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Chapter 2: Problem 97

Use the following information to answer the next nine exercises: The population parameters below describe the full-time equivalent number of students (FTES) each year at Lake Tahoe Community College from 1976鈥�1977 through 2004鈥�2005. \(\bullet \mu=1000 \mathrm{FTES}\) \(\bullet\) median \(=1,014 \mathrm{FTES}\) \(\bullet \quad \sigma=474 \mathrm{FTES}\) \(\cdot\) first quartile \(=528.5\) FTES \(\cdot\) third quartile \(=1,447.5\) FTES \(\cdot n=29\) years What percent of the FTES were from 528.5 to 1447.5? How do you know?

Short Answer

Expert verified
50% of the FTES are between 528.5 and 1447.5 because this range represents the interquartile range, covering the middle 50% of the data.

Step by step solution

01

Understanding the Question

We need to determine the percentage of full-time equivalent students (FTES) that fall between the first quartile (528.5 FTES) and the third quartile (1447.5 FTES).
02

Recall Quartile Range

The range between the first quartile ( Q1 ) and the third quartile ( Q3 ) is called the interquartile range (IQR). It covers the middle 50% of the data in a dataset.
03

Determine Percentage from Quartile Information

Since quartiles split the data into four equal parts, 25% of the data lies between each section. Therefore, the range from the first quartile to the third quartile represents the middle 50% of the data.

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Key Concepts

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

Quartiles
When analyzing data, quartiles act as tools that divide a dataset into four equal parts. Splitting data like this helps us gain insight into how the values are spread across the dataset. Here's how it works:
  • First Quartile (Q1): Represents 25% of the data. This is the point below which 25% of the data falls.
  • Second Quartile: Also known as the median, splits the dataset into two equal parts.
  • Third Quartile (Q3): Marks the 75% mark, below which 75% of the data is found.
  • Interquartile Range (IQR): This is calculated by subtracting the first quartile from the third quartile (IQR = Q3 - Q1). The IQR shows the range within which the central 50% of the data points lie. Specifically, for Lake Tahoe Community College's FTES data, the IQR is 1,447.5 - 528.5.
Understanding quartiles and the IQR is invaluable in identifying the spread and central tendency of the dataset.
Median
In statistics, the median is a measure of central tendency that divides a dataset into two equal halves. Since it lies in the middle of an ordered dataset, it is less impacted by outliers or extreme values, making it a more robust indicator compared to the mean. For example, in the context of the full-time equivalent students at Lake Tahoe Community College, the median is 1,014 FTES. This means that if we were to list all the annual FTES numbers from smallest to largest, the median would be the middle number in this sequence when the data is ordered.
  • The median ensures that 50% of the data lies below it and 50% lies above it, giving a clear central point about the dataset's distribution.
    • This makes the median particularly useful in understanding a dataset where the average might be skewed by unusual values.
    Population Parameters
    Population parameters are numerical values that summarize data for an entire population. These measurements offer a broad understanding of various aspects of a dataset without examining every single individual in the population.
    • Mean (\(\mu\)): The arithmetic average of all data points, providing an overall sense of the center of data. For the FTES at Lake Tahoe Community College, the mean FTES was 1,000.
    • Standard Deviation (\(\sigma\)): Indicates data variability, showing how much the values deviate from the mean. Here, the FTES standard deviation was 474, meaning there was a significant range of deviation from the average yearly FTES.
    • Quartiles and Median: As discussed, these define "cut-off points" that segment the data into manageable parts.
    Using population parameters like these allows researchers to derive meaningful conclusions about the overall structure and characteristics of a large dataset such as FTES.
    Full-time Equivalent Students (FTES)
    Full-time equivalent students (FTES) is a measure used in educational institutions to signify the number of full-time students in a more manageable format. Instead of just counting enrolled students, FTES takes into account both full-time and part-time students, rendering a more accurate representation of student attendance and workload. For instance, at Lake Tahoe Community College:
    • Each part-time student's workload is converted into a fraction of a full-time student.
    • The total FTES number provides a clear understanding of the actual educational load and resources needed.
    • Helping institutions allocate resources, analyze student numbers precisely, and plan accordingly becomes simpler with FTES.
    The FTES measure offers a fair perspective, enabling better decision-making and strategic planning, providing vital insights into student demographics across several academic years.

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    Most popular questions from this chapter

    One hundred teachers attended a seminar on mathematical problem solving. The attitudes of a representative sample of 12 of the teachers were measured before and after the seminar. A positive number for change in attitude indicates that a teacher's attitude toward math became more positive. The 12 change scores are as follows: 3; 8; 鈥�1; 2; 0; 5; 鈥�3; 1; 鈥�1; 6; 5; 鈥�2 a. What is the mean change score? b. What is the standard deviation for this population? c. What is the median change score? d. Find the change score that is 2.2 standard deviations below the mean.

    a. For runners in a race, a low time means a faster run. The winners in a race have the shortest running times. Is it more desirable to have a finish time with a high or a low percentile when running a race? b. The 20th percentile of run times in a particular race is 5.2 minutes. Write a sentence interpreting the 20th percentile in the context of the situation. c. A bicyclist in the 90th percentile of a bicycle race completed the race in 1 hour and 12 minutes. Is he among the fastest or slowest cyclists in the race? Write a sentence interpreting the 90th percentile in the context of the situation.

    In a recent issue of the IEEE Spectrum, 84 engineering conferences were announced. Four conferences lasted two days. Thirty-six lasted three days. Eighteen lasted four days. Nineteen lasted five days. Four lasted six days. One lasted seven days. One lasted eight days. One lasted nine days. Let \(X=\) the length (in days) of an engineering conference. a. Organize the data in a chart. b. Find the median, the first quartile, and the third quartile. c. Find the 65th percentile. d. Find the 10th percentile. e. Construct a box plot of the data. f. The middle 50\(\%\) of the conferences last from _____ days to _____ days. g. Calculate the sample mean of days of engineering conferences. h. Calculate the sample standard deviation of days of engineering conferences. i. Find the mode. j. If you were planning an engineering conference, which would you choose as the length of the conference: mean; median; or mode? Explain why you made that choice. k. Give two reasons why you think that three to five days seem to be popular lengths of engineering conferences.

    Use the following information to answer the next three exercises: State whether the data are symmetrical, skewed to the left, or skewed to the right. When the data are symmetrical, what is the typical relationship between the mean and median?

    Use the following information to answer the next three exercises: State whether the data are symmetrical, skewed to the left, or skewed to the right. 87; 87; 87; 87; 87; 88; 89; 89; 90; 91
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