Download 5.8 Normal Distributions 5.9 The 68–95–99.7 Rule

5.5 Describing Spread: The Quartiles 5.6 The Five-Number Summary and Boxplots
20. In chronological order, here are the percents of the popular vote won by each successful candidate in the last 15
presidential elections, starting with 1948:
a. Make a stemplot of the winners' percents.
b. What is the median percent of the vote won by the successful candidate in presidential elections?
c. It is called an election a landslide if the winner's percent falls at or above the third quartile. Find the third quartile.
Which elections were landslides?
d. Find the range.
21. Here are those charges (in dollars) of the tuition and fees charged by the 55 four-year colleges in the state of
Massachusetts., arranged in increasing order.
Find the five-number summary and make a boxplot. What distinctive feature of the histogram do these summaries miss?
Remember that numerical summaries are not a substitute for looking at the data.
27. How much oil the wells in a given field will ultimately produce is key information in deciding whether to drill more
wells. Here are the estimated total amounts of oil recovered from 64 wells in the Devonian Richmond Dolomite area of
the Michigan basin, in thousands of barrels. [J. Marcus Jobe and Hutch Jobe, A statistical approach for additional infill
development, Energy Exploration and Exploitation, 18 (2000): 89–103.]
a. Make a histogram and describe its main features.
b. Find the mean and median of the amounts recovered. Explain how the relationship between the mean and the
median reflects the shape of the distribution.
c. Give the five-number summary and explain briefly how it reflects the shape of the distribution.
5.7 Describing Spread: The Standard Deviation
32. The level of various substances in the blood influences our health. Here are measurements of the level of phosphate
in the blood of a patient, in milligrams of phosphate per deciliter of blood, made on six consecutive visits to a clinic.
5.6 5.2 4.6 4.9 5.7 6.4
a. Find the mean.
b. Find the standard deviation.
33. The mean x and standard deviation s measure center and spread but are not a complete description of a distribution.
Data sets with different shapes can have the same mean and standard deviation. To demonstrate this fact, use your
calculator to find x and s for these two small data sets. Then make a stemplot of each and comment on the shape of each
distribution.
5.8 Normal Distributions
5.9 The 68–95–99.7 Rule
37. Some teachers graded “on a curve” based on the belief that classroom test scores are normally distributed. One way
of doing this is to assign a “C” to all scores within 1 standard deviation of the mean. Then, the teacher would assign a “B”
to all scores between 1 and 2 standard deviations above the mean, an “A” to all scores more than 2 standard deviations
above the mean, and use symmetry to define the regions for “D” and “F” on the left side of the normal curve. If 200
students take an exam, determine the number of students who would receive a B.
38. The length of human pregnancies from conception to birth varies according to a distribution that is approximately
normal, with mean 266 days and standard deviation 16 days. Draw a normal curve for this distribution on which the
mean and standard deviation are correctly located. (Hint: First draw the curve, then mark the axis.)
39. Figure 5.22 shows a smooth curve used to describe a distribution that is not symmetric. The mean and median do
not coincide. Which of the points marked is the mean of the distribution, and which is the median? Explain your answer.
FIGURE 5.22 A curve describing a skewed distribution, for Exercise 39.
41. Bigger animals tend to carry their young longer before birth. The length of horse pregnancies from conception to
birth varies according to a roughly normal distribution, with mean 336 days and standard deviation 3 days. Use the 68–
95–99.7 rule to answer the following questions.
a. Almost all (99.7%) horse pregnancies fall in what range of lengths?
b. What percent of horse pregnancies are longer than 339 days?
42. Scores on the three-section SAT Reasoning college entrance test for the class of 2007 were roughly normal, with
mean 1511 and standard deviation 194.
a. What was the range of the middle 68% of SAT scores?
b. How high must a student score to be in the top 2.5% of SAT scores?
43. What are the quartiles of SAT Reasoning scores, according to the distribution in Exercise 42?
45. The yearly rate of return on the Standard & Poor's 500 (an index of 500 large-cap corporations) is approximately
normal. From January 1956 through September 2007, the S&P 500 had a mean yearly return of 10.51%, with a standard
deviation of about 15.51%. Take this normal distribution to be the distribution of yearly returns over a long period.
a. In what range do the middle 95% of all yearly returns lie?
b. Stocks can go down as well as up. What are the worst 2.5% of annual returns?
46. What is the range of the middle 50% of annual returns on stocks, according to the distribution given in the previous
exercise? (Hint: What two numbers mark off the middle 50% of any distribution?)
47. The concentration of the active ingredient in capsules of a prescription painkiller varies according to a normal
distribution with μ = 10% and σ = 0.2%.
a. What is the median concentration? Explain your answer.
b. What range of concentrations covers the middle 95% of all the capsules?
c. What range covers the middle half of all capsules?
48. Answer the following questions for the painkiller in Exercise 47.
a. What percent of all capsules have a concentration of active ingredient higher than 10.4%?