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Chapter 15: Problem 40

Refer to SAT test scores for \(2014 .\) A total of \(N=1,672,395\) college-bound students took the SAT in 2014 Assume that the test scores are sorted from lowest to highest and that the sorted data set is \(\left\\{d_{1}, d_{2}, \ldots, d_{1,672,395}\right\\}\). (a) Determine the position of the third quartile \(Q_{3}\). (b) Determine the position of the 60 th percentile.

Short Answer

Expert verified
The position of the third quartile \((Q_{3})\) is approximately 1,254,297th and the position of the 60th percentile is approximately 1,003,437th.

Step by step solution

01

Calculation of Third Quartile Position.

The third quartile, \(Q_{3}\), is the median of the second half of the data. To find its position, simply calculate \(0.75 * N\) where N is the total number of students. For the given problem, calculate \(0.75 * 1,672,395\).
02

Rounding Off

The calculated number in step 1 may not always be an integer. However, the position in a dataset must be an integer. Therefore, if the calculated number is not an integer, round it off to the nearest integer.
03

Calculation of 60th Percentile Position.

To find the position of the 60th percentile, calculate \(0.60 * N\), where N is the total number of students. Hence, for the given problem, calculate \(0.60 * 1,672,395\).
04

Rounding Off

As with the third quartile, the position of the 60th percentile must be an integer. Therefore, if the calculated number is not an integer, round it off to the nearest integer.

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Key Concepts

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

Quartiles
Quartiles are critical in dividing a data set into four equal parts, giving a snapshot of how the data is spread. They are particularly useful for understanding the distribution of scores, like SAT test scores.
  • The first quartile (\(Q_1\)) represents the 25th percentile, indicating that 25% of the data falls below this value.
  • The second quartile (\(Q_2\)), also known as the median, represents the 50th percentile. Half of the data falls below this point.
  • The third quartile (\(Q_3\)) stands at the 75th percentile, meaning 75% of the data is below this quartile.
It is important to note that \(Q_3\) marks the top 25% of the students when test scores are considered. Calculating the position of \(Q_3\) involves multiplying 0.75 by the total number of data points, as mentioned in the exercise.
Percentiles
Percentiles rank data on a scale of 1 to 100, showing the percentage of scores below a given value. They help to assess an individual's performance compared to a broader group.
  • The 60th percentile, for instance, indicates that 60% of scores are below, and 40% are above.
  • This measure is beneficial in educational assessments, like SAT scores, where they show where a student stands in comparison to peers.
To find a specific percentile's position, multiply the desired percentile (expressed as a decimal) by the total number of scores, as demonstrated for the 60th percentile in the exercise.
Data Sorting
Sorting data efficiently is crucial to the analysis process. It involves organizing data points, such as student scores, in a certain order鈥攖ypically from lowest to highest.
  • Sorting allows statisticians and educators to easily identify key statistics, such as quartiles and percentiles.
  • In the SAT example, sorting helps to locate the third quartile and the 60th percentile positions accurately.
Properly sorted data ensures that calculations based on position, like those needed for quartiles and percentiles, are accurate and meaningful.
Educational Assessment
Educational assessments like the SAT are designed to evaluate students' readiness for college-level work. They often rely on statistical measures to provide insight into student performance.
  • These assessments are not only critical for individual evaluation but also hold significance in larger educational policy-making.
  • Quartiles and percentiles are often used to summarize and report scores, making it easier to understand general trends and individual standings.
By breaking scores into quartiles and percentiles, educators can better support students' needs and measure educational outcomes effectively.

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Most popular questions from this chapter

Refer to Table \(15-13,\) which gives the home-to-school distance \(d\) (rounded to the nearest half-mile) for each of the 27 kindergarten students at Cleansburg Elementary School. $$ \begin{array}{c|c|c|c} \begin{array}{c} \text { Student } \\ \text { ID } \end{array} & \boldsymbol{d} & \begin{array}{c} \text { Student } \\ \text { ID } \end{array} & \boldsymbol{d} \\ \hline 1362 & 1.5 & 3921 & 5.0 \\ \hline 1486 & 2.0 & 4355 & 1.0 \\ \hline 1587 & 1.0 & 4454 & 1.5 \\ \hline 1877 & 0.0 & 4561 & 1.5 \\ \hline 1932 & 1.5 & 5482 & 2.5 \\ \hline 1946 & 0.0 & 5533 & 1.5 \\ \hline 2103 & 2.5 & 5717 & 8.5 \\ \hline 2877 & 1.0 & 6307 & 1.5 \\ \hline 2964 & 0.5 & 6573 & 0.5 \\ \hline 3491 & 0.0 & 8436 & 3.0 \\ \hline 3588 & 0.5 & 8592 & 0.0 \\ \hline 3711 & 1.5 & 8964 & 2.0 \\ \hline 3780 & 2.0 & 9205 & 0.5 \\ \hline & & 9658 & 6.0 \\ \hline \end{array} $$ Draw a bar graph for the home-to-school distances for the kindergarteners at Cleansburg Elementary School using the following class intervals: Very close: Less than 1 mile Close: 1 mile up to and including 1.5 miles Nearby: 2 miles up to and including 2.5 miles Not too far: 3 miles up to and including 4.5 miles Far: 5 miles or more

Refer to Table \(15-13,\) which gives the home-to-school distance \(d\) (rounded to the nearest half-mile) for each of the 27 kindergarten students at Cleansburg Elementary School. $$ \begin{array}{c|c|c|c} \begin{array}{c} \text { Student } \\ \text { ID } \end{array} & \boldsymbol{d} & \begin{array}{c} \text { Student } \\ \text { ID } \end{array} & \boldsymbol{d} \\ \hline 1362 & 1.5 & 3921 & 5.0 \\ \hline 1486 & 2.0 & 4355 & 1.0 \\ \hline 1587 & 1.0 & 4454 & 1.5 \\ \hline 1877 & 0.0 & 4561 & 1.5 \\ \hline 1932 & 1.5 & 5482 & 2.5 \\ \hline 1946 & 0.0 & 5533 & 1.5 \\ \hline 2103 & 2.5 & 5717 & 8.5 \\ \hline 2877 & 1.0 & 6307 & 1.5 \\ \hline 2964 & 0.5 & 6573 & 0.5 \\ \hline 3491 & 0.0 & 8436 & 3.0 \\ \hline 3588 & 0.5 & 8592 & 0.0 \\ \hline 3711 & 1.5 & 8964 & 2.0 \\ \hline 3780 & 2.0 & 9205 & 0.5 \\ \hline & & 9658 & 6.0 \\ \hline \end{array} $$ (a) Make a frequency table for the distances in Table \(15-13 .\) (b) Draw a line graph for the data in Table \(15-13\).

Refer to the data set shown in Table \(15-12 .\) The table shows the scores on a Chem 103 test consisting of 10 questions worth 10 points each. $$ \begin{array}{c|c|c|c} \begin{array}{c} \text { Student } \\ \text { ID } \end{array} & \text { Score } & \begin{array}{c} \text { Student } \\ \text { ID } \end{array} & \text { Score } \\ \hline 1362 & 50 & 4315 & 70 \\ \hline 1486 & 70 & 4719 & 70 \\ \hline 1721 & 80 & 4951 & 60 \\ \hline 1932 & 60 & 5321 & 60 \\ \hline 2489 & 70 & 5872 & 100 \\ \hline 2766 & 10 & 6433 & 50 \\ \hline 2877 & 80 & 6921 & 50 \\ \hline 2964 & 60 & 8317 & 70 \\ \hline 3217 & 70 & 8854 & 100 \\ \hline 3588 & 80 & 8964 & 80 \\ \hline 3780 & 80 & 9158 & 60 \\ \hline 3921 & 60 & 9347 & 60 \\ \hline 4107 & 40 & & \end{array} $$ Suppose that the grading scale for the test is \(\mathrm{A}: 80-100 ; \mathrm{B}:\) \(70-79 ; \mathrm{C}: 60-69 ; \mathrm{D}: 50-59 ;\) and \(\mathrm{F}: 0-49\) (a) Make a frequency table for the distribution of the test grades. (b) Draw a relative frequency bar graph for the test grades.

Table \(15-16\) shows the percentage of U.S. working married couples in which the wife's income is higher than the husband's \((1999-2009) .\) (a) Draw a pictogram for the data in Table \(15-16\). Assume you are trying to convince your audience that things are looking great for women in the workplace and that women's salaries are catching up to men's very quickly. (b) Draw a different pictogram for the data in Table \(15-16\), where you are trying to convince your audience that women's salaries are catching up with men's very slowly. $$ \begin{array}{l|c|c|c|c|c|c} \text { Year } & 1999 & 2000 & 2001 & 2002 & 2003 & 2004 \\ \hline \text { Percent } & 28.9 & 29.9 & 30.7 & 31.9 & 32.4 & 32.6 \\ \hline \text { Year } & 2005 & 2006 & 2007 & 2008 & 2009 & \\ \hline \text { Percent } & 33.0 & 33.4 & 33.5 & 34.5 & 37.7 & \end{array} $$

Consider the data set \\{-5,7,4,8,2,8,-3,-6\\} (a) Find the first quartile \(Q_{1}\) of the data set. (b) Find the third quartile \(Q_{3}\) of the data set. (c) Consider the data set \\{-5,7,4,8,2,8,-3,-6,2\\} obtained by adding one more data point to the original data set. Find the first and third quartiles of this data set.
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