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Chapter 3: Problem 12

The following data represent the weight (in grams) of a random sample of 25 Tylenol tablets. $$ \begin{array}{lllll} \hline 0.608 & 0.601 & 0.606 & 0.602 & 0.611 \\ \hline 0.608 & 0.610 & 0.610 & 0.607 & 0.600 \\ \hline 0.608 & 0.608 & 0.605 & 0.609 & 0.605 \\ \hline 0.610 & 0.607 & 0.611 & 0.608 & 0.610 \\ \hline 0.612 & 0.598 & 0.600 & 0.605 & 0.603 \\ \hline \end{array} $$ (a) Construct a box plot. (b) Use the box plot and quartiles to describe the shape of the distribution.

Short Answer

Expert verified
1. Organize data, 2. Calculate quartiles: Q1 = 0.603, Median = 0.608, Q3 = 0.610, 3. Min = 0.598, Max = 0.612, 4. Create the box plot, 5. Analyze shape: Symmetric distribution.

Step by step solution

01

- Organize the Data

Sort the data in ascending order to find the quartiles more easily.
02

- Calculate the Quartiles

Determine the first quartile (Q1), the median (Q2), and the third quartile (Q3).
03

- Determine Minimum and Maximum

Identify the minimum and maximum values from the dataset.
04

- Construct the Box Plot

Draw a number line that encompasses the minimum and maximum values identified. Then, plot Q1, Q2 (median), and Q3, and draw the box and whiskers to complete the plot.
05

- Analyze Shape of Distribution

Use the box plot and the quartiles to describe the shape of the distribution, including skewness and presence of outliers if any.

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Key Concepts

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

Box Plot
A box plot, also known as a box-and-whisker plot, visually displays the distribution of a dataset through five key summary statistics. These key points are the minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum. By offering a clear picture of the data's range, spread, and central tendency, box plots are essential tools in statistical data analysis.
To construct a box plot:
  • Draw a number line that includes the minimum and maximum values of your data set.
  • Plot Q1, the median (Q2), and Q3 as short vertical lines on the number line.
  • Draw a box between the Q1 and Q3 lines.
  • Add 'whiskers' by drawing lines from Q1 to the minimum value and from Q3 to the maximum value.
This visualization is particularly useful for identifying skewness and potential outliers in your data.
Quartiles
Quartiles are key values that divide a data set into four equal parts. These parts help us understand the distribution and spread of data. The first quartile, Q1, is the median of the first half of the data and marks the 25th percentile. The second quartile, Q2, is the overall median, marking the 50th percentile. The third quartile, Q3, is the median of the second half of the data, representing the 75th percentile.
To find the quartiles of a dataset:
  • Sort the data in ascending order.
  • Identify the median (Q2).
  • Determine Q1 by finding the median of the lower half of the data, excluding Q2 if the dataset has an odd number of elements.
  • Determine Q3 by finding the median of the upper half of the data, again excluding Q2 if the dataset has an odd number of elements.
These quartiles help build the box in a box plot and are crucial for summarizing data.
Distribution Shape
The shape of a distribution tells you how data are spread out along a number line. By using a box plot, we can get a quick visual interpretation of this shape.
The distribution can be:
  • Symmetrical: If the box plot is roughly equal on either side of the median.
  • Left-skewed: If the left whisker is longer than the right whisker, indicating that the data have a long left tail.
  • Right-skewed: If the right whisker is longer than the left whisker, indicating that the data have a long right tail.
Understanding the shape of the distribution is crucial for interpreting data and making further statistical inferences.
Skewness
Skewness refers to the asymmetry in the distribution of data. It helps us understand where most of the data points lie in relation to the mean and median.
Types of Skewness:
  • Symmetrical: Data points are evenly distributed around the mean.
  • Left-skewed (or negatively skewed): The left tail is longer, and the bulk of the data is concentrated on the right. The mean is typically less than the median.
  • Right-skewed (or positively skewed): The right tail is longer, and the bulk of the data is concentrated on the left. The mean is typically greater than the median.
By looking at the box plot, you can quickly assess the skewness of the data. For instance, if your median is closer to Q3 than Q1, the data is left-skewed.
Outliers Analysis
Outliers are data points that differ significantly from other observations. They can affect the mean and standard deviation, thus skewing the data.
To identify outliers using a box plot:
  • Calculate the Interquartile Range (IQR), which is Q3 - Q1.
  • Outliers are typically any data points that fall below Q1 - 1.5 * IQR or above Q3 + 1.5 * IQR.
Identifying outliers is essential because:
  • They could indicate variability in measurement.
  • They might be errors that need to be corrected.
  • They could provide insights into the peculiarities of the data set.
By carefully analyzing outliers, you can make more accurate and robust statistical analyses.

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Most popular questions from this chapter

The Empirical Rule The Stanford-Binet Intelligence Quotient (IQ) measures intelligence. IQ scores have a bellshaped distribution with a mean of 100 and a standard deviation of \(15 .\) (a) What percentage of people has an IQ score between 70 and \(130 ?\) (b) What percentage of people has an IQ score less than 70 or greater than \(130 ?\) (c) What percentage of people has an IQ score greater than \(130 ?\)

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Babies born after a gestation period of 32-35 weeks have a mean weight of 2600 grams and a standard deviation of 660 grams. Babies born after a gestation period of 40 weeks have a mean weight of 3500 grams and a standard deviation of 470 grams. Suppose a 34 -week gestation period baby weighs 2400 grams and a 40 -week gestation period baby weighs 3300 grams. What is the \(z\) -score for the 34 -week gestation period baby? What is the \(z\) -score for the 40 -week gestation period baby? Which baby weighs less relative to the gestation period?

In December \(2014,\) the average price of regular unleaded gasoline excluding taxes in the United States was \(\$ 3.06\) per gallon, according to the Energy Information Administration. Assume that the standard deviation price per gallon is \(\$ 0.06\) per gallon to answer the following. (a) What minimum percentage of gasoline stations had prices within 3 standard deviations of the mean? (b) What minimum percentage of gasoline stations had prices within 2.5 standard deviations of the mean? What are the gasoline prices that are within 2.5 standard deviations of the mean? (c) What is the minimum percentage of gasoline stations that had prices between \(\$ 2.94\) and \(\$ 3.18 ?\)

The following data represent the amount of time (in minutes) a random sample of eight students took to complete the online portion of an exam in Sullivan's Statistics course. Compute the mean, median, and mode time. $$ 60.5,128.0,84.6,122.3,78.9,94.7,85.9,89.9 $$
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