Suppose IQ scores were obtained for 20 randomly selected sets of twins. The 20 pairs of measurements yield x overbarequals97.81​, y overbarequals95.7​, requals0.860​, ​P-valueequals​0.000, and ModifyingAbove y with caret equals 0.92 plus 0.97 x​, where x represents the IQ score of the twin born second. Find the best predicted value of ModifyingAbove y with caret given that the twin born second has an IQ of 98​? Use a significance level of 0.05.
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Part 1
The best predicted value of ModifyingAbove y with caret is 
  
enter your response here.
​(Round to two decimal places as​ needed.)

We are tasked with predicting the best value of \( \hat{y} \) (the predicted IQ score of the twin born first) given the IQ score of the twin born second, which is 98.

The regression equation provided is:

\[
\hat{y} = 0.92 + 0.97x
\]

where \( x \) is the IQ score of the twin born second, and \( \hat{y} \) is the predicted IQ score of the twin born first.

### Step 1: Substituting \( x = 98 \) into the regression equation:
\[
\hat{y} = 0.92 + 0.97(98)
\]

### Step 2: Calculating the value of \( \hat{y} \):
\[
\hat{y} = 0.92 + 0.97 \times 98 = 0.92 + 95.06 = 95.98
\]

Thus, the best predicted value of \( \hat{y} \) is **95.98**.

### Conclusion:
The best predicted IQ score for the twin born first is \( 95.98 \).

Let me know if you'd like more details!

Here are five follow-up questions to deepen your understanding:

1. What is the significance of the \( P \)-value being 0.000 in this context?
2. How would the prediction change if the correlation \( r \) were lower?
3. Why is the Pearson correlation coefficient important for this problem?
4. How would we interpret the slope of the regression line in this scenario?
5. What would happen to the prediction if the IQ of the second twin was significantly higher, say 110?

**Tip:** The \( P \)-value indicates whether the correlation is statistically significant at the given level of significance (e.g., 0.05).

Suppose IQ scores were obtained for 20 randomly selected sets of twins. The 20 pairs of measurements yield x̄ = 97.81, ȳ = 95.7, r = 0.860, P-value = 0.000, and ŷ = 0.92 + 0.97x, where x represents the IQ score of the twin born second. Find the best predicted value of ŷ given that the twin born second has an IQ of 98? Use a significance level of 0.05.

This problem involves predicting the IQ score of one twin using the linear regression formula ŷ = 0.92 + 0.97x, with x representing the IQ score of the second twin. With a Pearson correlation coefficient of r = 0.860 and a P-value = 0.000, we use regression analysis to find the best predicted value of ŷ when the second twin's IQ is 98.

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