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Chapter 2: Problem 71

Infant mortality Europe For Western Europe, the infant mortality rates reported by the Human Development Report 2006 were Sweden \(3,\) Finland \(3,\) Spain \(3,\) Belgium \(4,\) Denmark 4, France \(4,\) Germany \(4,\) Greece \(4,\) Italy \(4,\) Norway \(4,\) Portugal \(4,\) Netherlands \(5,\) Switzerland \(5,\) UK 5 . Show that \(Q 1=Q 2=Q 3=4\). (The quartiles, like the median, are less useful when the data are highly discrete.)

Short Answer

Expert verified
Quartiles are all 4 (\(Q_1 = 4, Q_2 = 4, Q_3 = 4\)).

Step by step solution

01

Organize the Data Set

First, organize the data in ascending order: 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5. This arrangement helps us visualize the data and locate the quartiles effectively.
02

Calculate the Position of Quartiles

To find the first quartile \(Q_1\), second quartile \(Q_2\) (median), and third quartile \(Q_3\), we use the formula to find the positions. With 15 points, use \( Q1 = \frac{1}{4}(n+1) \), \( Q2 = \frac{2}{4}(n+1) \), \( Q3 = \frac{3}{4}(n+1) \). For \(n = 15\), the positions are: \(Q1 = \frac{1}{4}(16) = 4\), \(Q2 = \frac{2}{4}(16) = 8\), \(Q3 = \frac{3}{4}(16) = 12\).
03

Identify the Quartile Values

From Step 1, identify \(Q_1, Q_2,\) and \(Q_3\) from the sorted list at the positions found in Step 2. The values at positions 4, 8, and 12 are 4, 4, and 4 respectively. Thus, \(Q_1 = 4\), \(Q_2 = 4\), and \(Q_3 = 4\).

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Key Concepts

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

Infant Mortality
Infant mortality refers to the death of infants before their first birthday. It is a crucial indicator of the overall health and wellbeing of a population. The rates are often expressed as the number of deaths per 1,000 live births. Here are some key factors influencing infant mortality:
  • Healthcare Access: Adequate access to healthcare for mothers and infants greatly reduces mortality rates.
  • Nutrition: Proper nutrition during pregnancy and infancy is vital for healthy development.
  • Socioeconomic Status: Higher income families generally have better access to resources that promote infant health.
In Western Europe, the numbers vary, but typically remain low compared to global averages. This is attributed to high-quality healthcare systems and supportive social programs. Understanding infant mortality helps nations develop strategies to save young lives and improve public health.
Quartiles
Quartiles are statistical tools used to divide a data set into four equal parts. They help in understanding the spread and skewness of data, and they are particularly useful in descriptive statistics. Let's discuss the three main quartiles:
  • First Quartile (Q1): This marks the 25th percentile. It is the median of the first half of the data set.
  • Second Quartile (Q2): Also known as the median, it is the midpoint of the data, dividing it into two equal parts. It corresponds to the 50th percentile.
  • Third Quartile (Q3): Denoting the 75th percentile, this is the median of the second half of the data set.
When data points are repetitive or highly discrete, like the infant mortality rates in the problem, multiple quartiles can have the same value. This shows that the quartiles may not always provide deep insights into such datasets, as was evident in the exercise where all quartiles equaled 4.
Data Analysis
Data analysis is the process of systematically applying statistical tools to describe, illustrate, and evaluate data. In the context of our infant mortality exercise, data analysis helps in making sense of the numbers. The goal is to extract meaningful insights for decision-making. Here鈥檚 how data analysis works:
  • Data Organization: First, organize raw data logically, making it easier to analyze, such as arranging infant mortality rates in ascending order.
  • Statistical Techniques: Use methods like calculating quartiles to understand the distribution and central tendencies of the data.
  • Interpretation: Analyze the results to draw conclusions. For instance, recognizing that quartiles may not be informative with highly discrete data.
  • Reporting: Present the analyzed data in a clear format to communicate findings effectively.
Effective data analysis is pivotal for uncovering trends and making predictions, particularly when planning healthcare interventions or evaluating public health outcomes.

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Most popular questions from this chapter

Infant mortality Africa The Human Development Report \(2006,\) published by the United Nations, showed infant mortality rates (number of infant deaths per 1000 live births) by country. For Africa, some of the values reported were: South Africa 54 , Sudan 63 , Ghana 68 , Madagascar 76 , Senegal 78 , Zimbabwe 79 , Uganda \(80,\) Congo 81 , Botswana 84, Kenya 96, Nigeria 101, Malawi 110 , Mali \(121,\) Angola \(154 .\) a. Find the first quartile (Q1) and the third quartile (Q3). b. Find the interquartile range (IQR). Interpret it.

Male heights According to a recent report from the U.S. National Center for Health Statistics, for males aged \(25-34\) years, \(2 \%\) of their heights are 64 inches or less, \(8 \%\) are 66 inches or less, \(27 \%\) are 68 inches or less, \(39 \%\) are 69 inches or less, \(54 \%\) are 70 inches or less, \(68 \%\) are 71 inches or less, \(80 \%\) are 72 inches or less, \(93 \%\) are74 inches or less, and \(98 \%\) are 76 inches or less. These are called cumulative percentages. a. Which category has the median height? Explain why. b. Nearly all the heights fall between 60 and 80 inches, with fewer than \(1 \%\) falling outside that range. If the heights are approximately bell-shaped, give a rough approximation for the standard deviation of the heights. Explain your reasoning.

Health insurance In \(2004,\) the five-number summary of positions for the distribution of statewide percentage of people without health insurance had a minimum of \(8.9 \%\) (Minnesota), \(\mathrm{Q} 1=11.6,\) Median \(=14.2,\) \(\mathrm{Q} 3=17.0,\) and maximum of \(25.0 \%\) (Texas) (Statistical Abstract of the United States, 2006 ). a. Do you think the distribution is symmetric, skewed right, or skewed left? Why? b. Which is most plausible for the standard deviation: \(-16,0,4,15,\) or \(25 ?\) Why? Explain what is unrealistic about the other values.

Female body weight The College Athletes data file on the text CD has data for 64 female college athletes. The data on weight (in pounds) are roughly bell shaped with \(\bar{x}=133\) and \(s=17\) a. Give an interval within which about \(95 \%\) of the weights fall. b. Identify the weight of an athlete who is three standard deviations above the mean in this sample. Would this be a rather unusual observation? Why?

Categorical or quantitative? Identify each of the following variables as categorical or quantitative. a. Number of children in family b. Amount of time in football game before first points scored c. College major (English, history, chemistry,...) d. Type of music (rock, jazz, classical, folk, other)
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