Statistics Practice Problems

Problem 1

1. Diego and Gregory began arguing about who did better on their tests, but they couldn't decide who did better given that they took different tests. Diego took a test in Math and earned a 77, and Gregory took a test in Science and earned a 69.6. Use the fact that all the students' test grades in the Math class had a mean of 72.4 and a standard deviation of 9.3, and all the students' test grades in Science had a mean of 62.9 and a standard deviation of 10.8 to answer the following questions.

a) Calculate the z-score for Diego's test grade.
z=z=

b) Calculate the z-score for Gregory's test grade.
z=z=

c) Which person did relatively better?

• Diego
• Gregory
• They did equally well.

2. Suppose that the distance of fly balls hit to the outfield (in baseball) is normally distributed with a mean of 257 feet and a standard deviation of 45 feet. Let X be the distance in feet for a fly ball.

a. What is the distribution of X? X ~ N(,)

b. Find the probability that a randomly hit fly ball travels less than 279 feet. 

c. Find the 90th percentile for the distribution of distance of fly balls. Round to 2 decimal places. 

3. The patient recovery time from a particular surgical procedure is normally distributed with a mean of 4 days and a standard deviation of 1.6 days. Let X be the recovery time for a randomly selected patient. Round all answers to 4 decimal places where possible.

a. What is the distribution of X? X ~ N(,)

b. What is the median recovery time? days

c. What is the Z-score for a patient that took 5.6 days to recover? 

d. What is the probability of spending more than 3.4 days in recovery? 

e. What is the probability of spending between 4.7 and 5.2 days in recovery? 

f. The 70th percentile for recovery times is ----- days.

4. Los Angeles workers have an average commute of 31 minutes. Suppose the LA commute time is normally distributed with a standard deviation of 14 minutes. Let X represent the commute time for a randomly selected LA worker. Round all answers to 4 decimal places where possible.

a. What is the distribution of X? X ~ N(,)

b. Find the probability that a randomly selected LA worker has a commute that is longer than 32 minutes. 

c. Find the 70th percentile for the commute time of LA workers

Problem 2

5. Private nonprofit four-year colleges charge, on average, $26,424 per year in tuition and fees. The standard deviation is $7,494. Assume the distribution is normal. Let X be the cost for a randomly selected college. Round all answers to 4 decimal places where possible.

a. What is the distribution of X? X ~ N(,)

b. Find the probability that a randomly selected Private nonprofit four-year college will cost less than 29,762 per year. 

c. Find the 71th percentile for this distribution. $ (Round to the nearest dollar.)

6. On a planet far far away from Earth, IQ of the ruling species is normally distributed with a mean of 113 and a standard deviation of 14. Suppose one individual is randomly chosen. Let X = IQ of an individual.

a. What is the distribution of X? X ~ N(,)

b. Find the probability that a randomly selected person's IQ is over 82.  c. A school offers special services for all children in the bottom 3% for IQ scores. What is the highest IQ score a child can have and still receive special services? Round your answer to 2 decimal places.

d. Find the Inter Quartile Range (IQR) for IQ scores. Round your answers to 2 decimal placesRound your answer to 4 decimal places.

6. Suppose that the weight of seedless watermelons is normally distributed with mean 6.7 kg. and standard deviation 1.5 kg. Let X be the weight of a randomly selected seedless watermelon. Round all answers to 4 decimal places where possible.

a.  What is the distribution of X? X ~ N(,)

b.  What is the median seedless watermelon weight?  kg.
 
c.  What is the Z-score for a seedless watermelon weighing 7.2 kg? 

d.  What is the probability that a randomly selected watermelon will weigh more than 6 kg? 

e.  What is the probability that a randomly selected seedless watermelon will weigh between 6.5 and 7.3 kg? 

f.  The 80th percentile for the weight of seedless watermelons is ----  kg.

7. The mean height of an adult giraffe is 18 feet. Suppose that the distribution is normally distributed with standard deviation 1 feet. Let X be the height of a randomly selected adult giraffe. Round all answers to 4 decimal places where possible.

a. What is the distribution of X? X ~ N(,)

b. What is the median giraffe height?

c. What is the Z-score for a giraffe that is 19 foot tall? 

d. What is the probability that a randomly selected giraffe will be shorter than 17.5 feet tall? 

e. What is the probability that a randomly selected giraffe will be between 18.3 and 19.1 feet tall? 

f. The 85th percentile for the height of giraffes is ----- ft.

8. Terri Vogel, an amateur motorcycle racer, averages 129.96 seconds per 2.5 mile lap (in a 7 lap race) with a standard deviation of 2.28 seconds . The distribution of her race times is normally distributed. We are interested in one of her randomly selected laps. (Source: log book of Terri Vogel) Let X be the number of seconds for a randomly selected lap. Round all answers to 4 decimal places where possible.

a. What is the distribution of X? X ~ N(,)

b. Find the proportion of her laps that are completed between 126.42 and 128.85 seconds. 

c. The fastest 2% of laps are under ----- seconds.

d. The middle 60% of her laps are from ---- seconds to -----

9. According to a study done by UCB students, the height for Martian adult males is normally distributed with an average of 68 inches and a standard deviation of 2.3 inches. Suppose one Martian adult male is randomly chosen. Let X = height of the individual. Round all answers to 4 decimal places where possible.

a. What is the distribution of X? X ~ N(,)

b. Find the probability that the person is between 65.9 and 67.9 inches. 

c. The middle 40% of Martian heights lie between what two numbers?
Low: ---- inches
High: ---- inches

10. The average student loan debt for college graduates is $25,600. Suppose that that distribution is normal and that the standard deviation is $13,850. Let X = the student loan debt of a randomly selected college graduate. Round all probabilities to 4 decimal places and all dollar answers to the nearest dollar.

a.  What is the distribution of X? X ~ N(,)

b   Find the probability that the college graduate has between $28,450 and $36,350 in student loan debt. 

c.  The middle 20% of college graduates' loan debt lies between what two numbers?
     Low: $
     High: $