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Chapter 15: Problem 10

In \(9-14,\) find the median and the first and third quartiles for each set of data values. \(34,35,35,36,38,40,42,43,43,43,44,46,48,50\)

Short Answer

Expert verified
Median = 42.5, Q1 = 36, Q3 = 46.

Step by step solution

01

Arrange the Data

Ensure that the data set is in ascending order. The given data is: \[34, 35, 35, 36, 38, 40, 42, 43, 43, 43, 44, 46, 48, 50\]Since it is already sorted, we can move to the next step.
02

Find the Median

The median is the middle number of the sorted data set. Since there are 14 numbers (an even set), the median is the average of the 7th and 8th numbers in the data set:\[\text{Median} = \frac{42 + 43}{2} = 42.5\]
03

Identify the Lower Half

To find the quartiles, we first split the data into two halves. The lower half includes the first seven data points:\[34, 35, 35, 36, 38, 40, 42\]
04

Find the First Quartile (Q1)

The first quartile is the median of the lower half. There are 7 numbers:\[ \text{Q1} = 36 \]since 36 is the 4th number in this subset.
05

Identify the Upper Half

The upper half includes the last seven data points:\[43, 43, 43, 44, 46, 48, 50\]
06

Find the Third Quartile (Q3)

The third quartile is the median of the upper half. There are 7 numbers:\[ \text{Q3} = 46 \]since 46 is the 4th number in this subset.

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Key Concepts

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

Median
The median of a data set is an important statistic that represents the middle value when all data points are ordered from the smallest to largest. If the number of data points is odd, the median is the middle number. However, if the data set has an even number of points, such as in this exercise with 14 numbers, the median is calculated as the average of the two middle numbers.

In our example, the data set is:
  • 34
  • 35
  • 35
  • 36
  • 38
  • 40
  • 42
  • 43
  • 43
  • 43
  • 44
  • 46
  • 48
  • 50
To find the median, average the 7th and 8th values in the sorted list: that is, \ \( \text{Median} = \frac{42 + 43}{2} = 42.5 \ \). This means that half of the data points are below 42.5 and half are above.
Data Set
A data set is a collection of numbers or values that relate to a particular subject. In this example, our data set consists of 14 numbers. Presenting your data set in a clear and logical order is the first step towards any statistical analysis, like calculating the median or quartiles.

When working with data sets, remember to:
  • Arrange the data in ascending order for easier calculation of medians and quartiles.
  • Check for any outliers or anomalies, as they can affect statistical calculations.
  • Understand what each number represents to provide context to your analysis results.
Working with data sets is a fundamental skill in statistics, and organizing your data properly can make solving problems more straightforward and error-free.
First Quartile (Q1)
The first quartile, or Q1, represents the value below which 25% of the data falls. To find Q1 in a sorted data set, you focus on the lower half of data points. This helps to understand the distribution of data in the lower segment.

In our example, the lower half of the data includes:
  • 34
  • 35
  • 35
  • 36
  • 38
  • 40
  • 42
Since this half contains 7 numbers, Q1 is the 4th number, effectively dividing the lower segment into two equal parts. Therefore, we find: \ \( \text{Q1} = 36 \ \). This value, 36, shows that 25% of the data values are less than or equal to 36.
Third Quartile (Q3)
The third quartile, or Q3, marks the point below which 75% of the data falls, essentially demarcating the upper quarter of the data distribution. To determine Q3, we look exclusively at the upper half of the data set.

Considering our data set, the upper half is composed of the following numbers:
  • 43
  • 43
  • 43
  • 44
  • 46
  • 48
  • 50
This subdivision, like the lower half, also consists of 7 numbers. Consequently, Q3 is positioned at the 4th value, which is 46. Thus, \ \( \text{Q3} = 46 \ \), indicating that 75% of the population in the data set falls below this value.

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Most popular questions from this chapter

In \(15-19\) a. Draw a scatter plot for each data set. Based on the scatter plot, would the correlation coefficient be close to \(-1,0,\) or 1\(?\) Explain. c. Use a calculator to find the correlation coefficient for each set of data. The table below shows the five-day forecast and the actual high temperature for the fifth day over the course of 18 days. The temperature is given in degrees Fahrenheit.

In \(7-14,\) for each of the given correlation coefficients, describe the linear correlation as strong positive, moderate positive, none, moderate negative, or strong negative. \(r=0.3\)

In order to improve customer relations, an auto-insurance company surveyed 100 people to determine the length of time needed to complete a report form following an auto accident. The result of the survey is summarized in the following table showing the number of minutes needed to complete the form. Find the mean and median amount of time needed to complete the form. $$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|}\hline \text { Minutes } & {26-30} & {31-35} & {36-40} & {41-45} & {46-50} & {51-55} & {56-60} & {61-65} & {66-70} \\\ \hline \text { Frequency } & {2} & {8} & {12} & {15} & {10} & {24} & {26} & {1} & {2} \\ \hline\end{array} $$

The ages of all of the students in a science class are shown in the table. Find the variance and the standard deviation. \(\begin{array}{|c|c|}\hline \text { Age } & {\text { Frequency }} \\ \hline 18 & {1} \\ {17} & {2} \\ {16} & {9} \\ {15} & {9} \\ \hline\end{array}\)

In \(11-14,\) select the numeral that precedes the choice that best completes the statement or answers the question. The playing life of a Euclid mp3 player is normally distributed with a mean of \(30,000\) hours and a standard deviation of 500 hours. Matt's mp3 player lasted for \(31,500\) hours. His \(\mathrm{mp} 3\) player lasted longer than what percent of other Euclid mp3 players? $$ \begin{array}{llll}{\text { (1) } 68 \%} & {\text { (2) } 95 \%} & {\text { (3) } 99.7 \%} & {\text { (4) more than } 99.8 \%}\end{array} $$
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