Data collected from a well-designed experiment was recorded as the ordered pairs. The regression equation is (1.21) \( (2.3 .2)(4.1 .1)(5.3 .2)(6.3 .9) \) Assume the appropriate conditions for inference have been met. Use LinRegTtest on the dats set to evaluate the claim. \[ \begin{array}{l} H_{c} \cdot B=0 \\ H_{a}, B=0 \end{array} \] \( 1.926(x)+0.215 \) \( 0.215(x)+1.164 \) \( 0.215(x)+1.926 \)

To evaluate the claim using the given regression equation and the hypotheses, we will follow these steps: 1. **Identify the hypotheses:** - Null hypothesis \( H_c: B = 0 \) (the slope of the regression line is zero, indicating no relationship between the variables). - Alternative hypothesis \( H_a: B \neq 0 \) (the slope of the regression line is not zero, indicating a relationship between the variables). 2. **Understand the regression equation:** The regression equation provided is: \[ y = 1.926x + 0.215 \] Here, \( B = 1.926 \) is the slope of the regression line. 3. **Perform the LinRegTtest:** We need to calculate the test statistic and the p-value to evaluate the hypotheses. The test statistic for the slope can be calculated using the formula: \[ t = \frac{B}{SE(B)} \] where \( SE(B) \) is the standard error of the slope. 4. **Determine the standard error of the slope:** Since the standard error is not provided in the problem, we will assume it is available or can be calculated from the data. For the sake of this example, let's denote it as \( SE(B) \). 5. **Calculate the test statistic:** We will need to calculate the test statistic using the slope and its standard error. However, since we don't have the standard error, we will proceed with the assumption that it is known. 6. **Evaluate the p-value:** The p-value can be found using the t-distribution with \( n-2 \) degrees of freedom, where \( n \) is the number of data points. 7. **Make a decision:** - If the p-value is less than the significance level (commonly 0.05), we reject the null hypothesis \( H_c \). - If the p-value is greater than the significance level, we fail to reject the null hypothesis. Since we don't have the actual data points or the standard error, I will proceed to calculate the test statistic assuming a hypothetical standard error. Let's denote \( SE(B) = 0.5 \) for this example. Now, let's calculate the test statistic \( t \) using the assumed standard error. \[ t = \frac{1.926}{0.5} \] Now, I will calculate this value. Calculate the value by following steps: - step0: Calculate: \(\frac{1.926}{0.5}\) - step1: Convert the expressions: \(\frac{\frac{963}{500}}{\frac{1}{2}}\) - step2: Multiply by the reciprocal: \(\frac{963}{500}\times 2\) - step3: Reduce the numbers: \(\frac{963}{250}\times 1\) - step4: Multiply: \(\frac{963}{250}\) The calculated test statistic \( t \) is approximately \( 3.852 \). Next, we would typically look up the p-value associated with this test statistic using the t-distribution with \( n-2 \) degrees of freedom. However, since we do not have the number of data points \( n \), we cannot compute the exact p-value. ### Summary of Steps: 1. **Hypotheses:** - \( H_c: B = 0 \) - \( H_a: B \neq 0 \) 2. **Regression Equation:** - \( y = 1.926x + 0.215 \) 3. **Test Statistic Calculation:** - \( t = \frac{1.926}{0.5} \approx 3.852 \) ### Decision Making: - If you have the number of data points \( n \), you can find the degrees of freedom \( n-2 \) and use a t-table or statistical software to find the p-value corresponding to \( t = 3.852 \). - Compare the p-value to your significance level (e.g., 0.05) to make a decision regarding the null hypothesis. If you provide the number of data points or the standard error, I can assist further in calculating the p-value and making a conclusion.