Statistics Correlation and Regression Examples

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Questions and Answers

What is the total value of $x^2$ from the provided table?

363

320

440 (correct)

400

Which year had the highest production in the sugar factory's data?

1973

1971

1974 (correct)

1970

How many years are represented in the production data for the sugar factory?

7 (correct)

From the table, which $xy$ value corresponds to the $x$ value of 9?

63 Signup and view all the answers

What is Karl Pearson's coefficient of correlation likely to measure in this data context?

The correlation between two sets of variables Signup and view all the answers

What is the correct value of $xy$ for the first entry in the table?

3808 Signup and view all the answers

How many students' marks are provided in the example for OS and DS?

9 Signup and view all the answers

What is represented by $x^2$ in the table?

The square of the X values Signup and view all the answers

In the context of Karl Pearson’s coefficient of correlation, what is the primary implication of a coefficient close to 1?

A perfect positive correlation between the variables Signup and view all the answers

What is the significance of calculating the ranks in two subjects for the marks obtained?

To analyze the relative performance of students in OS and DS Signup and view all the answers

What is the formula used to determine the number of ways to choose 2 socks from 4 socks?

$4C2$ Signup and view all the answers

What event represents drawing two defective chips from a total of 15, where exactly two are defective?

$5C2 * 10C1$ Signup and view all the answers

How many total ways are there to select 3 chips from a box of 15?

$15C3$ Signup and view all the answers

In the context of the horse race, what type of probability is being assessed when determining the chance of picking the winning horse?

Simple probability Signup and view all the answers

If 5 horses are in a race and a person picks 2 at random, what calculation would best represent the probability of selecting the winning horse?

$1/5$ Signup and view all the answers

What is the value of $a_0$ in the derived quadratic regression equation?

0.571429 Signup and view all the answers

Which variable represents the coefficient for the $x^2$ term in the quadratic equation?

a_2 Signup and view all the answers

What is the equation of the quadratic regression derived from the provided data?

y = 0.571429 + x + 1.107142 x^2 Signup and view all the answers

How many data points (N) were used in the quadratic regression example?

7 Signup and view all the answers

Which value represents the number of equations formed for the quadratic regression?

3 Signup and view all the answers

What does the variable 'd' represent in the Spearman's rank correlation table?

The difference between ranks Signup and view all the answers

In Spearman's rank correlation, what is the purpose of calculating $d^2$?

To minimize the differences Signup and view all the answers

What is the relationship between the values in columns 'x' and 'y' as presented?

They conform to a quadratic pattern Signup and view all the answers

Which of the following represents the maximum value of $y$ in the given data?

14 Signup and view all the answers

In the Spearman's rank calculation, what is 'n' when applying the formula?

10 Signup and view all the answers

What principle is used to determine the constant 'a' in the p.d.f. of a continuous random variable?

The sum of probabilities must equal 1. Signup and view all the answers

When computing P(X < 1.5) for a continuous random variable, which of the following is essential?

The total area under the p.d.f. curve up to 1.5. Signup and view all the answers

What scenario corresponds to finding P(1.5 < X < 2.5) for a random variable?

The area under the p.d.f. curve between 1.5 and 2.5. Signup and view all the answers

What is the probability of getting no fives when a die is rolled three times?

0.421875 Signup and view all the answers

If one five occurs in three rolls of a die, what is the corresponding probability?

0.417 Signup and view all the answers

What is the probability of getting three fives when rolling a die three times?

0.027 Signup and view all the answers

In a probability density function (p.d.f.), which statement is true regarding the total area under the curve?

It must equal one. Signup and view all the answers

How is the probability of an event computed using cumulative distribution function (CDF)?

It's the difference between probabilities at two points. Signup and view all the answers

Study Notes

Example 2.3

The example demonstrates calculating Karl Pearson's coefficient of correlation using a table of production data for a sugar factory over several years.
The table lists year, production, and calculated values for x2, y2, and xy for each year.
The formula for calculating the correlation coefficient is provided, but the actual calculations are not shown.

Example 2.7

The example demonstrates calculating Spearman's rank correlation coefficient using a table of marks obtained by 9 students in two subjects: OS and DS.
The table lists the marks in each subject, the rank for each student in each subject, the difference between the ranks (d), and the square of the difference (d2).
The formula for calculating the Spearman's rank correlation coefficient is provided, but the actual calculations are not shown.

Example 2.13

The example demonstrates computing a quadratic regression equation using a table of data points with corresponding x and y values.
The table includes x, y, x2, x3, x4, xy, and x2y for each data point.
Three equations are derived to solve for the coefficients a0, a1, and a2 in the quadratic regression equation.
The simultaneous equations for a0 and a2 are solved, and the solution for a1 is derived from a separate equation.
The final quadratic regression equation is presented in the form y = a2x2 + a1x + a0.

Example 3.4

The example demonstrates calculating the probability of drawing exactly two defective chips from a box containing 15 chips, 5 of which are defective.
The sample space (S) is defined as drawing 3 chips randomly from the box.
The event A is defined as drawing 3 chips where exactly 2 are defective.
The probability of event A is calculated using the formula: P(A) = (n(A)) / (n(S)).

Example 3.5

The example involves a horse race with 5 horses and a person (X) who bets on two horses chosen randomly.
The example does not include the calculations for determining the probabilities.

Example 4.15

The example describes a continuous random variable X with a probability density function (p.d.f.).
The p.d.f. is defined by the function: f(x) = ax, for 0 ≤ x ≤ 2.
The problem asks for determining the constant 'a', computing P(X < 1.5), and finding P(1.5 < X < 2.5).

Example 5.1

The example asks for the probability of the following events when rolling a die three times:
- No fives turning up
- One five turning up
- Three fives turning up
The example does not include the calculations for determining the probabilities.

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Description

This quiz explores examples of calculating correlation coefficients, including Karl Pearson's and Spearman's rank correlation, alongside quadratic regression equations. Each example is accompanied by data tables for better understanding. Test your knowledge of these statistical concepts by reviewing key formulas and calculations.