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Chapter 13: Problem 10

Rank each set of data. $$ 11.7,18.6,41.7,11.7,16.2,5.1,31.4,5.1,14.3 $$

Short Answer

Expert verified
5.1 = 1.5, 11.7 = 3.5, 14.3 = 5, 16.2 = 6, 18.6 = 7, 31.4 = 8, 41.7 = 9.

Step by step solution

01

List and Identify the Data

Start by noting down the given set of numbers: 11.7, 18.6, 41.7, 11.7, 16.2, 5.1, 31.4, 5.1, 14.3. Make sure you identify any repeating values, as ties need special consideration when ranking.
02

Order the Data in Ascending Order

Arrange the numbers from the smallest to the largest value: 5.1, 5.1, 11.7, 11.7, 14.3, 16.2, 18.6, 31.4, 41.7.
03

Assign Ranks to Ordered Data

Assign ranks starting from 1 for the lowest number. Since 5.1 repeats, assign both as rank 1.5 (the average of ranks 1 and 2). Similarly, both 11.7s receive rank 3.5 (the average of ranks 3 and 4). Continue assigning ranks to the rest of the numbers: 14.3 as 5, 16.2 as 6, 18.6 as 7, 31.4 as 8, and 41.7 as 9.

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Key Concepts

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

Understanding Ascending Order
To rank data effectively, the first step is organizing the values in ascending order. Simply put, this means arranging the numbers from smallest to largest.

For instance, if we have the numbers 11.7, 5.1, and 18.6, sorting them in ascending order would result in 5.1, 11.7, and 18.6.

Why is ascending order important? This arrangement helps us compare and rank the values more intuitively, as you're working from the lowest to the highest. Remembering this straightforward procedure is crucial for any ranking process.
Handling Ties in Ranking
Sometimes, in a list of numbers, certain values can repeat鈥攖hese are called ties. When ranking data, ties require special attention to ensure fairness and accuracy.

Here's how you handle ties:
  • Identify the repeated values.
  • Instead of assigning them consecutive ranks, calculate the average of those ranks and assign the average rank to all tied values.
For example, if the values 11.7 and 11.7 appear in positions 3 and 4, find the average: \[ \frac{3 + 4}{2} = 3.5 \]In this way, both instances of 11.7 receive a rank of 3.5. Handling ties in this manner ensures that the ranking system reflects the actual data accurately and equitably.
Assigning Ranks
Once your data is organized in ascending order and you've resolved any ties, you're ready to assign ranks to each value.

This process begins with ranking the smallest value as 1. Simply continue assigning increasing ranks for each subsequent number.

When you encounter tied values, use the average rank as discussed. Consider the following sequence:
  • 5.1, 5.1 (both receive rank 1.5)
  • 11.7, 11.7 (both receive rank 3.5)
  • 14.3 receives 5
  • 16.2 as 6, 18.6 as 7, and so on
Assigning ranks like this ensures clarity and precision, helping make data analysis straightforward and meaningful.

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Most popular questions from this chapter

Perform these steps. a. Find the Spearman rank correlation coefficient. b. State the hypotheses. c. Find the critical value. Use \(\alpha=0.05\). d. Make the decision. e. Summarize the results. Use the traditional method of hypothesis testing unless otherwise specified. Cyber School Enrollments Shown are the numbers of students enrolled in cyber school for five randomly selected school districts and the per-pupil costs for the cyber school education. At \(\alpha=0.10\), is there a relationship between the two variables? How might this information be useful to school administrators? $$ \begin{array}{l|ccccc} \text { Number of students } & 10 & 6 & 17 & 8 & 11 \\ \hline \text { Per-pupil cost } & 7200 & 9393 & 7385 & 4500 & 8203 \end{array} $$

When \(n \geq 30,\) the formula \(r=\frac{\pm z}{\sqrt{n-1}}\) can be used to find the critical values for the rank correlation coefficient. For example, if \(n=40\) and \(\alpha=0.05\) for a two-tailed test, $$ r=\frac{\pm 1.96}{\sqrt{40-1}}=\pm 0.314 $$ Hence, any \(r_{s}\) greater than or equal to +0.314 or less than or equal to -0.314 is significant. Find the critical \(r\) value for each (assume that the test is two-tailed). $$ n=60, \alpha=0.10 $$

Use the Kruskal-Wallis test and perform these steps. a. State the hypotheses and identify the claim. b. Find the critical value. c. Compute the test value. d. Make the decision. e. Summarize the results. Use the traditional method of hypothesis testing unless otherwise specified. Job Offers for Chemical Engineers A recent study recorded the number of job offers received by randomly selected, newly graduated chemical engineers at three colleges. The data are shown here. At \(\alpha=0.05,\) is there a difference in the average number of job offers received by the graduates at the three colleges? $$ \begin{array}{ccc} \text { College A } & \text { College B } & \text { College C } \\ \hline 6 & 2 & 10 \\ 8 & 1 & 12 \\ 7 & 0 & 9 \\ 5 & 3 & 13 \\ 6 & 6 & 4 \end{array} $$

Perform these steps. a. Find the Spearman rank correlation coefficient. b. State the hypotheses. c. Find the critical value. Use \(\alpha=0.05\). d. Make the decision. e. Summarize the results. Use the traditional method of hypothesis testing unless otherwise specified. Subway and Commuter Rail Passengers Six cities are randomly selected, and the number of daily passenger trips (in thousands) for subways and commuter rail service is obtained. At \(\alpha=0.05,\) is there a relationship between the variables? Suggest one reason why the transportation authority might use the results of this study. $$ \begin{array}{l|rrrrrr} \text { City } & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \text { Subway } & 845 & 494 & 425 & 313 & 108 & 41 \\ \hline \text { Rail } & 39 & 291 & 142 & 103 & 33 & 38 \end{array} $$

Use the Kruskal-Wallis test and perform these steps. a. State the hypotheses and identify the claim. b. Find the critical value. c. Compute the test value. d. Make the decision. e. Summarize the results. Use the traditional method of hypothesis testing unless otherwise specified. Speaking Confidence Fear of public speaking is a common problem for many individuals. A researcher wishes to see if educating individuals on the aspects of public speaking will help people be more confident when they speak in public. She designs three programs for individuals to complete. Group A studies the aspects of writing a good speech. Group \(\mathrm{B}\) is given instruction on delivering a speech. Group \(\mathrm{C}\) is given practice and evaluation sessions on presenting a speech. Then each group is given a questionnaire on selfconfidence. The scores are shown. At \(\alpha=0.05\), is there a difference in the scores on the tests? $$ \begin{array}{ccc} \text { Group A } & \text { Group B } & \text { Group C } \\ \hline 22 & 18 & 16 \\ 25 & 24 & 17 \\ 27 & 25 & 19 \\ 26 & 27 & 23 \\ 33 & 29 & 18 \\ 35 & 31 & 31 \\ 30 & 17 & 15 \\ 36 & 15 & 36 \end{array} $$
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