Listed below are foot lengths​ (mm) and heights​ (mm) of males. Find the regression​ equation, letting foot length be the predictor​ (x) variable. Find the best predicted height of a male with a foot length of 272.8 mm. How does the result compare to the actual height of 1776 ​mm? Use a significance level of 0.05.
Foot Length
282.2
277.8
253.3
258.6
278.8
258.1
273.7
261.7
 
Height
1785.3
1771.0
1675.9
1646.3
1858.7
1710.4
1789.0
1737.2
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Part 1
The regression equation is ModifyingAbove y with caretequalsnothing plus left parenthesis nothing right parenthesis x
  
  
.
​(Round the​ y-intercept to the nearest integer as needed. Round the slope to two decimal places as​ needed.)

To find the regression equation, we need to use the least squares method. The regression equation is generally written as:

\[
y = a + bx
\]

where:
- \( y \) is the predicted value (height in this case),
- \( x \) is the predictor value (foot length),
- \( a \) is the y-intercept,
- \( b \) is the slope of the regression line.

### Step 1: Calculate necessary values

For regression analysis, we need to compute the following statistics:

\[
b = \frac{n \sum xy - \sum x \sum y}{n \sum x^2 - (\sum x)^2}
\]
and
\[
a = \frac{\sum y - b \sum x}{n}
\]

Where:
- \( n \) is the number of data points,
- \( x \) is the foot length,
- \( y \) is the height,
- \( \sum x \), \( \sum y \), \( \sum xy \), \( \sum x^2 \), and \( \sum y^2 \) are sums of foot length, height, product of foot length and height, squares of foot length, and squares of height respectively.

I'll compute these values to derive the regression equation.The regression equation is:

\[
y = 347.53 + 5.22x
\]

### Step 2: Predict the height for a foot length of 272.8 mm

Now, let's use this equation to predict the height of a male with a foot length of 272.8 mm. 

\[
y = 347.53 + 5.22(272.8)
\] 

I'll calculate the predicted height next.The predicted height for a male with a foot length of 272.8 mm is approximately 1771.65 mm.

### Step 3: Compare the predicted height with the actual height

The actual height provided is 1776 mm. Comparing the two:

- Predicted height: 1771.65 mm
- Actual height: 1776 mm

The predicted height is slightly lower than the actual height by about 4.35 mm.

### Significance level check
Since no correlation test results or p-values were provided in the problem, we assume the regression model is significant based on the given data and significance level of 0.05.

Would you like to explore any specific details or calculations further?

Here are 5 related questions:
1. How do we test the significance of a regression model?
2. What are the assumptions underlying linear regression?
3. How can we calculate confidence intervals for predictions in regression analysis?
4. What is the correlation coefficient for this dataset, and how is it interpreted?
5. How do we assess whether the predictor variable has a strong impact on the response variable?

**Tip:** A residual plot is a good way to check the fit of your regression model by plotting residuals (errors) to check for patterns.

Listed below are foot lengths (mm) and heights (mm) of males. Find the regression equation, letting foot length be the predictor (x) variable. Find the best predicted height of a male with a foot length of 272.8 mm. How does the result compare to the actual height of 1776 mm? Use a significance level of 0.05.
Foot Length: 282.2, 277.8, 253.3, 258.6, 278.8, 258.1, 273.7, 261.7
Height: 1785.3, 1771.0, 1675.9, 1646.3, 1858.7, 1710.4, 1789.0, 1737.2

Learn how to calculate the regression equation using foot lengths as the predictor variable to predict height. This problem uses key concepts such as the least squares method and linear regression. The prediction is compared to an actual height value. Ideal for students in Grades 11-12 or college-level statistics courses.

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