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Chapter 3: Problem 18

Room and board costs for selected schools are summarized in this distribution. Find the approximate cost of room and board corresponding to each of the following percentiles by constructing a percentile graph. $$ \begin{array}{lr} \text { Costs (in dollars) } & \text { Frequency } \\ \hline 3000.5-4000.5 & 5 \\ 4000.5-5000.5 & 6 \\ 5000.5-6000.5 & 18 \\ 6000.5-7000.5 & 24 \\ 7000.5-8000.5 & 19 \\ 8000.5-9000.5 & 8 \\ 9000.5-10.000 .5 & 5 \end{array} $$ a. 30th percentile b. 50 th percentile c. 75 th percentile d. 90th percentile Using the same data, find the approximate percentile rank of each of the following costs. e. 5500 f. 7200 g. 6500 h. 8300

Short Answer

Expert verified
a. $5000.5-6000.5$; b. $6000.5-7000.5$; c. $7000.5-8000.5$; d. $8000.5-9000.5$. Percentage ranks: e. ~34.1; f. ~64.1; g. ~49.7; h. ~78.3.

Step by step solution

01

Calculate Cumulative Frequency

First, we need to determine the cumulative frequency for each interval. Add up the frequency values from the beginning up to the current interval to find the cumulative frequency. - For $3000.5-4000.5$: 5 - For $4000.5-5000.5$: 5 + 6 = 11 - For $5000.5-6000.5$: 11 + 18 = 29 - For $6000.5-7000.5$: 29 + 24 = 53 - For $7000.5-8000.5$: 53 + 19 = 72 - For $8000.5-9000.5$: 72 + 8 = 80 - For $9000.5-10.0005$: 80 + 5 = 85
02

Calculate Total Frequency

Add all the frequencies together to find the total number of observations, which is needed to calculate the percentile position.\[ 5 + 6 + 18 + 24 + 19 + 8 + 5 = 85 \]
03

Find Percentile Ranks for Specific Percentiles

We aim to find specific percentiles: P30, P50, P75, and P90 using the cumulative frequency list. Use the formula for finding the percentile rank position:\[ \text{Percentile Rank Position} = \frac{P}{100} \times \text{Total Frequency} \]- 30th percentile (P30): \( \frac{30}{100} \times 85 = 25.5 \)- 50th percentile (P50): \( \frac{50}{100} \times 85 = 42.5 \)- 75th percentile (P75): \( \frac{75}{100} \times 85 = 63.75 \)- 90th percentile (P90): \( \frac{90}{100} \times 85 = 76.5 \)
04

Determine Costs from Percentile Ranks

Using cumulative frequency, determine the cost interval for each percentile: - P30: Position 25.5 falls within 5000.5-6000.5 (cumulative frequency just over 25 is 29) - P50: Position 42.5 falls within 6000.5-7000.5 (cumulative frequency just over 42 is 53) - P75: Position 63.75 falls within 7000.5-8000.5 (cumulative frequency just over 63 is 72) - P90: Position 76.5 falls within 8000.5-9000.5 (cumulative frequency just over 76 is 80)
05

Calculate Percentile Rank for Given Costs

We have specific costs: 5500, 7200, 6500, 8300. Use the cumulative frequency distribution to calculate the percentile rank:- Cost 5500: Falls in interval 5000.5-6000.5. Cumulative frequency at the beginning of this class is 11. Rank = \( \frac{11 + \frac{(5500 - 5000.5)}{1000}}{85} \times 100 \approx 34.1 \)- Cost 7200: Falls in interval 7000.5-8000.5. Cumulative frequency at the beginning of this class is 53. Rank = \( \frac{53 + \frac{(7200 - 7000.5)}{1000}}{85} \times 100 \approx 64.1 \)- Cost 6500: Falls in interval 6000.5-7000.5. Cumulative frequency at the beginning of this class is 29. Rank = \( \frac{29 + \frac{(6500 - 6000.5)}{1000}}{85} \times 100 \approx 49.7 \)- Cost 8300: Falls in interval 8000.5-9000.5. Cumulative frequency at the beginning of this class is 72. Rank = \( \frac{72 + \frac{(8300 - 8000.5)}{1000}}{85} \times 100 \approx 78.3 \)

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Key Concepts

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

Cumulative Frequency
Cumulative frequency is an essential concept in data analysis, especially when constructing a cumulative frequency distribution. It helps you to understand how many observations lie below a particular value in a data set. To calculate cumulative frequency:
  • Start with the first frequency value in your frequency distribution table.
  • Add each subsequent frequency to the sum of the previous frequencies.
This step-by-step addition gives the cumulative frequency. This concept is useful for determining percentile ranks because it tells us the number of observations below a certain value, making it easier to understand the spread and distribution of data.
In our example, cumulative frequency helps us determine at what cost intervals certain percentiles (like the 30th or 50th) lie.
Frequency Distribution
Frequency distribution is a way of organizing data to show how often each value occurs in a data set. It helps to observe patterns, such as which intervals have the most or least observations.
  • In a frequency distribution table, values range over specific intervals, known as bins.
  • Each bin or interval has an associated frequency that indicates the number of observations within that range.
This concept is foundational for creating cumulative frequency distributions and identifying various percentiles. By looking at the frequency distribution of room and board costs, we can determine at which intervals the data is concentrated, such as the interval between $6000.5 and $7000.5 having the highest frequency, indicating many schools fall within this cost range.
Percentile Rank
The percentile rank of a data point tells you the position of that point relative to the entire data set. It is a measure that indicates the percentage of scores that a particular value is above.
  • To find the percentile rank, you first identify the total number of observations.
  • Determine how many observations fall below the data point in question.
  • Use the formula: \( \text{Percentile Rank} = \left( \frac{\text{Number of Scores Below} + \text{Fraction of the Interval}}{\text{Total Number of Observations}} \right) \times 100 \)
For given costs like \(5500 or \)7200, the percentile rank predicts their position in the list of sorted data. For instance, a cost of $5500 falls around the 34.1st percentile, meaning it exceeds just over 34% of the data.
Data Analysis
Data analysis is the process of examining, cleaning, and modeling data to discover useful information, draw conclusions, and support decision-making. It involves several steps, each with distinct purposes and techniques.
  • Exploratory Data Analysis (EDA) helps understand underlying patterns through charts, diagrams, and summary statistics.
  • Descriptive statistics like mean, median, mode, and standard deviation provide insights into data variations and patterns.
  • Data visualization effectively communicates findings, making abstract numbers tangible through visual means like graphs and histograms.
In our exercise, data analysis helps us not only determine specific percentiles but also understand room and board cost distributions across schools. Techniques like constructing frequency tables, calculating cumulative frequency, and determining percentile ranks are all part of a comprehensive data analysis strategy, enabling us to make data-driven conclusions.

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Most popular questions from this chapter

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