Normal Distribution
1. Suppose that the speed at which cars go on the freeway is normally distributed with mean 69 mph and standard deviation 8 miles per hour. Let X be the speed for a randomly selected car. Round all answers to 4 decimal places where possible.
a. What is the distribution of X? X ~ N(,)
b. If one car is randomly chosen, find the probability that it is traveling more than 68 mph.
c. If one of the cars is randomly chosen, find the probability that it is traveling between 72 and 76 mph.
2. In the 1992 presidential election, Alaska’s 40 election districts averaged 1803 votes per district for President Clinton. The standard deviation was 552. (There are only 40 election districts in Alaska.) The distribution of the votes per district for President Clinton was bell-shaped. Let X = number of votes for President Clinton for an election district. (Source: The World Almanac and Book of Facts) Round all answers except part e. to 4 decimal places.
a. What is the distribution of X? X ~ N(,)
b. Is 1803 a population mean or a sample mean?
c. Find the probability that a randomly selected district had fewer than 1657 votes for President Clinton.
d. Find the probability that a randomly selected district had between 1983 and 2184 votes for President Clinton.
e. Find the third quartile for votes for President Clinton. Round your answer to the nearest whole number.
3. Suppose that the weight of seedless watermelons is normally distributed with mean 6.9 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 8.3 kg?
d. What is the probability that a randomly selected watermelon will weigh more than 7.5 kg?
e. What is the probability that a randomly selected seedless watermelon will weigh between 7.2 and 7.8 kg?
f. The 90th percentile for the weight of seedless watermelons is kg.
4. On a planet far far away from Earth, IQ of the ruling species is normally distributed with a mean of 102 and a standard deviation of 18. 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 112. Round your answer to 4 decimal places.
c. A school offers special services for all children in the bottom 4% 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 places.
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Q3:
IQR:
5. Suppose that the distance of fly balls hit to the outfield (in baseball) is normally distributed with a mean of 264 feet and a standard deviation of 44 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 226 feet. Round to 4 decimal places.
c. Find the 80th percentile for the distribution of distance of fly balls. Round to 2 decimal places. feet
6. The average THC content of marijuana sold on the street is 10.2%. Suppose the THC content is normally distributed with standard deviation of 1%. Let X be the THC content for a randomly selected bag of marijuana that is sold on the street. 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 bag of marijuana sold on the street will have a THC content greater than 9.9.
c. Find the 77th percentile for this distribution. %
7. Los Angeles workers have an average commute of 32 minutes. Suppose the LA commute time is normally distributed with a standard deviation of 12 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 30 minutes.
c. Find the 75th percentile for the commute time of LA workers. minutes
8. 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 5.5 days to recover?
d. What is the probability of spending more than 4.5 days in recovery?
e. What is the probability of spending between 4.1 and 4.6 days in recovery?
f. The 90th percentile for recovery times is days.
9. Terri Vogel, an amateur motorcycle racer, averages 130 seconds per 2.5 mile lap (in a 7 lap race) with a standard deviation of 2.3 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 131.82 and 133.19 seconds.
c. The fastest 2% of laps are under seconds.
d. The middle 70% of her laps are from seconds to seconds.
10. The average American man consumes 9.5 grams of sodium each day. Suppose that the sodium consumption of American men is normally distributed with a standard deviation of 1 grams. Suppose an American man is randomly chosen. Let X = the amount of sodium consumed. Round all numeric answers to 4 decimal places where possible.
a. What is the distribution of X? X ~ N(,)
b. Find the probability that this American man consumes between 10.5 and 12.4 grams of sodium per day.
c. The middle 30% of American men consume between what two weights of sodium?
Low:
High: