Using the sample data below, create a confidence interval for $\beta$ to see if there is evidence that there is a positive correlation between $x$ and $y$ with $\alpha =0.01$.

• A. $(-0.75,1.37)$; there is not enough evidence to suggest that there is a positive correlation between $x$ and $y$ with $\alpha=0.01$.
• B. $\left(-0.75,1.37\right)$; there is enough evidence to suggest that there is a positive correlation between $x$ and $y$ with $\alpha=0.01$.
• C. $(0.75,1.37)$; there is not enough evidence to suggest that there is a positive correlation between $x$ and $y$ with $\alpha=0.01$.
• D. $\left(0.75,1.37\right)$; there is enough evidence to suggest that there is a positive correlation between $x$ and $y$ with $\alpha=0.01$.

Answer: (A)

Verified step by step guidance

1. Step 1: Calculate the sample size \( n \) by counting the number of paired observations in the data. Here, count how many \( (x, y) \) pairs are given.
2. Step 2: Compute the sample means \( \bar{x} \) and \( \bar{y} \) by summing all the \( x \) values and dividing by \( n \), and similarly for the \( y \) values.
3. Step 3: Calculate the sample variances \( s_x^2 \) and \( s_y^2 \), and the sample covariance \( s_{xy} \) using the formulas: \[ \displaystyle s_x^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2 \] \[ \displaystyle s_y^2 = \frac{1}{n-1} \sum_{i=1}^n (y_i - \bar{y})^2 \] \[ \displaystyle s_{xy} = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y}) \]
4. Step 4: Calculate the slope estimate \( b \) of the regression line, which is the point estimate for \( \beta \), using: \[ \displaystyle b = \frac{s_{xy}}{s_x^2} \]
5. Step 5: Construct the confidence interval for \( \beta \) at the \( \alpha = 0.01 \) significance level using the formula: \[ \displaystyle b \pm t_{\alpha/2, n-2} \times SE_b \] where \( t_{\alpha/2, n-2} \) is the critical value from the t-distribution with \( n-2 \) degrees of freedom, and \( SE_b \) is the standard error of \( b \), calculated as: \[ \displaystyle SE_b = \frac{s}{\sqrt{\sum (x_i - \bar{x})^2}} \] with \( s \) being the standard error of the estimate from the residuals.