Answered

A data set comparing a woman's shoe size to her height is represented by the table.




Shoe Size		Height (inches)

7.5		63

8		70.5

12		68

7		71

9		69.5

10		70

12		75

12.5		$63^{-}$

14		72






Using technology, calculate the line of best fit. Identify and interpret the slope in this scenario.


AThe slope of the line of best fit is 0.25. Each time the shoe size increases by one, the height increases by 0.25 inches.
BThe slope of the line of best fit is 0.25. Each time the shoe size increases by 0.25, the height increases by one inch.
CThe slope of the line of best fit is 66.6. Each time the shoe size increases by one, the height increases by 66.6 inches.
DThe slope of the line of best fit is 66.6. Each time the shoe size increases by 66.6, the height increases by one inch.

Verified Answer :

  • The line of best fit \(y = mx + b\) is determined by the least squares method using the provided shoe size and height data.
  • The slope \(m\) is approximately \(0.25\), while the y-intercept \(b\) is approximately \(66.6\).
  • The slope of the line of best fit is \(0.25\), meaning that each time the shoe size increases by one, the height increases by \(0.25\) inches.
  • The correct option is the first one: \(\boxed{\text{The slope of the line of best fit is 0.25. Each time the shoe size increases by one, the height increases by 0.25 inches.}}\)

Explanation

  1. Data Analysis and Calculating Sums

To find the line of best fit \(y = mx + b\) for the given data, we first need to perform a regression analysis on the provided shoe size (\(x\)) and height (\(y\)) data.
The given data set consists of 9 pairs: \((7.5, 63), (8, 70.5), (12, 68), (7, 71), (9, 69.5), (10, 70), (12, 75), (12.5, 63), (14, 72)\).
First, we calculate the following necessary sums:

  • \(n = 9\)
  • \(\sum x = 92\)
  • \(\sum y = 622\)
  • \(\sum xy = 6370.5\)
  • \(\sum x^2 = 990.5\)
  1. Calculating the Line of Best Fit

Using the formula for the slope of the line of best fit \(m = \frac{n \sum xy - \sum x \sum y}{n \sum x^2 - (\sum x)^2}\):
\[m = \frac{9(6370.5) - (92)(622)}{9(990.5) - (92)^2}\]
\[m = \frac{57334.5 - 57224}{8914.5 - 8464} = \frac{110.5}{450.5} \approx 0.2453\]
Rounding to the nearest meaningful value provided in the options, the slope is \(0.25\).
Next, we calculate the y-intercept using the formula \(b = \frac{\sum y - m \sum x}{n}\):
\[b = \frac{622 - 0.2453(92)}{9} \approx \frac{622 - 22.5676}{9} \approx 66.6036\]
Rounding to the nearest meaningful value, we find \(b \approx 66.6\). The line of best fit is approximately \(y = 0.25x + 66.6\).

  1. Interpreting the Results

The slope of the line of best fit is approximately \(0.25\). In linear regression, the slope represents the change in the dependent variable (\(y\), height) for a one-unit change in the independent variable (\(x\), shoe size). Therefore, a slope of \(0.25\) means that for every increase of one shoe size unit, the predicted height increases by \(0.25\) inches. This corresponds to the first provided option.

Examples

In real-world data science, finding the line of best fit allows us to make predictions based on trends. For example, if a store owner wants to predict customer demand based on temperature, they can collect data on daily sales and outside temperatures. By calculating the line of best fit, they can estimate how much their sales might change if the temperature rises by 5 degrees, enabling better stock management and planning. Linear regression is a foundational tool for turning raw observations into actionable business insights.

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