What percent of a standard normal distribution N | Complete Solution

1. What percent of a standard normal distribution N(μ = 0, = 1) is found in each region? Be sure to draw a graph.
(a) Z > 1.13 (b) Z <0.18 (c) Z >8 (d) |Z|<0.5

2. In triathlons, it is common for racers to be placed into age and gender groups. Friends Leo and Mary both completed the Hermosa Beach Triathlon, where Leo competed in the Men, Ages 30 - 34 group while Mary competed in the Women, Ages 25 - 29 group. Leo completed the race in 1:22:28 (4948 seconds), while Mary completed the race in 1:31:53 (5513 seconds). Obviously Leo finished faster, but they are curious about how they did within their respective groups. Can you help them? Here is some information on the performance of their groups:

•     The finishing times of the Men, Ages 30 - 34 group has a mean of 4313 seconds with a standard deviation of 583 seconds. 


•     The finishing times of the Women, Ages 25 - 29 group has a mean of 5261 seconds with a standard deviation of 807 seconds. 


•     The distributions of finishing times for both groups are approximately Normal. 


3. In Exercise 3.4 we saw two distributions for triathlon times: N(μ = 4313, = 583) for Men, Ages 30 - 34 and N(μ = 5261, = 807) for the Women, Ages 25 - 29 group. Times are listed in seconds. Use this information to compute each of the following:

a. The cutoff time for the fastest 5% of athletes in the men’s group, i.e. those who took the shortest 5% of time to finish.

b. The cutoff time for the slowest 10% of athletes in the women’s group.

4. Heights of 10 year olds, regardless of gender, closely follow a normal distribution with mean 55 inches and standard deviation 6 inches.

•     (a)  What is the probability that a randomly chosen 10 year old is shorter than 48 inches? 


•     (b)  What is the probability that a randomly chosen 10 year old is between 60 and 65 inches? 


•     (c)  If the tallest 10% of the class is considered “very tall”, what is the height cuto↵ for “very tall”? 


•     (d)  The height requirement for Batman the Ride at Six Flags Magic Mountain is 54 inches. What percent of 10 year olds cannot go on this ride? 


5. Find the standard deviation of the distribution in the following situations.

•     (a)  MENSA is an organization whose members have IQs in the top 2% of the population. IQs are normally distributed with mean 100, and the minimum IQ score required for admission to MENSA is 132. 


•     (b)  Cholesterol levels for women aged 20 to 34 follow an approximately normal distribution with mean 185 milligrams per deciliter (mg/dl). Women with cholesterol levels above 220 mg/dl are considered to have high cholesterol and about 18.5% of women fall into this category. 


6. We learned in Exercise 3.26 that about 90% of American adults had chickenpox before adulthood. We now consider a random sample of 120 American adults.

•     (a)  How many people in this sample would you expect to have had chickenpox in their childhood? And with what standard deviation? 


•     (b)  Would you be surprised if there were 105 people who have had chickenpox in their childhood? 


•     (c)  What is the probability that 105 or fewer people in this sample have had chickenpox in their childhood? How does this probability relate to your answer to part (b)? 


7. Sickle cell anemia is a genetic blood disorder where red blood cells lose their flexibility and assume an abnormal, rigid, “sickle” shape, which results in a risk of various complications. If both parents are carriers of the disease, then a child has a 25% chance of having the disease, 50% chance of being a carrier, and 25% chance of neither having the disease nor being a carrier. If two parents who are carriers of the disease have 3 children, what is the probability that

(a) two will have the disease? (b) none will have the disease?

(c) at least one will neither have the disease nor be a carrier? (d) the first child with the disease will the be 3rd child?

8. Suppose you roll a single six-sided die with numbers 1 - 6 printed on the sides. Assume that each side has an equal probability of being rolled.

•     (a)  Create a probability distribution for the number showing on one roll of a die. 


•     (b)  Calculate the mean of this probability distribution. 


•     (c)  If you were to roll such a die over and over while recording the number on the face each time, what do expect would be the mean from all these numbers? 


9. Suppose you take a 10 question True or False quiz and you guess on every problem. You only get 2 correct. Is this an unusually low number of correct guesses. Use the criteria that a number (x) is unusually low if P (x or less) <=0.05.

10. The number of hours spent online by college students is claimed to be 22.5 hours per week with a standard deviation of 2.1 hours.

•     (a)  Suppose we randomly select 50 college students, what is the probability that the mean number of hours online is greater than 25. 


•     (b)  Suppose I survey the 50 game programming students and the mean number of hours spent online is 25 hours per week. Is this unusual? Why or why not?