# 1. Print out data in columns
height = [134, 138, 142, 146, 150, 154, 158, 162, 166, 170, 174, 178]
fev = [1.7, 1.9, 2., 2.1, 2.2, 2.5, 2.7, 3.,3.1, 3.4, 3.8, 3.9]
print ("height FEV")
for i in range(len(height)):
print(f'{height[i]} {fev[i]}')

```
height FEV
134 1.7
138 1.9
142 2.0
146 2.1
150 2.2
154 2.5
158 2.7
162 3.0
166 3.1
170 3.4
174 3.8
178 3.9
```

# 2. Import libraries - NumPy and matplotlib
import numpy as np
import matplotlib.pyplot as plt

# 3. Plot the data; add labels to axes and plot title
plt.scatter(height, fev)
plt.title("Height vs FEV for boys aged 10-15")
plt.xlabel("Height in cm")
plt.ylabel("FEV")

# 4. Input functions to find interpolating polynomial
def compute_int_coeff (n, x, y) :
coeff = [] # initialize list
for i in range (0,n+1): # loop to calculate n+1 coefficients
numerator = y[i]
denominator = 1. # because denominator is a product we must initialize to 1
for j in range (0,n+1):
if ( i != j) : # denominator is product of all terms except
# when x_i=x_j which would give 0
denominator = denominator * (x[i] - x[j])
coeff.append (numerator / denominator) # add term to list
return (coeff)
#
#
def eval_int_polynomial (n, x, x_eval, coeff) :
my_sum = 0. # formula is sum of n+1 terms so initialize
for i in range (0,n+1) : # loop to form sum of terms
product = 1. # initialize product in numerator of L_i
for j in range (0, n+1) : # loop to form product of terms in numerator
if ( i != j ) : # include all terms except when i = j
product = product * (x_eval - x[j])
# when this j loop is complete we have the numerator of L_i for one term
yi_times_Li = product * coeff[i] # multiplies numerator times C_i (y_i/ product)
my_sum = my_sum + yi_times_Li # keeps running tally of sum of terms
return (my_sum)

# 5. Plot the interpolating polynomial from a height of 134 cm to 178 cm
n1 = len(height)-1
coeff = compute_int_coeff(n1,height,fev)
x_plt = np.linspace(134,178)
y_plt = []
for i in range (0,len(x_plt)):
xx = x_plt[i]
n = len(height)-1
poly_at_x = eval_int_polynomial (n, height, xx, coeff)
y_plt.append (poly_at_x)
plt.plot(x_plt, y_plt)

# 6. Use the polynomial to predict the FEV for a height of 165 cm and of 175 cm.
# Which do you think is a better prediction?
x_predict = [165.,175.]
n_predict = 2
for i in range(n_predict) :
xx = x_predict[i]
poly_at_x = eval_int_polynomial (n, height, xx, coeff)
print (f"the polynomial evaluated at x={xx} is {poly_at_x}")
print ("I believe the prediction at 165 cm is the better prediction because it is closer to the data already given.")

```
the polynomial evaluated at x=165.0 is 3.1044129974208774
the polynomial evaluated at x=175.0 is 3.3490242131985744
I believe the prediction at 165 cm is the better prediction because it is closer to the data already given.
```

# Import libraries
import matplotlib.pyplot as plt
import numpy as np

# Plot piecewise linear interpolating polynomial and data; use
y_interp = np.interp(x_plt,height,fev)
plt.plot(x_plt,y_interp)
plt.xlabel("heights 134-178")
plt.ylabel("interpolated FEV values")

# Use the piecewise polynomial to predict the FEV for a height of 165 cm and of 176 cm.
# How does this compare with 11th degree polynomial which predicted 3.104 for 165 cm and
# 2.697 for 176 cm
x_predict = [165.,176.]
n_predict = 2
for i in range(n_predict) :
xx = x_predict[i]
eval_at_x = np.interp(xx,height,fev)
print (f"the polynomial evaluated at x={xx} is {eval_at_x}")

```
the polynomial evaluated at x=165.0 is 3.075
the polynomial evaluated at x=176.0 is 3.8499999999999996
```