91Ó°ÊÓ

The U.S. Census Bureau provides statistics on family life in the United States, including the age at the time of first marriage, current marital status, and size of household (U.S. Census Bureau website, March 20,2006 ). The following data show the age at the time of first marriage for a sample of men and a sample of women. $$\begin{array}{lcccccccc} \text { Men } & 26 & 23 & 28 & 25 & 27 & 30 & 26 & 35 & 28 \\ & 21 & 24 & 27 & 29 & 30 & 27 & 32 & 27 & 25 \\ \text { Women } & 20 & 28 & 23 & 30 & 24 & 29 & 26 & 25 & \\ & 22 & 22 & 25 & 23 & 27 & 26 & 19 & & \end{array}$$ a. Determine the median age at the time of first marriage for men and women. b. Compute the first and third quartiles for both men and women. c. Twenty-five years ago the median age at the time of first marriage was 25 for men and 22 for women. What insight does this information provide about the decision of when to marry among young people today?

Step by step solution

01

Organize the Data

Separate the data for men and women and list the ages in ascending order. - **Men:** 21, 23, 24, 25, 25, 26, 26, 27, 27, 27, 28, 28, 29, 30, 30, 32, 35 - **Women:** 19, 20, 22, 22, 23, 23, 24, 25, 25, 26, 26, 27, 28, 29, 30

02

Find the Median Age

To find the median, locate the middle value in each ordered list. - **Men:** There are 17 ages. The median is the 9th value: 27. - **Women:** There are 15 ages. The median is the 8th value: 25.

03

Compute the Quartiles for Men

To find the quartiles, locate the 25th and 75th percentiles. - **Men:** - **1st Quartile (Q1):** Median of the first 8 values: (24 + 25) / 2 = 24.5. - **3rd Quartile (Q3):** Median of the last 8 values: (28 + 29) / 2 = 28.5.

04

Compute the Quartiles for Women

Find the quartiles using the same method as for men. - **Women:** - **1st Quartile (Q1):** Median of the first 7 values: 22. - **3rd Quartile (Q3):** Median of the last 7 values: 27.

05

Analyze the Changes over 25 Years

Compare the current medians to those from 25 years ago. - **Men:** Median is now 27, up from 25. - **Women:** Median is now 25, up from 22. This suggests that both men and women are choosing to marry later than they did 25 years ago.

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Key Concepts

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

Median

The median is a simple yet powerful tool in descriptive statistics. It represents the middle value in a data set when the numbers are arranged in ascending or descending order.
For example, when finding the median age of men and women at their first marriage from the provided census data, we first sorted the ages.

Once sorted:

• For men, with 17 entries, the median is the 9th value, which is 27.
• For women, with 15 entries, the median is the 8th value, resulting in a median of 25.

The beauty of the median is its resistance to extreme values that might skew an average. It offers a clear picture of where the center of your data lies.

Quartiles

Quartiles are incredibly useful for understanding the spread and distribution of data. They divide the dataset into four equal parts.

• The first quartile (Q1), also called the lower quartile, is the median of the first half of the data.
• The third quartile (Q3), or the upper quartile, is the median of the second half of the data.

Let's look at the first and third quartiles for both men and women from the exercise:

• For men: With the data ordered, Q1 is the median of the first eight values: \[ Q1 = \frac{24 + 25}{2} = 24.5 \] And Q3 is the median of the last eight values:\[ Q3 = \frac{28 + 29}{2} = 28.5 \]
• For women: Q1 is the median of the first seven values, which is 22.Q3 is the median of the last seven values, which is 27.

Quartiles help detect outliers and comprehend the data others might miss, offering deep insights into variability.

Census Data Analysis

Census data analysis is key to understanding societal trends. By analyzing patterns like the age of marriage, we gain insight into shifting cultural norms and behaviors.

Looking at the historical comparison provided:

• 25 years ago, the median age for first marriages was 25 for men and 22 for women.
• Currently, the median is 27 for men and 25 for women.

This shift signifies a trend where individuals marry later in life.

This could reflect broader social changes, such as increased focus on career or education before marriage, or varying societal expectations and values. Understanding these trends through census data empowers policymakers and social scientists to make informed decisions.