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

Welcome back. In this problem, we'd like to use sample data below to create a confidence interval for beta, the slope of the population's regression line, to see if there's evidence that there is a positive correlation between x and y with alpha equals 0.01. So let's start by finding our confidence level for this confidence interval, which will be one minus alpha. Because alpha is point zero one, this is one minus point zero one, which is point nine nine. So once we build our confidence interval for beta, we'll be 99% certain that the actual value for beta or that actual slope for a population's regression line will be inside that confidence interval. So if the confidence interval only includes positive values, we'd be 99% sure that beta would actually be positive, and that would be enough evidence to suggest positive linear correlation between x and y. So what we're trying to do in this problem is see if the confidence interval is entirely positive. Alright. Let's pull out our TI 80 fours to create that confidence interval. We're gonna start by clicking on stat and clicking on enter and then putting in our data, making sure that our data for x is in l one and our data for y is in l two. As you can see, I've already done that, but definitely pause the video, take a second to do it on your own. And once that's all set, that's step one done, and we're ready to move on to step two. Clicking on the stat button, scrolling to the right to the test menu and scrolling down to option g, the lin reg t int. Once there, we'll click enter and that's step two. All set. For step three, we just wanna make sure we have the correct information in our menu. We need our x list to be l one where we put the x data and our y list to be l two where we put the y data. We want the frequency at one, which is the default, and we need to change our c level to match the problem, which in this case should be point nine nine. We can skip reg e q and scroll down to calculate and click enter. And that's step three. All set. Our confidence interval will appear in the viewing window, and we can see that it stretches from approximately negative 0.75 to about 1.37. So as we can see, there is a negative lower bound, which means that our interval includes some negatives and zero. So it's not entirely positive. So we can say, no, there is not enough evidence to suggest that there is a positive correlation between x and y. Great work.

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Multiple Choice

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

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

$$\left$(-0.75,1.37$\right$)$ ; there is enough evidence to suggest that there is a positive correlation between $x$ and $y$ with $$\alpha$=0.01$ .

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

$$\left$(0.75,1.37$\right$)$ ; there is enough evidence to suggest that there is a positive correlation between $x$ and $y$ with $$\alpha$=0.01$ .

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Verified step by step guidance

Step 1: Calculate the means of the sample data for both variables $x$ and $y$. Use the formulas: \[\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i \quad \text{and} \quad \bar{y} = \frac{1}{n} \sum_{i=1}^n y_i\] where $n$ is the number of data points.

Step 2: Compute the slope estimate $\hat{\beta}$ of the regression line, which represents the estimated correlation between $x$ and $y$. Use the formula: \[\hat{\beta} = \frac{\sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y})}{\sum_{i=1}^n (x_i - \bar{x})^2}\]

Step 3: Calculate the standard error of the slope estimate $SE_{\hat{\beta}}$. First, find the residual sum of squares (RSS) by computing the differences between observed $y_i$ and predicted values $\hat{y}_i = \hat{\beta} x_i + \hat{\alpha}$, then use: \[SE_{\hat{\beta}} = \sqrt{\frac{\text{RSS} / (n-2)}{\sum_{i=1}^n (x_i - \bar{x})^2}}\] where $\hat{\alpha}$ is the intercept estimate.

Step 4: Determine the critical value $t^*$ from the $t$-distribution for a two-tailed test with significance level $\alpha = 0.01$ and degrees of freedom $df = n - 2$.

Step 5: Construct the confidence interval for $\beta$ using the formula: \[\hat{\beta} \pm t^* \times SE_{\hat{\beta}}\] Interpret the interval to check if it contains zero. If zero is included, there is not enough evidence to conclude a positive correlation at the $\alpha=0.01$ significance level.