Download Problem of the Day The heights of adult American males are

Problem of the Day
The heights of adult American males are approximately normally distributed with mean 69.5 inches and standard
deviation 2.5 inches.
a) What percent of adult American males are between 67 in. and 74.5 in. tall? Hint: Draw a normal curve and use
the Empirical Rule from last class.
b) In a group of 2000 adult American males, about how many would you expect to be taller than 6 ft (or 72 in.)?
Ex: If the mean of a data set is 20, the standard deviation is 1.5, and the distribution of the data values is approximately
normal, about 95% of the data values fall in what interval centered on the mean?
a) 18.5 to 21.5
b) 17 to 23
c) 15.5 to 24.5
d) 14 to 26
Ex: An hourly wage is normally distributed with a mean of $6.75 and a standard deviation of $0.55. What is the
probability that an employee’s hourly wage is not between $5.65 and $7.85?
a) 0.025
b) 0.05
c) 0.34
d) 0.68
Ex: A town has 685 households. The number of people per household is normally distributed with a mean, 𝜇, of 3.67 and
a standard deviation, 𝜎, of 0.34. Approximately how many households have between 2.99 and 4.01 people?
a) 493 households
b) 520 households
c) 558 households
d) 575 households
Ex: The distribution of heights of adult American men is approximately normal with mean 69 inches and standard
deviation 2.5 inches.
a. What percent of men are taller than 74 inches?
b. Between what heights do the middle 95% of men fall?
c. What percent of men are shorter than 66.5 inches?
Z-Scores
The z-score indicates the number of standard deviations a value lies above/below the mean of a population. When
finding the z-score, we use the formula 𝑧 =
standard deviation of the population.
𝑥−𝜇
𝜎
where x is a data point, μ is the mean of the population, and σ is the
Ex: In a given population, the weights of newborns are normally distributed about the mean 3250 g. The standard
deviation of the population is 500 g.
1. What is the z-score of a newborn weighing 2500 g?
2. What is the z-score of a newborn weighing 4500 g?
3. What is the percentage of newborns weighing below 2500 g?
4. What is the percentage of newborns weighing below 4500 g?
5. What is the percentage of newborns weighing above 4500 g?
6. What is the probability that a newborn weighs between 2270 g and 4230 g?
Ex: Suppose the heights (in inches) of women (ages 20-29) in the United States are normally distributed with a mean of
64.1 inches and a standard deviation of 2.75 inches.
a. Find the percent of women who are no more than 65 inches tall.
b. Find the probability that a randomly chosen woman is between 60 inches and 63 inches tall.
On Your Own: Suppose the heights (in inches) of adult females (ages 20-29) in the United States are normally distributed
with a mean of 64.1 inches and a standard deviation of 2.75 inches.
a. Find the percent of women who are at least 66 inches tall.
b. Find the percent of women who are less than or equal to 61.6 inches tall.
Ex: The scores of a reference population on the Wechsler Intelligence Scale for Children (WISC) are normally distributed
with 𝜇 = 100 and 𝜎 = 15. A school district classifies children as “gifted” if their WISC score exceeds 135. There are 1300
sixth-graders in the school district. About how many of them are gifted?
Ex: The scores on a recent test are normally distributed. John’s test score of 69 was 1 standard deviation below the
mean. Betty’s test score of 99 was 3 standard deviations above the mean. What are the mean and standard deviation for
the test score distribution?
a) The mean is 76.5, and the standard deviation is 7.5
b) The mean is 79, and the standard deviation is 10
c) The mean is 84, and the standard deviation is 15
d) The mean is 91, and the standard deviation is 2.5