The table below shows my weight for the first few days after I was born. Let 𝑊W represent my weight, in ounces, at 𝑡t days old.

Day

Weight (ounces)

1

48

4

55

5

61

8

69

9

70

10

70

Demos ProTip: Did you know that you can copy and paste this table into Desmos?

a) Use technology and all the data points to calculate the regression ("best fit") linear model for 𝑊W. Round your result to the nearest hundredth.

𝑊=W= 

b) If I could leave the hospital when I weighed 5 pounds (80 ounces), according to your model, on what day was I allowed to go home (round to the nearest whole number)?

The model predicts that I was allowed to go home on day  .

c)I really went home on day 24. According to your model, how much did I weigh on day 24 (to the nearest whole ounce)?

The model predicts that on day 24 I weighed  ounces.

Let's go through this step-by-step:

### a) Calculate the linear regression model for \( W \) (weight in ounces) as a function of \( t \) (days).

The data points we have are:

| Day (\( t \)) | Weight (\( W \)) (oz) |
|---------------|------------------------|
| 1             | 48                     |
| 4             | 55                     |
| 5             | 61                     |
| 8             | 69                     |
| 9             | 70                     |
| 10            | 70                     |

To calculate the regression line \( W = mt + b \) (where \( m \) is the slope and \( b \) is the y-intercept), I will apply linear regression using the given points.

Let me compute that for you.The linear regression model for \( W \), based on the data, is:

\[
W = 2.63t + 45.94
\]

where:
- \( m = 2.63 \) is the slope (rate of weight gain per day),
- \( b = 45.94 \) is the y-intercept (initial weight at day 0).

### b) When could you go home (i.e., weigh 80 ounces)?

To find the day \( t \) when your weight reached 80 ounces, we use the equation:
\[
80 = 2.63t + 45.94
\]
Solving for \( t \).According to the model, you would have been allowed to go home on **day 13**.

### c) Predicted weight on day 24

Now, let's use the model to predict your weight on day 24. We substitute \( t = 24 \) into the equation:
\[
W = 2.63(24) + 45.94
\]According to the model, your predicted weight on day 24 would have been **109 ounces**.

Would you like any further details or explanations? 

Here are 5 related questions you can explore:
1. How would the model change if more data points were added?
2. What does the slope represent in this context, and why is it important?
3. Can a quadratic or another type of regression model better fit the data?
4. What are some factors that could cause deviations from the predicted model in real life?
5. How can you determine the accuracy of this regression model?

**Tip:** In statistics, checking the residuals of a regression model helps assess how well the model fits the data!

The table below shows my weight for the first few days after I was born. Let W represent my weight, in ounces, at t days old.

| Day | Weight (ounces) |
|-----|-----------------|
| 1   | 48              |
| 4   | 55              |
| 5   | 61              |
| 8   | 69              |
| 9   | 70              |
| 10  | 70              |

a) Use technology and all the data points to calculate the regression ('best fit') linear model for W. Round your result to the nearest hundredth.

b) If I could leave the hospital when I weighed 5 pounds (80 ounces), according to your model, on what day was I allowed to go home (round to the nearest whole number)?

c) I really went home on day 24. According to your model, how much did I weigh on day 24 (to the nearest whole ounce)?

Learn how to calculate a linear regression model to predict weight growth over time using data from the first 10 days after birth. This problem covers algebraic concepts and data analysis, including the linear regression equation, slope calculation, and solving for unknown variables. Suitable for students in Grades 9-12.

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