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Chapter 2: Problem 1

Do Students Need Summer Development? For 108 randomly selected college applicants, the following frequency distribution for entrance exam scores was obtained. Construct a histogram, frequency polygon, and ogive for the data. (The data for this exercise will be used for Exercise 13 in this section. $$ \begin{array}{rr} \text { Class limits } & \text { Frequency } \\ \hline 90-98 & 6 \\ 99-107 & 22 \\ 108-116 & 43 \\ 117-125 & 28 \\ 126-134 & 9 \\ & \text { Total } 108 \end{array} $$ Applicants who score above 107 need not enroll in a summer developmental program. In this group, how many students do not have to enroll in the developmental program?

Short Answer

Expert verified
80 students do not need the developmental program.

Step by step solution

01

Identify Students Needing Development

Examine the frequency distribution table: Students with scores above 107 are not required to enroll in the summer program. This refers to students in the 108-116, 117-125, and 126-134 score ranges.
02

Count Students Not Needing Development

Add up the frequencies for students scoring above 107. Specifically, add frequencies from the intervals 108-116, 117-125, and 126-134: 43 (for 108-116) + 28 (for 117-125) + 9 (for 126-134).
03

Calculate Total

Compute the total number of students by summing up the calculated frequencies: 43 + 28 + 9 = 80. Thus, 80 students scored above 107 and do not need to enroll in the summer program.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Histograms
A histogram is a type of graph that helps us visualize the distribution of numerical data. Imagine you have a set of numbers, and you want to see how often each number appears. A histogram turns this information into a series of bars.

Here鈥檚 how it works:
  • Each bar on the histogram represents a range of values, known as a class interval.
  • For our exercise, each class interval represents a range of entrance exam scores, such as 90-98 or 99-107.
  • The height of each bar corresponds to the frequency, which tells us how many numbers fit into that range.
Using this, you can quickly see which ranges have the most scores by looking for the tallest bars. It's like getting a bird鈥檚-eye view of where most students' scores fall, giving you insight into the data鈥檚 pattern.
Frequency Polygons Explained
A frequency polygon is another way to display frequency distribution data. It鈥檚 similar to a histogram but instead of using bars, it uses lines to connect points.
  • Each point on a frequency polygon represents the frequency of a class interval.
  • You place these points at the midpoint of each class interval. For example, if your interval is 90-98, the midpoint is 94.
  • You then draw lines to connect each point in order from left to right.
This creates a simple line graph that helps to show trends over class intervals. It provides a clearer picture of the distribution shape and can be smoother to interpret compared to the blocky appearance of a histogram.

Frequency polygons are handy for comparing two different data sets as they allow you to overlay multiple lines.
What is an Ogive?
An ogive is a graph that can help you understand cumulative frequencies. It's useful when you want to know how many values fall below a certain point. Think of it as climbing a staircase where each step represents the cumulative frequency as you move upwards.

To make an ogive:
  • You plot a point for the upper boundary of each class interval against the cumulative frequency.
  • The cumulative frequency is the total number of observations that fall below the upper boundary of the interval.
  • Finally, you connect the dots with lines from left to right to form the ogive curve.
An ogive allows you to see how fast the accumulation of data points is happening across intervals. It鈥檚 particularly useful for finding medians, percentiles, and for understanding overall distribution.
Frequency Distribution Fundamentals
A frequency distribution is a summary of how often different values occur within a data set. It organizes data into categories or intervals, allowing for easier analysis and understanding of the data's structure. For entrance exam scores:
  • Each class interval, like 90-98 or 108-116, covers a specific score range.
  • The frequency tells you the number of applicants whose scores fall within that range.
Frequency distribution provides a structured overview of the data allowing you to easily see where most values concentrate.
Making sense of frequency distributions helps in creating other graphs like histograms, frequency polygons, and ogives.
It鈥檚 particularly vital when determining conditions, such as deciding which students might need additional programs based on their score distributions.

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Most popular questions from this chapter

Percentage of People Who Completed 4 or More Years of College Listed by state are the percentages of the population who have completed 4 or more years of a college education. Construct a frequency distribution with 7 classes. $$ \begin{aligned} &\begin{array}{llllllllll} 21.4 & 26.0 & 25.3 & 19.3 & 29.5 & 35.0 & 34.7 & 26.1 & 25.8 & 23.4 \\ 27.1 & 29.2 & 24.5 & 29.5 & 22.1 & 24.3 & 28.8 & 20.0 & 20.4 & 26.7 \\ 35.2 & 37.9 & 24.7 & 31.0 & 18.9 & 24.5 & 27.0 & 27.5 & 21.8 & 32.5 \\ 33.9 & 24.8 & 31.7 & 25.6 & 25.7 & 24.1 & 22.8 & 28.3 & 25.8 & 29.8 \\ 23.5 & 25.0 & 21.8 & 25.2 & 28.7 & 33.6 & 33.6 & 30.3 & 17.3 & 25.4 \end{array}\\\ &\text { Source: New York Times Almanac. } \end{aligned} $$

State which graph (Pareto chart, time series graph, or pie graph) would most appropriately represent the given situation. a. The number of students enrolled at a local college for each year during the last 5 years b. The budget for the student activities department at a certain college for a specific year c. The means of transportation the students use to get to school d. The percentage of votes each of the four candidates received in the last election e. The record temperatures of a city for the last 30 years f. The frequency of each type of crime committed in a city during the year

The number of faculty listed for a sample of private colleges that offer only bachelor's degrees is listed below. Use these data to construct a frequency distribution with 7 classes, a histogram, a frequency polygon, and an ogive. Discuss the shape of this distribution. What proportion of schools have 180 or more faculty? $$ \begin{array}{rlllrrll} 165 & 221 & 218 & 206 & 138 & 135 & 224 & 204 \\ 70 & 210 & 207 & 154 & 155 & 82 & 120 & 116 \\ 176 & 162 & 225 & 214 & 93 & 389 & 77 & 135 \\ 221 & 161 & 128 & 310 & & & & \end{array} $$

Show frequency distributions that are incorrectly constructed. State the reasons why they are wrong. $$ \begin{array}{lc} \text { Class } & \text { Frequency } \\ \hline 10-19 & 1 \\ 20-29 & 2 \\ 30-34 & 0 \\ 35-45 & 5 \\ 46-51 & 8 \end{array} $$

Ages of Declaration of Independence Signers The ages of the signers of the Declaration of Independence are shown. (Age is approximate since only the birth year appeared in the source, and one has been omitted since his birth year is unknown.) Construct a grouped frequency distribution and a cumulative frequency distribution for the data, using 7 classes. \(\begin{array}{llllllllllll}41 & 54 & 47 & 40 & 39 & 35 & 50 & 37 & 49 & 42 & 70 & 32 \\ 44 & 52 & 39 & 50 & 40 & 30 & 34 & 69 & 39 & 45 & 33 & 42 \\ 44 & 63 & 60 & 27 & 42 & 34 & 50 & 42 & 52 & 38 & 36 & 45 \\ 35 & 43 & 48 & 46 & 31 & 27 & 55 & 63 & 46 & 33 & 60 & 62 \\ 35 & 46 & 45 & 34 & 53 & 50 & 50 & & & & & \end{array}\) Source: The Universal Almanac.
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