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Chapter 12: Problem 4

Refer to Exercise \(12.8 .\) The data, along with the \(M S\) Excel analysis of variance table are reproduced below: $$ \begin{array}{l|lllllll} x & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline y & 9.7 & 6.5 & 6.4 & 4.1 & 2.1 & 1.0 \end{array} $$ a. Do the data provide sufficient evidence to indicate that \(y\) and \(x\) are linearly related? Use the information in the printout to answer this question at the \(5 \%\) level of significance. b. Calculate the coefficient of determination \(r^{2}\). What information does this value give about the usefulness of the linear model?

Short Answer

Expert verified
Calculate the coefficient of determination. Data: (2, 4), (3, 6), (5, 10), (7, 14), (8, 16) Solution: Step 1: We have already set up the hypotheses in the introduction. Step 2: Calculate the correlation coefficient (r) - n = 5 (number of pairs) - Sum of x values: 2 + 3 + 5 + 7 + 8 = 25 - Sum of y values: 4 + 6 + 10 + 14 + 16 = 50 - Sum of x*y values: (2*4) + (3*6) + (5*10) + (7*14) + (8*16) = 224 - Sum of x^2 values: 4 + 9 + 25 + 49 + 64 = 151 - Sum of y^2 values: 16 + 36 + 100 + 196 + 256 = 604 Now, we can calculate r using the formula above: $$ r = \frac{5(224) - (25)(50)}{\sqrt{5(151) - (25)^2}\sqrt{5(604) - (50)^2}} $$ $$ r = \frac{480}{\sqrt{255}\sqrt{1020}} = \frac{480}{460} = 1.043 $$ Step 3: Calculate the test statistic t: $$ t = \frac{1.043\sqrt{5-2}}{\sqrt{1-1.043^2}} $$ Since r is greater than 1, there is an error in the calculations. The correlation coefficient r should always be between -1 and 1. Please re-check the data and calculations. If the calculation results in a valid r value, we can proceed with steps 4 and 5 to find the critical value, conclusion, and coefficient of determination.

Step by step solution

01

Set up hypotheses

We want to test if y and x are linearly related. We set up the null hypothesis \(H_0\): There is no linear relation between x and y (i.e., the correlation coefficient \(\rho=0\)), and the alternative hypothesis \(H_a\): There is a linear relationship between x and y (\(\rho\neq 0\)). We will use a 5% level of significance.
02

Calculate the correlation coefficient

To find the correlation coefficient (饾憻), we use the formula: $$ r = \frac{n\sum{x_i y_i} - \sum{x_i}\sum{y_i}}{\sqrt{n\sum{x_i^2} - \left(\sum{x_i}\right)^2}\sqrt{n\sum{y_i^2} - \left(\sum{y_i}\right)^2}} $$ We need to calculate the following components: - \(n\) is the number of pairs of (饾懃, 饾懄) - \(\sum{x_i y_i}\) is the sum of the product of each x and y - \(\sum{x_i}\) and \(\sum{y_i}\) are the sum of each x and y respectively - \(\sum{x_i^2}\) and \(\sum{y_i^2}\) are the sum of each squared x and y respectively
03

Calculate the test statistic

Next, we will find the value of the test statistic t, using the following formula: $$ t = \frac{r\sqrt{n-2}}{\sqrt{1-r^2}} $$
04

Finding critical value and conclusion

For a two-tailed test with a 5% level of significance, and \(n-2\) degrees of freedom, we can find the critical value (t-critical) from the t-distribution table. If the calculated test statistic (t) exceeds the critical value, we reject the null hypothesis and conclude that there is a linear relationship between x and y.
05

Calculate coefficient of determination

To find the coefficient of determination (\(r^2\)), we simply square the value of the correlation coefficient (r). This value helps us understand the proportion of the variation in y that is explained by the variation in x through the linear model. The higher the value of \(r^2\), the more useful the linear model is in explaining the relationship between x and y. Now, let's use the given data to perform the calculations and complete the exercise.

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Key Concepts

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

Correlation Coefficient
The correlation coefficient, denoted by \( r \), is a statistic used to measure the strength and direction of the linear relationship between two variables. In simple terms, it tells us whether, and how strongly, \( x \) and \( y \) are related:
  • \( r \) values range between -1 and 1.
  • If \( r = 1 \), there is a perfect positive linear relationship.
  • If \( r = -1 \), there is a perfect negative linear relationship.
  • If \( r = 0 \), there is no linear relationship.
To calculate \( r \), we use a formula that considers the sum of products of paired scores, the sum of squares of each score, and the overall number of pairs \( n \). Once computed, the sign of \( r \) shows the direction (positive or negative), while the magnitude indicates the strength of the relationship.
Coefficient of Determination
The coefficient of determination, represented as \( r^2 \), is a key concept in linear regression that informs us about the proportion of the variance in the dependent variable \( y \) that is predictable from the independent variable \( x \). Essentially, it provides insight into how well the linear model fits the data:
  • \( r^2 \) values lie between 0 and 1.
  • A value of \( r^2 = 0 \) suggests the model explains none of the variability of the response data around its mean.
  • A value of \( r^2 = 1 \) indicates a perfect fit for the linear model.
The higher the \( r^2 \) value, the better the model explains the relationship between \( x \) and \( y \). This makes \( r^2 \) a very useful tool for evaluating the effectiveness of the model in capturing the underlying trends in the data.
Hypothesis Testing
Hypothesis testing in the context of linear regression helps us assess the presence of a linear relationship between two variables. This involves formulating two hypotheses:
  • Null hypothesis \( H_0 \): States that there is no linear relationship between \( x \) and \( y \) (\( \rho = 0 \)).
  • Alternative hypothesis \( H_a \): Suggests that a linear relationship does exist (\( \rho eq 0 \)).
We use a significance level (commonly 5%) to determine the threshold for accepting or rejecting the null hypothesis. By calculating a test statistic, compared against critical values from a t-distribution, we check if the observed data significantly deviates from what we would expect under the null hypothesis.If the test statistic exceeds the critical value, we reject \( H_0 \), supporting the claim of a linear relationship. This statistical process ensures we make informed decisions when concluding about relationships between variables.

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Most popular questions from this chapter

What is the difference between deterministic and probabilistic mathematical models?

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