The accompanying table shows the average SAT Math and Critical Reading scores for students in a sample of states. A scatterplot for these two variables suggests a linear trend. Complete parts​ (a) through​ (c) below.

SAT Math 

Score

SAT Critical Reading Score

575575

579579

504504

499499

485485

490490

592592

588588

461461

462462

494494

494494

487487

488488

563563

557557

440440

441441

506506

495495

587587

582582

450450

463463

597597

592592

563563

580580

605605

591591

599599

596596

524524

523523

473473

468468

600600

589589

510510

502502

544544

551551

530530

525525

508508

487487

494494

494494

463463

467467

503503

509509

486486

470470

502502

489489

504504

498498

Question content area bottom

Part 1

a. Find and report the value for the correlation coefficient and the regression equation for predicting the math score from the critical reading score. Then find the predicted math score for a state with a critical reading score of 561561.

The correlation coefficient is enter your response here.

​(Round to three decimal places as​ needed.)

To complete part (a), we need to calculate both the correlation coefficient and the regression equation. Here's the step-by-step process:

### Steps:
1. **Correlation Coefficient**: This measures the strength and direction of a linear relationship between two variables (Math score and Critical Reading score). The formula for the correlation coefficient \( r \) is:

\[
r = \frac{n \sum xy - \sum x \sum y}{\sqrt{[n \sum x^2 - (\sum x)^2][n \sum y^2 - (\sum y)^2]}}
\]

Where:
- \( x \) is the Critical Reading score,
- \( y \) is the Math score,
- \( n \) is the number of data points.

2. **Regression Equation**: The equation of the regression line is given by:

\[
\hat{y} = b_0 + b_1 x
\]

Where:
- \( \hat{y} \) is the predicted Math score,
- \( x \) is the Critical Reading score,
- \( b_1 \) is the slope of the regression line, calculated by:

\[
b_1 = \frac{n \sum xy - \sum x \sum y}{n \sum x^2 - (\sum x)^2}
\]

- \( b_0 \) is the y-intercept, calculated by:

\[
b_0 = \bar{y} - b_1 \bar{x}
\]

Where \( \bar{x} \) and \( \bar{y} \) are the means of the Critical Reading and Math scores, respectively.

3. **Predicted Math Score**: Once we have the regression equation, we can substitute the given Critical Reading score (561) into the equation to find the predicted Math score.

Let me calculate the correlation coefficient, the regression equation, and the predicted Math score.### Results:

- The **correlation coefficient** is approximately \( 0.986 \), indicating a very strong positive linear relationship between SAT Math and Critical Reading scores.
  
- The **regression equation** for predicting the Math score based on the Critical Reading score is:

\[
\hat{y} = 0.766 + 1.004x
\]

Where \( x \) is the Critical Reading score.

- For a Critical Reading score of 561, the **predicted Math score** is approximately \( 563.92 \).

Would you like more details on these calculations, or any other clarifications?

Here are some follow-up questions:
1. What does a correlation coefficient close to 1 imply about the relationship between the variables?
2. How would you interpret the slope of the regression equation?
3. What impact would outliers have on this regression model?
4. Could we use this regression model to predict scores for a Critical Reading score outside the range of our data? Why or why not?
5. How would you assess the accuracy of the prediction made by this regression model?

**Tip**: When making predictions using regression models, it’s important to ensure the input values (like Critical Reading score) fall within the range of the data used to build the model to avoid extrapolation errors.

The accompanying table shows the average SAT Math and Critical Reading scores for students in a sample of states. A scatterplot for these two variables suggests a linear trend. Complete parts (a) through (c) below.

SAT Math Score
SAT Critical Reading Score
575 579
504 499
485 490
592 588
461 462
494 494
487 488
563 557
440 441
506 495
587 582
450 463
597 592
563 580
605 591
599 596
524 523
473 468
600 589
510 502
544 551
530 525
508 487
494 494
463 467
503 509
486 470
502 489
504 498

Part 1

Find and report the value for the correlation coefficient and the regression equation for predicting the math score from the critical reading score. Then find the predicted math score for a state with a critical reading score of 561.

This problem involves calculating the correlation coefficient and performing regression analysis for SAT Math and Critical Reading scores. Learn how to predict SAT Math scores using a linear model and interpret the strength of the relationship between the two variables. Suitable for high school and college students studying statistics.

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