1. Normal Distribution Questions
    1. Melatonin and Sleep
    2. Reproductive Habits of Roe Deer
    3. Acorn Length
    4. Plowing Snow
    5. Raccoons Consuming Corn
    6. NFL Running Backs
    7. Stock Returns
    8. HMF in Honey
    9. Size of Farms in England
    10. Painted Turtle Hatchling Size
    11. Moose Hunting
    12. Lengths of Pine Needles
  2. General (Non-Normal Distribution) Questions
    1. Coin Purse
    2. Greenhouse Plants

 

Normal Distribution Questions

In all of the following questions, I urge you to follow these steps when answering probability questions:
  1. Identify the distribution that would be used to answer the question (population or sampling).
  2. Identify characteristics of the required distribution; i.e., whether normal or not and, if normal, the center and dispersion (SD or SE).
  3. Identify whether the question can be answered or not (can be if normal, cannot be if not normal).
  4. Answer to the question; use distrib() if the probability can be computed (i.e., normal distribution) or explain why the probability cannot be computed (note specifically which distribution is not normal).
A reminder to use these steps will NOT be provided on future quizzes, but you should get in the habit of following them.

 

Melatonin and Sleep

MIT researchers examined the effect of melatonin on sleep. Suppose that we know that men given a placebo fell asleep according to a N(15,5) distribution.

  1. What is the probability, with 4 randomly selected men, of observing a mean of more than 12 mins?
  2. What is the probability that a randomly selected man will fall asleep in more than 12 mins?
  3. What is the probability that 25 randomly selected will have a mean time to fall asleep of more than 15.5 minutes?

Reproductive Habits of Roe Deer

Researchers on Storfosna Island, Norway examined the reproductive habits of Roe Deer (Capreolus pygargus) in the northern extremities of the island (Andersen and Linnell 2000). The researchers found that the distribution of number of fawns born to a female between 1991 and 1994 was extremely right-skewed with a mean of 2.2 and a standard deviation of 0.46 fawns. Answer the questions below assuming that these values represent the entire population of Roe Deer. [Note: if you decide that a question cannot be answered, then describe your reasoning very specifically.]

  1. What is the probability that a Roe Deer has more than 2 fawns?
  2. What is the probability that a sample of 10 Roe Deer will have an average of more than 2 fawns?
  3. What is the probability that a sample of 35 Roe Deer will have an average of more than 2 fawns?
  4. What is the probability that a sample of 35 Roe Deer will have a mean between 2.0 and 2.3 fawns?
  5. What is the most common 90% of sample means for n=35 Roe Deer?
  6. What is the mean such that 20% of all samples of n=35 Roe Deer have a smaller mean?

Acorn Length

Suppose that it is known that the distribution of lengths of acorns is slightly right-skewed with a mean of 21 mm and a standard deviation of 6 mm.

  1. What is the probability that an acorn will have a length greater than 25 mm?
  2. What is the probability that the mean length of a sample of 20 acorns will be greater than 25 mm?
  3. What is the probability that the mean length of a sample of 50 acorns will be between 20 and 25 mm?
  4. What is the probability that the mean length of a sample of 6 acorns will be less than 20 mm?
  5. What is the probability that an acorn will have a length less than 20 mm?

Plowing Snow

Suppose that it is known that the distribution of times spent plowing a single city block on snow days is right-skewed with a mean of 45 seconds and a standard deviation of 28 seconds.

  1. What is the probability that in a sample of 5 blocks that the mean is greater than 60 seconds?
  2. What is the probability that it will take longer than 60 seconds for one block?
  3. What is the probability that in a sample of 40 blocks will have a mean between 40 and 50 seconds?
  4. What is the Q1 for the mean plow time in samples of 40 blocks?

Raccoons Consuming Corn

The WI Department of Natural Resources is examining the amount of domestic corn consumed by raccoons per week. Assume that the amount eaten is slightly right-skewed, with a mean of 8 kg, and a standard deviation of 2 kg.

  1. What is the probability that a raccoon consumes more than 13 kg per week?
  2. What is the probability that a sample of 25 raccoons have a mean corn consumption of more than 10 kg per week?
  3. What is the probability that a sample of 60 raccoons have a TOTAL corn consumption of more than 510 kg per week?

NFL Running Backs

Suppose that it is known that the number of yards gained per game for the primary running back on a National Football League team is slightly left-skewed with a mean of 82 yards and a standard deviation of 26 yards.

  1. What is the probability that a running back will gain more than 100 yards in a single game?
  2. What is the probability that a running back will average more than 100 yards per game in a 16-game season?
  3. What is the probability that a running back will average between 70 and 90 yards per game in a 16-game season?
  4. What is the probability that a running back will average more than 70 yards per game over two 16-game seasons?
  5. What is the top 25% of yards gained by a running back in a single game?
  6. What is the top 5% of mean yards gained by a running back in a 16-game season?

Stock Returns

Suppose that the average annual rate of return for a wide array of available stocks is approximately normally distributed with a mean of 4.2 with a standard deviation of 4.9.

  1. What is the probability that five randomly selected stocks produce a positive average rate of return?
  2. What is the probability that a randomly selected stock produces a positive rate of return?
  3. What is the probability that ten randomly selected stocks produce a less than 2% average rate of return?
  4. The top 10% of stocks produce what rate of return?
  5. The top 10% of random samples of 10 stocks produce what average rate of return?

HMF in Honey

Renner (1970) examined the content of hydroxymethylfurfurol (HMF) in honey. HMF is an organic compound derived from cellulose without the use of fermentation and is a potential “carbon-neutral” source for fuels. This study found that the distribution of HMF in honey was extremely strongly right-skewed with a mean of 9.5 g/kg and a standard deviation of 13.5 g/kg.

  1. What is the probability that one kg of honey have more than 20 g of HMF?
  2. What is the probability that 20 samples of one kg of honey have an average of more than 20 g of HMF?
  3. What is the probability that 50 samples of one kg of honey have an average of less than 10 g of HMF?
  4. What are the 20% least common average amounts of HMF in 50 samples of one kg of honey?

Size of Farms in England

Allanson (1992) examined the size of farms in England in 1939 and 1989. He found the distribution of farm sizes in 1989 to be very right-skewed with a mean of 65.13 ha and a standard deviation of 108.71 ha.

  1. What are the 10% most common sizes of farms in England?
  2. What are the 10% most common average sizes in samples of 60 farms from England?
  3. What is the probability that the average size of 60 farms from England is less than 50 ha?
  4. What is the probability that a farm from England is greater than 50 ha?

Painted Turtle Hatchling Size

Janzen and Morjan (2002) examined the size of male and female painted turtles (Chrysemys picta) at hatching. They found in a sample of 77 turtles that size at hatching was very slightly right-skewed with a mean of 4.46 g with a standard deviation of 0.13 g. Assume that the results of this sample extend to the population to answer the questions below.

  1. What is the probability that a turtle will hatch in more than 7 days?
  2. What is the probability that a sample of 20 turtles will have an average number of days until hatching that is greater than 4.5 days?
  3. What is the probability that a sample of 50 turtles will have an average number of days until hatching that is greater than 4.5 days?
  4. What is the mean number of days until hatching such that 20% of samples of 50 turtles have a smaller mean?
  5. What are the most common 80% of times to hatching?

Moose Hunting

Assume that it is known that the distribution of time spent hunting (hours) by an individual Minnesota moose (Alces alces) hunter is approximately symmetric in shape with a mean of 40 hours and a standard deviation of 15 hours. Use this information to answer the questions below.

  1. Describe what an individual is in this problem.
  2. List the variable or variables in this problem and identify the type of variable for each.
  3. What is the probability that a hunter will spend more than 55 hrs hunting moose?
  4. What is the probability that the average hours spent hunting by a sample of 25 hunters is greater than 48 hrs?

Lengths of Pine Needles

Suppose that the length of all needles on a particularly large pine tree is known to be normally distributed with a mean of 75 mm and a standard deviation of 8 mm. Use this to answer the questions below.

  1. What is the probability that a randomly selected needle is between 70 and 80 mm long?
  2. What is the probability that a randomly selected needle is longer than 90 mm?
  3. What is the probability that a randomly selected needle is less than 50 mm long?

 

General (Non-Normal Distribution) Questions

Coin Purse

A coin purse contains 17 nickels and 15 dimes. Use this to answer the questions below.

  1. What is the probability of randomly selecting a nickel from this purse?
  2. What is the probability of randomly selecting a dime from this purse?
  3. What is the probability of randomly selecting a dime from this purse assuming that two nickels and three dimes have already been removed?

Greenhouse Plants

A very small green house contains 10 tomato, 12 pea, and 8 cauliflower plants. Use this to answer the questions below.

  1. What is the probability of randomly selecting a tomato plant from this greenhouse?
  2. What is the probability of randomly selecting a cauliflower plant from this greenhouse?
  3. What is the probability of randomly selecting a pea plant from this greenhouse assuming that all tomato plants had died and were removed from the greenhouse?