribution. Let the average salary be $2,000 per year with a standard deviation of $8,000. We randomly survey 1,000 residents of that country.

  1. How is it possible for the standard deviation to be greater than the average?
  2. Why is it more likely that the average of the 1,000 residents will be from $2,000 to $2,100 than from $2,100 to $2,200?
    8. NeverReady batteries has engineered a newer, longer lasting AAA battery. The company claims this battery has an average life span of 17 hours with a standard deviation of 0.8 hours. Your statistics class questions this claim. As a class, you randomly select 30 batteries and find that the sample mean life span is 16.7 hours. If the process is working properly, what is the probability of getting a random sample of 30 batteries in which the sample mean lifetime is 16.7 hours or less? Is the company’s claim reasonable?

     

    9. Your company has a contract to perform preventive maintenance on thousands of air-conditioners in a large city. Based on service records from previous years, the time that a technician spends servicing a unit averages one hour with a standard deviation of one hour. In the coming week, your company will service a simple random sample of 70 units in the city. You plan to budget an average of 1.1 hours per technician to complete the work. Will this be enough time?
    10. Suppose in a local Kindergarten through 12th grade (K – 12) school district, 53% of the population favor a charter school for grades K through five. A simple random sample of 300 is surveyed.
    1. Find the probability that less than 100 favor a charter school for grades K through 5.
    2. Find the probability that 170 or more favor a charter school for grades K through 5.
    3. Find the probability that no more than 140 favor a charter school for grades K through 5.
    4. Find the probability that there are fewer than 130 that favor a charter school for grades K through 5.
    5. Find the probability that exactly 150 favor a charter school for grades K through 5.

    11. Four friends, Janice, Barbara, Kathy and Roberta, decided to carpool together to get to school. Each day the driver would be chosen by randomly selecting one of the four names. They carpool to school for 96 days.

    1. Find the probability that Janice is the driver at most 20 days.
    2. Find the probability that Roberta is the driver more than 16 days.
    3. Find the probability that Barbara drives between  24 and 30 of those 96 days.

    12. A question is asked of a class of 200 freshmen, and 23% of the students know the correct answer. Suppose a sample of 50 students is taken.

    1. What is the mean and standard deviation of the distribution of the sample proportions?
    2. What is the distribution of the sample proportions?  Explain.
    3. What is the probability that more than 30% of the students answered correctly?
    4. What is the probability that less than 20% of the students answered correctly?
    5. What is the probability that between 21% and 25% of the students answered correctly?

    13. A virus attacks one in three of the people exposed to it. An entire large city is exposed.  Suppose a sample of 70 people in the city is taken.

    1. What is the mean and standard deviation of the distribution of the sample proportions?
    2. What is the distribution of the sample proportions?  Explain.
    3. What is the probability that between 21 and 40 of the people in the sample were exposed to the virus?
    4. What is the probability that more than 35% of the people in the sample were exposed to the virus?
    5. What is the probability that less than 25% of the people in the same were exposed to the virus?

    14. A game is played repeatedly. A player wins one-fifth of the time. Suppose a player plays the game 20 times.

    1. What is the mean and standard deviation of the distribution of the sample proportions?
    2. What is the distribution of the sample proportions?  Explain.
    3. What is the probability that the player wins at most 7 times?
    4. What is the probability that the player wins at least 30% of the time?
    5. What is the probability that the player wins less than 15% of the time?
    6. What is the probability that the player wins more than 10 times?

    15. A company inspects products coming through its production process, and rejects defective products. One-tenth of the items are defective. Suppose a sample of 40 items is taken.

    1. What is the mean and standard deviation of the distribution of the sample proportions?
    2. What is the distribution of the sample proportions?  Explain.
    3. What is the probability that fewer than 7 of the items in the sample are defective?
    4. What is the probability that more than 15% of the items in the sample are defective?
    5. What is the probability that at least 3 of the items in the sample are defective?
    6. What is the probability that at most 20% of the items in the sample are defective?

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