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Chapter 3: Problem 36

Consider a sample with data values of \(27,25,20,15,30,34,28,\) and \(25 .\) Provide the fivenumber summary for the data.

Short Answer

Expert verified
The five-number summary is: Minimum = 15, Q1 = 22.5, Median = 26, Q3 = 29, Maximum = 34.

Step by step solution

01

Organize the Data

First, list the data values in ascending order: 15, 20, 25, 25, 27, 28, 30, 34.
02

Identify the Minimum Value

The minimum value is the smallest value of the data set, which is 15.
03

Identify the Maximum Value

The maximum value is the largest value of the data set, which is 34.
04

Calculate the Median

The median is the middle number of the sorted data. Since there is an even number of data points (8), the median is the average of the 4th and 5th numbers: \( \frac{25 + 27}{2} = 26 \).
05

Determine the First Quartile (Q1)

The first quartile is the median of the first half of the data. The first half consists of \( 15, 20, 25, 25 \). The median of these numbers is \( \frac{20 + 25}{2} = 22.5 \).
06

Determine the Third Quartile (Q3)

The third quartile is the median of the second half of the data. The second half consists of \( 27, 28, 30, 34 \). The median of these numbers is \( \frac{28 + 30}{2} = 29 \).

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Key Concepts

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

Data analysis
Data analysis involves examining and organizing data to uncover useful information, suggest conclusions, and support decision-making. For the given dataset with values such as \(27, 25, 20\), and others, data analysis helps in extracting meaningful insights.

Analyzing a dataset often starts with organizing the data. It's crucial to arrange data in a logical order, as seen in the step-by-step solution where the data was sorted in ascending order.

The steps to analyze this data include:
  • Listing data in order, which aids in visual clarity.
  • Identifying crucial statistics like the minimum and maximum values, which help define the range of your data.
  • Grouping data into smaller sets helps in delving deeper into details, like finding median values or other quantiles.
This process provides a clearer picture and prepares the data for further statistical description.
Descriptive statistics
Descriptive statistics summarize and provide information about your data in a comprehensive way.

Using descriptive statistics involves calculating meaningful metrics to summarize the dataset. In our example, this includes:
  • Minimum and Maximum: These values (15 and 34) indicate the smallest and largest observations.
  • Median: Splitting the ordered dataset in half, the median (26) helps understand the center point of the data.
Descriptive statistics like these allow you to understand your data distribution at a glance and make it easier to compare to other datasets. These statistics form the basis of more complex analyses, making them a foundational tool in data description.
Quartiles
Quartiles are essential elements in statistics used to summarize a data set by dividing it into four equal parts. Each quartile represents a portion of the dataset. They provide insight into the spread and variability of the data.

For instance, in the given exercise, the dataset was divided into two main halves:
  • First Quartile (Q1): This is the median of the first half. A value of 22.5 indicates that 25% of the data lies below this value.
  • Third Quartile (Q3): This median of the latter half, calculated at 29, indicates that 75% of the data falls below this mark.
Utilizing quartiles provides a more detailed understanding of the data spread than simply knowing the minimum, median, and maximum. It helps identify possible outliers and the overall dispersion within the dataset. Recognizing these parts can lead to deeper insights about the dataset's structure, helping in identifying skewness or any central tendency within the data.

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Most popular questions from this chapter

Bloomberg Personal Finance (July/\mathrm{ August } 2 0 0 1 ) \text { included the following companies in its } recommended investment portfolio. For a portfolio value of \(\$ 25,000,\) the recommended dollar amounts allocated to each stock are shown. $$\begin{array}{lccc} \text { Company } & \begin{array}{c} \text { Portfolio } \\ \text { (\$) } \end{array} & \begin{array}{c} \text { Estimated Growth Rate } \\ \text { (\%) } \end{array} & \begin{array}{c} \text { Dividend Yield } \\ \text { (\%) } \end{array} \\ \text { Citigroup } & 3000 & 15 & 1.21 \\ \text { General Electric } & 5500 & 14 & 1.48 \\ \text { Kimberly-Clark } & 4200 & 12 & 1.72 \\ \text { Oracle } & 3000 & 0.00 & 0.96 \\ \text { Pharmacia } & 3000 & 20 & 2.48 \\ \text { SBC Communications } & 3800 & 12 & 0.00 \\ \text { WorldCom } & 2500 & 35 & 0.00 \\ \hline\end{array}$$ a. Using the portfolio dollar amounts as the weights, what is the weighted average estimated growth rate for the portfolio? b. What is the weighted average dividend yield for the portfolio?

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A bowler's scores for six games were \(182,168,184,190,170,\) and \(174 .\) Using these data as a sample, compute the following descriptive statistics. a. Range c. Standard deviation b. Variance d. Coefficient of variation

The grade point average for college students is based on a weighted mean computation. For most colleges, the grades are given the following data values: \(A(4), B(3), C(2)\) \(\mathrm{D}(1),\) and \(\mathrm{F}(0)\). After 60 credit hours of course work, a student at State University earned 9 credit hours of \(A, 15\) credit hours of \(B, 33\) credit hours of \(C,\) and 3 credit hours of \(D\) a. Compute the student's grade point average. b. Students at State University must maintain a 2.5 grade point average for their first 60 credit hours of course work in order to be admitted to the business college. Will this student be admitted?

A department of transportation's study on driving speed and mileage for midsize automobiles resulted in the following data. $$\begin{array}{l|llllllllll} \text { Driving Speed } & 30 & 50 & 40 & 55 & 30 & 25 & 60 & 25 & 50 & 55 \\ \hline \text { Mileage } & 28 & 25 & 25 & 23 & 30 & 32 & 21 & 35 & 26 & 25 \end{array}$$ Compute and interpret the sample correlation coefficient.
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