It looks like you've provided part of a problem with some numbers that might have been mistakenly typed or formatted, but I can help you with the concept of predicting the mean distance of a home run hit at 105 mph.

It seems you are working with a linear regression equation, likely of the form:

$\text{Distance} = a + b \times (\text{Speed})$

Where $a$ is the y-intercept and $b$ is the slope of the equation. From the way you've written it, it looks like you were trying to write something like:

$\text{Distance} = 3.836 \times (\text{Speed}) + 4.787$

If that's the case, we can predict the distance of a home run hit at 105 mph by substituting the speed (105 mph) into the equation.

Substituting $\text{Speed} = 105$ into the equation:

$\text{Distance} = 3.836 \times 105 + 4.787$

Let's calculate that. Given equation: Distance = 3.836 * Speed + 4.787 speed = 105 slope = 3.836 intercept = 4.787

Calculate distance

distance = slope * speed + intercept round(distance, 1)The predicted mean distance of all home runs hit at 105 mph is 407.6 feet (rounded to one decimal place).

Would you like more details or have any questions about this?

Here are some related questions for you to explore:

1. How does the slope of the regression equation affect the predicted distance?
2. What other factors might influence the distance of a home run besides speed?
3. How would the predicted distance change if the speed were 95 mph instead of 105 mph?
4. What does the y-intercept (4.787) represent in the context of the regression equation?
5. How do you interpret the result if the speed is zero?

Tip: Always ensure that your units (like mph or feet) are consistent across your equations when applying a formula.