https://file.aiquickdraw.com/mathbot/file/1725981034664m2yj28d8.jpg

Let's break down the problem and address each part:

### (i) Find the least squares regression line of mass on height.
The equation of the least squares regression line is:
\[ y = a + bx \]
Where:
- \( y \) is the dependent variable (mass),
- \( x \) is the independent variable (height),
- \( b \) is the slope of the line, and
- \( a \) is the y-intercept.

To find \( a \) and \( b \), we need to calculate the following:
- \( \bar{x} \): the mean of the heights,
- \( \bar{y} \): the mean of the masses,
- \( \sum x_i \), \( \sum y_i \), \( \sum x_i y_i \), \( \sum x_i^2 \), and the corresponding values to use in the formulas for slope \( b \) and intercept \( a \):
  \[
  b = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}
  \]
  \[
  a = \bar{y} - b\bar{x}
  \]

### (ii) Calculate the product moment correlation coefficient.
The product moment correlation coefficient, denoted by \( r \), is given by:
\[
r = \frac{n \sum xy - \sum x \sum y}{\sqrt{[n \sum x^2 - (\sum x)^2][n \sum y^2 - (\sum y)^2]}}
\]
Where:
- \( n \) is the number of data points,
- \( \sum x \), \( \sum y \), \( \sum xy \), \( \sum x^2 \), and \( \sum y^2 \) are the sums of the heights, masses, products, and squares of heights and masses respectively.

### (iii) Estimate the mass of a student if his height is 180 cm.
Using the regression equation from part (i), we can substitute \( x = 180 \) into the regression equation \( y = a + bx \) to estimate the mass.

### (iv) Rank the height and mass of the students and compute the Spearman’s rank correlation coefficient.
The Spearman's rank correlation coefficient is calculated using:
\[
r_s = 1 - \frac{6 \sum d_i^2}{n(n^2 - 1)}
\]
Where:
- \( d_i \) is the difference between the ranks of corresponding values of height and mass,
- \( n \) is the number of data points.

I will proceed by calculating the values for each part step by step. Let's first calculate the values for part (i), the least squares regression line.It seems like I can’t do more advanced data analysis right now. Please try again later. 

Would you like help understanding the steps involved, or do you want to proceed with manual calculations? Here are some related questions to consider:

1. How does the least squares regression line minimize errors in prediction?
2. Why is the product moment correlation coefficient useful in determining the strength of a linear relationship?
3. What are some limitations of using linear regression models in real-world data?
4. How is the Spearman's rank correlation different from the Pearson's correlation coefficient?
5. What factors could affect the reliability of predicting mass based on height?

**Tip**: Always double-check the data you input in formulas to avoid minor calculation errors, especially when handling large datasets!

A survey has been conducted to investigate whether the heights and masses of secondary school male students are related. Given the table of heights and masses of 7 students, find the least squares regression line, calculate the product moment correlation coefficient, estimate the mass of a student with a height of 180 cm, and compute Spearman's rank correlation coefficient.

In this problem, students are tasked with analyzing the relationship between the heights and masses of secondary school students. The solution includes calculating the least squares regression line, product moment correlation coefficient, and Spearman's rank correlation coefficient for 7 students. Suitable for Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation