Math / Data & Statistics

QuestionQuestion 9 of 10 (1 point) I Question Attempt 3 of Unlimited
Fast reactions: In a study of reaction times, the time to respond to a visual stimulus $(x)$ and the time to respond to an auditory stimulus ( $y$ ) were recorded for each of 7 subjects. Times were measured in thousandths of a second. The results are presented in the following table. \begin{tabular}{cc} \hline Visual & Auditory \\ \hline 161 & 159 \\ 176 & 163 \\ 178 & 201 \\ 188 & 193 \\ 201 & 197 \\ 203 & 206 \\ 211 & 189 \\ \hline \end{tabular} Send data to Excel
The least-squares regression line $\hat{y}=b_{0}+b_{1} x=51.2341+0.7203 x$ and $\Sigma(x-\bar{x})^{2}=1895.4286$ are known for this data. Construct a $95 \%$ confidence interval for the slope. Round the answers to at least four decimal places.
The $95 \%$ confidence interval is $\square$ $<\beta_{1}<$ $\square$ 1. Save For Later Submit Assig

Studdy Solution

STEP 1

1. The data consists of paired observations for visual and auditory reaction times.
2. The least-squares regression line is given by $\hat{y} = 51.2341 + 0.7203x$ .
3. The sum of squares of the deviations from the mean for $x$ is $\Sigma(x-\bar{x})^{2} = 1895.4286$ .
4. We need to construct a 95% confidence interval for the slope $\beta_1$ .

STEP 2

1. Calculate the standard error of the slope.
2. Determine the critical value for the t-distribution.
3. Construct the confidence interval for the slope.

STEP 3

Calculate the standard error of the slope ( $SE_{b_1}$ ) using the formula:
$SE_{b_1} = \sqrt{\frac{\Sigma(y_i - \hat{y}_i)^2}{(n-2) \cdot \Sigma(x-\bar{x})^2}}$
We need to calculate $\Sigma(y_i - \hat{y}_i)^2$ . First, compute the predicted values $\hat{y}_i$ for each $x_i$ , and then find the residuals $y_i - \hat{y}_i$ .

STEP 4

Calculate the predicted values $\hat{y}_i$ using the regression equation for each $x_i$ :
\begin{align} \hat{y}_1 &= 51.2341 + 0.7203 \times 161 \\ \hat{y}_2 &= 51.2341 + 0.7203 \times 176 \\ \hat{y}_3 &= 51.2341 + 0.7203 \times 178 \\ \hat{y}_4 &= 51.2341 + 0.7203 \times 188 \\ \hat{y}_5 &= 51.2341 + 0.7203 \times 201 \\ \hat{y}_6 &= 51.2341 + 0.7203 \times 203 \\ \hat{y}_7 &= 51.2341 + 0.7203 \times 211 \\ \end{align}
Calculate each $\hat{y}_i$ .

STEP 5

Calculate the residuals $y_i - \hat{y}_i$ and then $\Sigma(y_i - \hat{y}_i)^2$ .

STEP 6

Substitute $\Sigma(y_i - \hat{y}_i)^2$ and $\Sigma(x-\bar{x})^2$ into the formula for $SE_{b_1}$ .

STEP 7

Determine the critical value $t^*$ for a 95% confidence interval with $n-2$ degrees of freedom (where $n = 7$ ).

STEP 8

Construct the confidence interval for the slope $\beta_1$ using:
$b_1 \pm t^* \cdot SE_{b_1}$
Substitute the values for $b_1$ , $t^*$ , and $SE_{b_1}$ .
The 95% confidence interval for the slope $\beta_1$ is:
$\boxed{\text{Lower bound}} < \beta_1 < \boxed{\text{Upper bound}}$

Was this helpful?

Studdy Solution

STEP 1

STEP 2

1. Calculate the standard error of the slope.
2. Determine the critical value for the t-distribution.
3. Construct the confidence interval for the slope.

STEP 3

STEP 4

STEP 5

Calculate the residuals $y_i - \hat{y}_i$ and then $\Sigma(y_i - \hat{y}_i)^2$ .

STEP 6

Substitute $\Sigma(y_i - \hat{y}_i)^2$ and $\Sigma(x-\bar{x})^2$ into the formula for $SE_{b_1}$ .

STEP 7

Determine the critical value $t^*$ for a 95% confidence interval with $n-2$ degrees of freedom (where $n = 7$ ).

STEP 8

Construct the confidence interval for the slope $\beta_1$ using:
$b_1 \pm t^* \cdot SE_{b_1}$
Substitute the values for $b_1$ , $t^*$ , and $SE_{b_1}$ .
The 95% confidence interval for the slope $\beta_1$ is:
$\boxed{\text{Lower bound}} < \beta_1 < \boxed{\text{Upper bound}}$

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Studdy Solution

STEP 1

STEP 2

1. Calculate the standard error of the slope.2. Determine the critical value for the t-distribution.3. Construct the confidence interval for the slope.

STEP 3

STEP 4

STEP 5

Calculate the residuals yi−y^iy_i - \hat{y}_iyi​−y^​i​ and then Σ(yi−y^i)2\Sigma(y_i - \hat{y}_i)^2Σ(yi​−y^​i​)2.

STEP 6

Substitute Σ(yi−y^i)2\Sigma(y_i - \hat{y}_i)^2Σ(yi​−y^​i​)2 and Σ(x−xˉ)2\Sigma(x-\bar{x})^2Σ(x−xˉ)2 into the formula for SEb1SE_{b_1}SEb1​​.

STEP 7

Determine the critical value t∗t^*t∗ for a 95% confidence interval with n−2n-2n−2 degrees of freedom (where n=7n = 7n=7).

STEP 8

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. Calculate the standard error of the slope.
2. Determine the critical value for the t-distribution.
3. Construct the confidence interval for the slope.

Calculate the residuals $y_i - \hat{y}_i$ and then $\Sigma(y_i - \hat{y}_i)^2$ .

Substitute $\Sigma(y_i - \hat{y}_i)^2$ and $\Sigma(x-\bar{x})^2$ into the formula for $SE_{b_1}$ .

Determine the critical value $t^*$ for a 95% confidence interval with $n-2$ degrees of freedom (where $n = 7$ ).