To estimate the heights for a one-year-old and a thirteen-year-old, we first need to find the equation that models the relationship between age and height. It seems you mentioned part (d), which likely refers to an equation we should use to predict these values. However, since the details of part (d) were not included in your message, let's first focus on the data you provided.

Here are the given data points:

$\text{Age (years)}: 0, 2, 3, 5, 7, 10, 14$ $\text{Height (cm)}: 50.8, 83.8, 91.4, 106.6, 119.3, 137.1, 157.5$

Now, we need to create an equation that we can use to estimate the height for a one-year-old and a thirteen-year-old.

We can approach this problem by performing linear regression or using an appropriate curve-fitting method, such as quadratic regression, depending on how the data behaves. However, without part (d), I will proceed with a simple linear regression method, assuming the relationship between age and height is approximately linear. If a more complex curve (such as quadratic) is needed, let me know!

Steps for Linear Regression:

1. Create a Linear Model:

   • The linear regression formula is $y = mx + b$, where:
     • $y$ is the height (dependent variable),
     • $x$ is the age (independent variable),
     • $m$ is the slope,
     • $b$ is the y-intercept.

2. Use the provided data to calculate the slope ($m$) and the intercept ($b$).

Let me compute this for you now! numpy as np import scipy.stats as stats

Given data points (age in years, height in cm)

age = np.array([0, 2, 3, 5, 7, 10, 14]) height = np.array([50.8, 83.8, 91.4, 106.6, 119.3, 137.1, 157.5])

Perform linear regression to find the slope (m) and intercept (b)

slope, intercept, r_value, p_value, std_err = stats.linregress(age, height)

Calculate predicted heights for 1 year and 13 years old

predicted_height_1 = slope * 1 + intercept predicted_height_13 = slope * 13 + intercept

round(predicted_height_1, 1), round(predicted_height_13, 2)The estimated average heights based on the linear regression model are:

• For a one-year-old: 72.2 cm
• For a thirteen-year-old: 157.32 cm

Let me know if you'd like to see the detailed steps or if you need any other help!

Here are 5 related questions:

1. How do you calculate the slope and intercept for linear regression?
2. What is the difference between linear and quadratic regression?
3. How can we determine if a linear model is appropriate for the data?
4. What would happen if we used a non-linear model for this data?
5. How do we use the regression equation to predict values for different ages?

Tip: Always plot your data first to see if the relationship looks linear before using linear regression!