Exercises

In your homework script, make sure to include comments using the # for each problem and subproblem. For example:

# Exercise 1
# 1.a
qnorm(0.9, mean = 0, sd = 1)
# 1.b
dnorm(3, mean = 10, sd = 4)

1. Standard Normal Distribution (10 points)

Recall that the standard normal distribution has a mean = 0 and sd = 1. Use these values to answer the following questions.

You do not need to turn in your sketches of the distributions for credit, but you are stongly encouraged to draw these out to help check your answers.

  1. What is the probability of getting an observation of exactly 1.25?
  2. What is the probability of getting an observation of exactly -0.33?
  3. What is the probability of getting an observation of exactly 2.9?
  4. What is the probability of getting an observation of less than 2.5?
  5. What is the probability of getting an observation of less than -0.9?
  6. What is the probability of getting an observation of greater than -1?
  7. What is the probability of getting an observation between 1.4 and 3.5?
  8. What value represents the 10th percentile of this distribution?
  9. What value represents the 95th percentile of this distribution?
  10. What value represents the 5th percentile?

2. Normal Distribution: Age at childbirth (7 points)

In the last decade, the average age of a mother at childbirth is 26.4 years, with standard deviation 5.8 years. The distribution of age at childbirth is approximately normal.

  1. What age at childbirth puts a woman in the upper 2.5% of the age distribution? In other words, what is the 97.5 percentile of this age distribution?

  2. What proportion of women who give birth are 21 years of age or older?

  3. What is the probability that a woman will have a child at exactly 32.8 years old?

  4. What is the probability that a woman will have a child between the ages of 18 and 22?

  5. What is the probability that a woman will give birth at exactly the mean age of this distribution?

  6. What values bound approximately 95% of the data? (hint: \(\mu \pm x * \sigma\))

  7. What values bound exactly 95% of the data? (hint: use qnorm() twice. Think about what values to put in qnorm. We have 5% of the data left over, but where does the 5% need to be split between?)

  8. How many standard deviations (i.e., Z-score) from the mean is a woman who gives birth at the age of 35?

3. Triathlon times. (8 points)

In triathlons, it is common for racers to be placed into age and gender groups. The finishing times of men ages 30-34 has mean of 4,313 seconds with a standard deviation of 583 seconds. The finishing times of the women ages 25-29 has a mean of 5,261 seconds with a standard deviation of 807 seconds. The distribution of finishing times for both groups is approximately normal. Note that a better performance corresponds to a faster finish.

  1. If a man of the 30-34 age group finishes the race in 4,948 seconds, what percent of the triathletes in the group did he finish faster than?
  2. If a woman of the 25-29 age group finishes the race in 5,513 seconds, what percent of the triathletes in the group did she finish faster than?
  3. Calculate the cutoff time for the fastest 5% of athletes in the men’s group.
  4. Calculate the cutoff time for the slowest 10% of athletes in the women’s group.
  5. What is the standardized Z-score for a man that finishes in 4,460 seconds?
  6. What is the standardized Z-score for a woman that finishes in 5,105 seconds?
  7. Compare the Z-scores calculated in part e and f. Which person performed better based on their age and gender group? note be sure to put a comment in front of the text for your explanation to part g.
  8. What is the probability that a man will finish between 4,500 and 5,000 seconds?