Question

A test of reaction times was given to 50 people who want to join a police force. The results, measured to the nearest tenth of a second, are summarized in the "or more" cumulative percentage distribution shown in Table 4.32, where the lower implied boundary of a category was used in the cumulation. If the police department requires a reaction time of 0.5 seconds or less on this test, how many of the 50 candidates passed the test? How many were only $1 / 10$ of a second too slow? $$ \begin{aligned} &\text { Table } 4.30\\ &\begin{array}{lcc} \hline \text { 2d Exam } & \begin{array}{c} \text { Cumulative } \\ \text { frequency } \end{array} & \begin{array}{c} \text { Cumulative } \\ \text { percentage } \end{array} \\ \hline 45 \text { or more } & 64 & 100.0 \\ 50 \text { or more } & 63 & 98.4 \\ 55 \text { or more } & 63 & 98.4 \\ 60 \text { or more } & 59 & 92.2 \\ 65 \text { or more } & 56 & 87.5 \\ 70 \text { or more } & 51 & 79.7 \\ 75 \text { or more } & 48 & 75.0 \\ 80 \text { or more } & 42 & 65.6 \\ 85 \text { or more } & 31 & 48.4 \\ 90 \text { or more } & 22 & 34.4 \\ 95 \text { or more } & 5 & 7.8 \\ 100 & 0 & 0.0 \\ \hline \end{array} \end{aligned} $$ $$ \begin{aligned} &\text { Table 4.31 }\\ &\begin{array}{lr} \hline \text { Winnings (\$) } & \begin{array}{c} \text { Cumulative } \\ \text { percentage } \end{array} \\ \hline \text { Less than 1,500 } & 0.0 \\ \text { Less than 4,500 } & 42.9 \\ \text { Less than 10,500 } & 67.1 \\ \text { Less than 20,500 } & 77.1 \\ \text { Less than } 30,500 & 88.6 \\ \text { Less than 40,500 } & 92.9 \\ \text { Less than 45,500 } & 94.3 \\ \text { Less than } 81,500 & 94.3 \\ \text { Less than } 83,500 & 98.6 \\ \text { Less than 198,500 } & 98.6 \\ \text { Less than 200,500 } & 100.0 \\ \hline \end{array} \end{aligned} $$ $$ \begin{aligned} &\text { Table 4.32 }\\ &\begin{array}{lc} \hline \text { Reaction time (sec) } & \begin{array}{c} \text { Cumulative } \\ \text { percentage } \end{array} \\ \hline 0.15 \text { or more } & 100 \\ 0.25 \text { or more } & 98 \\ 0.35 \text { or more } & 94 \\ 0.45 \text { or more } & 86 \\ 0.55 \text { or more } & 76 \\ 0.65 \text { or more } & 50 \\ 0.75 \text { or more } & 30 \\ 0.85 \text { or more } & 18 \\ 0.95 \text { or more } & 8 \\ 1.05 \text { or more } & 0 \\ \hline \end{array} \end{aligned} $$

Name: Problem 23, Chapter 4: Schaum's Outline of Elements of Statistics I: Descriptive Statistics and Probability
Uploaded: 2023-10-11T21:42:58-08:00
Duration: 5 min 7 s
Channel: Sheryl Ezze

   A test of reaction times was given to 50 people who want to join a police force. The results, measured to the nearest tenth of a second, are summarized in the "or more" cumulative percentage distribution shown in Table 4.32, where the lower implied boundary of a category was used in the cumulation. If the police department requires a reaction time of 0.5 seconds or less on this test, how many of the 50 candidates passed the test? How many were only $1 / 10$ of a second too slow?
$$
\begin{aligned}
&\text { Table } 4.30\\
&\begin{array}{lcc}
\hline \text { 2d Exam } & \begin{array}{c}
\text { Cumulative } \\
\text { frequency }
\end{array} & \begin{array}{c}
\text { Cumulative } \\
\text { percentage }
\end{array} \\
\hline 45 \text { or more } & 64 & 100.0 \\
50 \text { or more } & 63 & 98.4 \\
55 \text { or more } & 63 & 98.4 \\
60 \text { or more } & 59 & 92.2 \\
65 \text { or more } & 56 & 87.5 \\
70 \text { or more } & 51 & 79.7 \\
75 \text { or more } & 48 & 75.0 \\
80 \text { or more } & 42 & 65.6 \\
85 \text { or more } & 31 & 48.4 \\
90 \text { or more } & 22 & 34.4 \\
95 \text { or more } & 5 & 7.8 \\
100 & 0 & 0.0 \\
\hline
\end{array}
\end{aligned}
$$
$$
\begin{aligned}
&\text { Table 4.31 }\\
&\begin{array}{lr}
\hline \text { Winnings (\$) } & \begin{array}{c}
\text { Cumulative } \\
\text { percentage }
\end{array} \\
\hline \text { Less than 1,500 } & 0.0 \\
\text { Less than 4,500 } & 42.9 \\
\text { Less than 10,500 } & 67.1 \\
\text { Less than 20,500 } & 77.1 \\
\text { Less than } 30,500 & 88.6 \\
\text { Less than 40,500 } & 92.9 \\
\text { Less than 45,500 } & 94.3 \\
\text { Less than } 81,500 & 94.3 \\
\text { Less than } 83,500 & 98.6 \\
\text { Less than 198,500 } & 98.6 \\
\text { Less than 200,500 } & 100.0 \\
\hline
\end{array}
\end{aligned}
$$
$$
\begin{aligned}
&\text { Table 4.32 }\\
&\begin{array}{lc}
\hline \text { Reaction time (sec) } & \begin{array}{c}
\text { Cumulative } \\
\text { percentage }
\end{array} \\
\hline 0.15 \text { or more } & 100 \\
0.25 \text { or more } & 98 \\
0.35 \text { or more } & 94 \\
0.45 \text { or more } & 86 \\
0.55 \text { or more } & 76 \\
0.65 \text { or more } & 50 \\
0.75 \text { or more } & 30 \\
0.85 \text { or more } & 18 \\
0.95 \text { or more } & 8 \\
1.05 \text { or more } & 0 \\
\hline
\end{array}
\end{aligned}
$$

Schaum's Outline of Elements of Statistics I: Descriptive Statistics and Probability

Stephen Bernstein,… 1st Edition

Chapter 4, Problem 23

Instant Answer

Solved by Expert Sheryl Ezze

10/11/2023

Step 1

5 seconds or less. According to Table 4.32, the cumulative percentage for 0.55 seconds or more is 76%. This means that 76% of the candidates have a reaction time of 0.55 seconds or more. Show more…

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A test of reaction times was given to 50 people who want to join a police force. The results, measured to the nearest tenth of a second, are summarized in the "or more" cumulative percentage distribution shown in Table 4.32, where the lower implied boundary of a category was used in the cumulation. If the police department requires a reaction time of 0.5 seconds or less on this test, how many of the 50 candidates passed the test? How many were only $1 / 10$ of a second too slow? $$ \begin{aligned} &\text { Table } 4.30\\ &\begin{array}{lcc} \hline \text { 2d Exam } & \begin{array}{c} \text { Cumulative } \\ \text { frequency } \end{array} & \begin{array}{c} \text { Cumulative } \\ \text { percentage } \end{array} \\ \hline 45 \text { or more } & 64 & 100.0 \\ 50 \text { or more } & 63 & 98.4 \\ 55 \text { or more } & 63 & 98.4 \\ 60 \text { or more } & 59 & 92.2 \\ 65 \text { or more } & 56 & 87.5 \\ 70 \text { or more } & 51 & 79.7 \\ 75 \text { or more } & 48 & 75.0 \\ 80 \text { or more } & 42 & 65.6 \\ 85 \text { or more } & 31 & 48.4 \\ 90 \text { or more } & 22 & 34.4 \\ 95 \text { or more } & 5 & 7.8 \\ 100 & 0 & 0.0 \\ \hline \end{array} \end{aligned} $$ $$ \begin{aligned} &\text { Table 4.31 }\\ &\begin{array}{lr} \hline \text { Winnings (\$) } & \begin{array}{c} \text { Cumulative } \\ \text { percentage } \end{array} \\ \hline \text { Less than 1,500 } & 0.0 \\ \text { Less than 4,500 } & 42.9 \\ \text { Less than 10,500 } & 67.1 \\ \text { Less than 20,500 } & 77.1 \\ \text { Less than } 30,500 & 88.6 \\ \text { Less than 40,500 } & 92.9 \\ \text { Less than 45,500 } & 94.3 \\ \text { Less than } 81,500 & 94.3 \\ \text { Less than } 83,500 & 98.6 \\ \text { Less than 198,500 } & 98.6 \\ \text { Less than 200,500 } & 100.0 \\ \hline \end{array} \end{aligned} $$ $$ \begin{aligned} &\text { Table 4.32 }\\ &\begin{array}{lc} \hline \text { Reaction time (sec) } & \begin{array}{c} \text { Cumulative } \\ \text { percentage } \end{array} \\ \hline 0.15 \text { or more } & 100 \\ 0.25 \text { or more } & 98 \\ 0.35 \text { or more } & 94 \\ 0.45 \text { or more } & 86 \\ 0.55 \text { or more } & 76 \\ 0.65 \text { or more } & 50 \\ 0.75 \text { or more } & 30 \\ 0.85 \text { or more } & 18 \\ 0.95 \text { or more } & 8 \\ 1.05 \text { or more } & 0 \\ \hline \end{array} \end{aligned} $$

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Key Concepts

Threshold Criteria and Decision Making

In many testing scenarios, a specific criterion or threshold is established to determine outcomes, such as passing a test. By using the cumulative frequency or percentage distribution, one can quickly ascertain not only how many individuals meet the criteria but also how many narrowly miss it, facilitating decisions based on quantitative benchmarks.

Grouped Data and Class Boundaries

When data is recorded in groups or intervals, such as reaction times measured to the nearest tenth, understanding the class boundaries is crucial. The lower boundary of a category represents the starting point of that interval, ensuring that the cumulation of data points is accurate and that the transition from one group to the next is well-defined.

Cumulative Percentage Distribution

This is the cumulative frequency expressed as a percentage of the total dataset. It simplifies the interpretation of how many individuals or data points lie below a specific value by converting raw frequencies into percentages, which can be particularly helpful when evaluating performance against a set criterion.

Cumulative Frequency Distribution

This concept involves summing the frequencies of data points in a dataset up to a given point. It is used to determine how many observations fall below or meet a particular threshold, making it easier to analyze grouped data such as test scores or reaction times.

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