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Chapter 3: Problem 19

The data shown represent the scores on a national achievement test for a group of 10th-grade students. Find the approximate percentile ranks of these scores by constructing a percentile graph. a. 220 b. 245 c. 276 d. 280 e. 300 \(\begin{array}{ll}{\text { Score }} & {\text { Frequency }} \\ \hline 196.5-217.5 & {5} \\ {217.5-238.5} & {17} \\ {238.5-259.5} & {17} \\\ {259.5-280.5} & {48} \\ {280.5-301.5} & {22} \\ {301.5-322.5} & {6}\end{array}\) For the same data, find the approximate scores that cor- respond to these percentiles. f. 15th g. 29th h. 43rd i. 65th j. 80th

Short Answer

Expert verified
Percentile ranks: 220 (13%), 245 (27%), 276 & 280 (52%), 300 (77%). Scores: 15th (217.5-238.5), 29th (238.5-259.5), 43rd & 65th (259.5-280.5), 80th (280.5-301.5).

Step by step solution

01

Calculate Cumulative Frequencies

First, we need to create cumulative frequencies for the given score ranges. We calculate these by adding the frequency of each class to the sum of the frequencies for all previous classes.\[\begin{array}{ll}\text{Class Range} & \text{Cumulative Frequency} \196.5-217.5 & 5 \217.5-238.5 & 22\ (5+17) \238.5-259.5 & 39\ (22+17) \259.5-280.5 & 87\ (39+48) \280.5-301.5 & 109\ (87+22) \301.5-322.5 & 115\ (109+6)\end{array}\]
02

Compute the Total Number of Students

Next, find the total number of students who took the test by adding up all the frequencies.\[5 + 17 + 17 + 48 + 22 + 6 = 115\]So, 115 students in total took the test.
03

Calculate Percentile Ranks for Scores

We now find the percentile rank for specific scores using the formula: \[\text{Percentile Rank} = \left( \frac{\text{Cumulative Frequency below the score} + 0.5 \times \text{Frequency of the score}}{\text{Total Frequency}} \right) \times 100\]For each score:- **220**: \[\frac{5 + 0.5 \times 17}{115} \times 100 = 13.04\%\]- **245**: \[\frac{22 + 0.5 \times 17}{115} \times 100 = 27.39\%\]- **276**: \[\frac{39 + 0.5 \times 48}{115} \times 100 = 52.17\%\]- **280**: \[\frac{39 + 0.5 \times 48}{115} \times 100 = 52.17\%\]- **300**: \[\frac{87 + 0.5 \times 22}{115} \times 100 = 77.39\%\]
04

Determine Scores for Given Percentiles

To find the score range corresponding to a given percentile, identify the interval where the cumulative frequency first exceeds the target percentile as a percentage of total students. - **15th percentile** corresponds to 17.25 students, which lies in the **217.5-238.5** range. - **29th percentile** corresponds to 33.35 students, which lies in the **238.5-259.5** range. - **43rd percentile** corresponds to 49.45 students, also in the **259.5-280.5** range. - **65th percentile** corresponds to 74.75 students, which lies in the **259.5-280.5** range. - **80th percentile** corresponds to 92 students, which is in the **280.5-301.5** range.

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Key Concepts

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

Understanding Frequency Distribution
Frequency Distribution is a method of organizing data to show how often each value occurs. Think of it as a summary of how many students scored within specific ranges on a test. In our example, the students' scores are grouped into specific score intervals such as 196.5-217.5, 217.5-238.5, and so forth. Each of these intervals has a corresponding count of students who scored within that range.
  • Purpose: Helps in making sense of large data sets by clustering similar values together.
  • Visualization: Often represented as histograms or frequency polygons for easier understanding of the data pattern.
By examining the frequency distribution, one can quickly get an overview of how scores are spread across a range of values. It's the first step in analyzing the data to find trends and patterns.
Breaking Down Cumulative Frequency
Cumulative Frequency is a progressive tally of the frequencies in a data set. It accumulates the total number of students scoring up to a certain range. This is vital for percentile calculations, as it helps in understanding the accumulation trend across score intervals.
  • Step by Step: Begin with the frequency of the first score range, then add this to the frequency of the next range, continuing until all classes are summed.
  • Example: For scores ranging from 196.5-217.5, the cumulative frequency is 5. Adding the next range, 217.5-238.5 with a frequency of 17, gives a cumulative frequency of 22.
Knowing how data accumulates across ranges can provide insight into how any score compares to others. This builds the foundation for calculating percentiles effectively.
Steps to Percentile Calculation
Percentile ranks offer a way to understand where a score lies in relation to the entire data set. Essentially, it represents the percentage of scores falling below a particular score. The formula considers both cumulative frequency and the halfway mark of the current interval.
  • Formula: \( \text{Percentile Rank} = \left( \frac{\text{Cumulative Frequency below the score} + 0.5 \times \text{Frequency of the score}}{\text{Total Frequency}} \right) \times 100 \)
  • Application: For a score of 220, calculate it as \( \left(\frac{5 + 0.5 \times 17}{115}\right) \times 100 = 13.04\% \). This means 13.04% of scores are below 220.
This process helps relate individual scores to the bigger picture, giving them context within the overall performance of the group.
Score Range Analysis in Percentiles
Score Range Analysis is about linking given percentiles with the corresponding score range. This is done by identifying the cumulative frequency cut-off for each percentile. It translates percentile positions back to a practical score range interpretation.
  • Approach: Using cumulative frequency, identify where the percentile kind-of lands when viewed as a fraction of the total entries.
  • Example: The 15th percentile corresponds to about 17.25 cumulative frequency, placing it in the 217.5-238.5 score range.
This analysis reveals the practical score that aligns with specific percentiles, aiding in understanding what scores are necessary to be in certain performance brackets.

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