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Chapter 8: Problem 3

Incomplete Kitchens The U.S. census questionnaire defines kitchens with complete facilities as those having a sink with piped water, a range, and a refrigerator. Homes that lack complete kitchen facilities have been rare in the United States for many years. The first census for which data were tabulated on this subject was in \(1970 .\) The table shows the percentage of housing units lacking complete kitchen facilities in the western United States. Percent of Western U.S. Homes with Incomplete Kitchens $$ \begin{array}{|l|c|c|c|} \hline \text { Year } & 1970 & 1980 & 1990 \\ \hline \text { Homes (\%) } & 3 & 2 & 1 \\ \hline \end{array} $$ a. Use the method of least squares to find the multivariable function \(f\) with inputs \(a\) and \(b\) for the best-fitting line \(y=a x+b,\) where \(x\) is years since \(1970 .\) b. Calculate the minimum value of \(f(a, b)\). Explain what this minimum value indicates about the relationship between and the best-fitting line. c. Write the linear model that best fits these data. d. In what year does the best-fitting line predict that no housing units will lack complete kitchen facilities?

Short Answer

Expert verified
a = -0.1, b = 3; the least squares method predicts zero incomplete kitchens by 2000.

Step by step solution

01

Define the Variables

To find the best-fitting line using the least squares method, we first define the variables. Let \( x \) be the number of years since 1970, and let \( y \) be the percentage of homes with incomplete kitchens. The data points are: \((0, 3)\), \((10, 2)\), and \((20, 1)\).
02

Calculate Means of x and y

Calculate the mean of \( x \) and \( y \). The values of \( x \) are 0, 10, and 20 which gives \( \bar{x} = \frac{0 + 10 + 20}{3} = 10 \). The values of \( y \) are 3, 2, and 1 which gives \( \bar{y} = \frac{3 + 2 + 1}{3} = 2 \).
03

Calculate the Slope a

The formula for the slope \( a \) is \( a = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \). Substituting the appropriate values, \( a = \frac{(0-10)(3-2) + (10-10)(2-2) + (20-10)(1-2)}{(0-10)^2 + (10-10)^2 + (20-10)^2} = \frac{-10 - 10}{200} = -0.1 \).
04

Calculate the Intercept b

Using the formula \( b = \bar{y} - a \bar{x} \), we substitute the values found: \( b = 2 - (-0.1) \times 10 = 2 + 1 = 3 \).
05

Write the Linear Model

The equation of the best-fitting line is \( y = -0.1x + 3 \). This linear model best describes the relationship between the year since 1970 and the percentage of homes with incomplete kitchens.
06

Predict the Year with 0% Incomplete Kitchens

To find out when the percentage of homes reaches 0, set \( y = 0 \) and solve for \( x \): \( 0 = -0.1x + 3 \), which gives \( x = \frac{3}{0.1} = 30 \). This means in the year 1970 + 30 = 2000.

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Key Concepts

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

Incomplete Kitchen Statistics
Statistics from the U.S. Census Bureau have shown a clear downward trend in the prevalence of homes lacking complete kitchen facilities. Homes considered incomplete are those without a sink, piped water, a range, or a refrigerator. In 1970, about 3% of homes in the western United States were classified as having incomplete kitchens. By 1980, this figure fell to 2%, and by 1990, it dropped to 1%.

This data is useful for various stakeholders, including policymakers, housing developers, and social scientists. By understanding these trends, initiatives can be formulated to address such housing deficiencies. These statistics serve as a baseline for improving housing standards. Overall, the decline indicates a positive trend towards improving living conditions. By tracking these figures, it becomes easier to identify significant improvements or areas needing further attention over several decades.
Linear Regression
Linear regression is a statistical method used to model the relationship between two variables by fitting a linear equation to the observed data. In this exercise, we apply the least squares method to find the best-fitting straight line for the data about incomplete kitchens.

Our goal is to find a line in the form of \( y = ax + b \), where \( x \) represents years since 1970, and \( y \) represents the percentage of homes with incomplete kitchen facilities. To begin, calculate the means of \( x \) (years since 1970) and \( y \) (percentages):
  • Mean \(\bar{x} = 10\)
  • Mean \(\bar{y} = 2\)

Next, determine the slope \( a \) using the slope formula \( a = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \). With \( a = -0.1 \), the negative slope implies a decrease in the percentage of incomplete kitchens over time.

Finally, calculate the intercept \( b = \bar{y} - a \bar{x} \), giving us \( b = 3 \). Combining these, the linear equation is \( y = -0.1x + 3 \). This model effectively describes the trend of decreasing incomplete kitchens between 1970 and 1990.
Data Analysis in Education
Data analysis is a crucial component in educational settings, enabling educators and students to understand and interpret statistical results. It involves the systematic application of statistical techniques to draw conclusions from data. In educational contexts, understanding these methods empowers learners to apply them to real-world situations like housing trends.

In this particular exercise focusing on incomplete kitchen statistics, students utilize linear regression through the least squares method. They learn to transform raw data (percentages of homes with incomplete kitchens) into a meaningful equation \( y = -0.1x + 3 \), enabling predictions and insights.

Furthermore, data analysis skills such as these are transferable, giving students the ability to analyze trends and make informed decisions in various disciplines. By practicing these skills, students improve their critical thinking and data literacy, essential competences in today's data-driven world. This seamless integration of statistics into education ensures students are better equipped for future challenges.

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Most popular questions from this chapter

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