Download Density Curve and Empirical Rule Homework

CP StatisticsX
Density Curves and the Normal Distribution
1. FINDING MEANS AND MEDIANS The figure below displays three density
curves, each with three points indicated. At which of these points on each curve the
mean and median fall?
(a) mean ______ median ______
(b) mean ______ median ______
(c) mean ______ median ______
For #2-10 Label the normal curve on which the mean and standard deviation are
correctly located and then use the Empirical Rule (or 68–95–99.7 rule) to answer
the questions.
2. MEN’S HEIGHTS 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?
(d) A height of 71.5 inches corresponds to what percentile of adult male American
heights?
3. IQ SCORES Scores on the Wechsler Adult Intelligence Scale (WAIS, a standard
“IQ test”) for the 20 to 34 age groups are approximately normally distributed with
μ = 110 and σ = 25.
(a) About what percent of people in this age group have scores above 110?
(b) About what percent have scores above 160?
(c) In what range do the middle 95% of all IQ scores lie?
4. HELMET SIZES The army reports that the distribution of head circumferences
among male soldiers is approximately normal with mean 22.8 inches and standard
deviation 1.1 inches.
(a) What percent of soldiers have head circumference greater than 23.9 inches?
(b) A head circumference of 23.9 inches would be what percentile?
(c) What percent of soldiers have head circumference between 21.7 inches and
23.9 inches?
5. WEIGHTS OF DISTANCE RUNNERS A study of elite distance runners found a
mean body weight of 63.1 kilograms, with a standard deviation of 4.8 kg.
Use the 68-95-99.7 rule to find intervals centered at the mean that will include
68%, 95%, and 99.7% of the weights of the runners.
6. SAT SAT (combined) scores of college-bound seniors in high school has the
normal distribution with mean 1050 and standard deviation 150.
(a) What is the middle 68% of the distribution of SAT score of a college-bound
senior?
(b) Find the percent of students that score between 900 and 1500.
(c) Find the value x such that the 16% of all seniors have SAT score below x.
(d) What is the 97.5th percentile of this distribution?
7. IQ Scores One measure of intelligence is the Stanford–Binet Intelligence
Quotient (IQ). IQ scores have a bell-shaped distribution with a mean of 100 and a
standard deviation of 15.
(a) What percentage of people has an IQ score between 70 and 130?
(b) What percentage of people has an IQ score less than 70 or greater than 130?
(c) What percentage of people has an IQ score greater than 130?
8. MATH SAT SAT Math scores have a bell-shaped distribution with a mean of 515
and a standard deviation of 114.
Source: College Board, 2007
(a) What percentage of SAT scores is between 401 and 629?
(b) What percentage of SAT scores is less than 401 or greater than 629?
(c) What percentage of SAT scores is greater than 743?
9. KIDNEYS The weight, in grams, of the pair of kidneys in adult males between
the ages of 40 and 49 have a bell-shaped distribution with a mean of 325 grams
and a standard deviation of 30 grams.
(a) About 95% of kidney pairs will be between what weights?
(b) What percentage of kidney pairs weighs between 235 grams and 415 grams?
(c) What percentage of kidney pairs weighs less than 235 grams or more than 415
grams?
(d) What percentage of kidney pairs weighs between 295 grams and 385 grams?
10. BOLT LENGTHS The distribution of the length of bolts has a bell shape with a
mean of 4 inches and a standard deviation of 0.007 inch.
(a) About 68% of bolts manufactured will be between what lengths?
(b) What percentage of bolts will be between 3.986 inches and 4.014 inches?
(c) If the company discards any bolts less than 3.986 inches or greater than 4.014
inches, what percentage of bolts manufactured will be discarded?
(d) What percentage of bolts manufactured will be between 4.007 inches and 4.021
inches?