Question
Patrick is studying the relationship between the time spent studying per day and the time spent reading per day and has collected the data shown in the table. The line of best fit for the data is
ˆ
y
=
0.135
x
+
36.6
.
Studying (Minutes)
Reading (Minutes)
60
45
80
47
100
50
120
53

(a) According to the line of best fit, the predicted number of minutes spent reading for someone who spent
122
minutes studying is
53.07
.

(b) Is it reasonable to use this line of best fit to make the above prediction?

Select the correct answer below:

The estimate, a predicted time of
53.07
minutes, is both unreliable and unreasonable.

The estimate, a predicted time of
53.07
minutes, is both reliable and reasonable.

The estimate, a predicted time of
53.07
minutes, is reliable but unreasonable.

The estimate, a predicted time of
53.07
minutes, is unreliable but reasonable.

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Let's break this down carefully.

The line of best fit is given by the equation:
\[
\hat{y} = 0.135x + 36.6
\]
where \( x \) is the number of minutes spent studying, and \( \hat{y} \) is the predicted number of minutes spent reading.

### Part (a)
The prediction for 122 minutes spent studying can be found using the line of best fit formula. 

Substitute \( x = 122 \) into the equation:
\[
\hat{y} = 0.135(122) + 36.6 = 16.47 + 36.6 = 53.07
\]
So, the predicted reading time is indeed **53.07 minutes**.

### Part (b)
To determine whether this estimate is reasonable and reliable, consider a few key points:

1. **Range of the data**: The given data points for studying range from 60 to 120 minutes. The prediction for 122 minutes falls slightly outside this range, but only by 2 minutes. Hence, it is not a significant extrapolation.

2. **Consistency of data**: The data shows a roughly linear relationship between studying and reading time. The equation reflects this trend, so the prediction based on this equation should be reliable.

3. **Reasonableness**: Since the prediction is based on a small extrapolation (just 2 minutes outside the data), it remains reasonable.

### Conclusion:
The estimate of 53.07 minutes is both **reliable and reasonable**, as it is based on a well-fitting linear model and is close to the observed data points.

Thus, the correct answer is:
**The estimate, a predicted time of 53.07 minutes, is both reliable and reasonable.**

---

Would you like more details or have any questions? Here are 5 related questions to deepen understanding:

1. What does "extrapolation" mean in the context of linear regression?
2. How do we determine if a line of best fit is appropriate for a given dataset?
3. Why is it important to check the range of data before making predictions?
4. Can the line of best fit be used to make predictions far outside the observed data range?
5. What factors could make a prediction unreliable even if the line of best fit seems accurate?

**Tip:** In linear regression, predictions made within the range of data are generally more reliable than those made outside the range (extrapolation).

Patrick is studying the relationship between the time spent studying per day and the time spent reading per day and has collected the data shown in the table. The line of best fit for the data is ŷ = 0.135x + 36.6. Studying (Minutes): 60, 80, 100, 120. Reading (Minutes): 45, 47, 50, 53. (a) According to the line of best fit, the predicted number of minutes spent reading for someone who spent 122 minutes studying is 53.07. (b) Is it reasonable to use this line of best fit to make the above prediction?

Understand how to predict the reading time for 122 minutes of studying using the line of best fit equation ŷ = 0.135x + 36.6. Explore the concepts of linear regression, prediction, and extrapolation with step-by-step reasoning. Suitable for students in Grades 9-12.

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Question Patrick is studying the relationship between the time spent studying per day and the time spent reading per day and has collected the data shown in the table. The line of best fit for the data is ˆ y

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