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The Practice of Statistics for AP Examination

Daren Starnes, Josh Tabor

$Math Studyset Vaia Explanations$ Math

6th Edition · 837 pages

ISBN: 9781319113339

Chapter 2: Q 37. (page 108)

The number of absences during the fall semester was recorded for each student in a large elementary school. The distribution of absences is displayed in the following cumulative relative frequency graph.
What is the interquartile range (IQR) for the distribution of absences?
a. 1
b. 2
c. 3
d. 5
e. 14

Short Answer

Expert verified

The correct option is (c) $3$

Step by step solution

01

Given information

02

Concept

We can investigate the location in a distribution using a cumulative relative frequency graph. You can estimate the percentile for an individual value using the completed graph, and vice versa.

03

Calculation

The difference between the first and third quartiles is the interquartile range.

The $1 s t$ quartile's characteristic demonstrates that $25 %$ of the data values are below it.

The $3 r d$ quartile's characteristic demonstrates that $75 %$ of the data values are below it.

From the graph, we can see that $2$

$1 s t$ quartile (or $25 %$ ) is at absences.

Such that

$Q_{1} = 2$

And

$3 r d$ quartile (or $75 %$ ) is at $5$ absences.

Such that

$Q_{2} = 5$

The difference between the 1st and 3rd quartile gives the interquartile range.

Thus,

$I Q R = Q_{3} - Q_{1} = 5 - 2 = 3$

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Most popular questions from this chapter

Measuring bone density Individuals with low bone density (osteoporosis) have a high risk of broken bones (fractures). Physicians who are concerned about low bone density in patients can refer them for specialized testing. Currently, the most common method for testing bone density is dual-energy $X -$ ray absorptiometry (DEXA). The bone density results for a patient who undergoes a DEXA test usually are reported in grams per square centimeter ( $g / {c m}^{2}$ ) and in standardized units. Judy, who is $25$ years old, has her bone density measured using DEXA. Her results indicate bone density in the hip of 948 g/cm2 and a standardized score of $z = - 1.45$ The mean bone density in the hip is 956 $g / {c m}^{2}$ in the reference population of $25 -$ year-old women like Judy.

a. Judy has not taken a statistics class in a few years. Explain to her in simple language what the standardized score reveals about her bone density.

b. Use the information provided to calculate the standard deviation of bone density in the reference population.

I think I can! Refer to Exercise 55. The locomotive’s manufacturer is considering two changes that could reduce the percent of times that the train arrives late. One option is to increase the mean adhesion of the locomotive. The other possibility is to decrease the variability in adhesion from trip to trip, that is, to reduce the standard deviation.

(a) If the standard deviation remains at $σ = 0.04$ , to what value must the manufacturer change the mean adhesion of the locomotive to reduce its proportion of late arrivals to less than $2 %$ of days? Show your work.

(b) If the mean adhesion stays at $μ = 0.37$ , how much must the standard deviation be decreased to ensure that the train will arrive late less than $2 %$ of the time? Show your worK

(c) Which of the two options (a) and (b) do you think is preferable?. justify your answer. (Be sure to consider the effect of these changes on the percent of days that the train arrives early to the switch point)

Comparing batting averages Three landmarks of baseball achievement are Ty Cobb’s batting average of $0.420$ in $1911,$ Ted Williams’s $0.406$ in $1941,$ and George Brett’s $0.390$ in $1980$ These batting averages cannot be compared directly because the distribution of major league batting averages has changed over the years. The distributions are quite symmetric, except for outliers such as Cobb, Williams, and Brett. While the mean batting average has been held roughly constant by rule changes and the balance between hitting

and pitching, the standard deviation has dropped over time. Here are the facts:

Find the standardized scores for Cobb, Williams, and Brett. Who had the best performance for the decade he played?

Foreign-born residents The cumulative relative frequency graph shows the distribution of the percent of foreign-born residents in the $50$ states.

a. Estimate the interquartile range (IQR) of this distribution. Show your method.

b. What is the percentile for Arizona, which had $15.1 %$ foreign-born residents that year?

c. Explain why the graph is fairly flat between $20 %$ and $27.5 %$

d. Draw the histogram that corresponds to this graph.

Used cars, cheap! Refer to Exercise $24$ Suppose each car’s price is reduced by 10% instead of $$ 500$ What effect will this discount have on each of the following characteristics of the resulting price distribution?

a. Shape

b. Median

c. Interquartile range (IQR)

See all solutions

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