Project 1

1.

Insurance data is a type of "supervised data" . This means that the data is labeled, meaning that each data point has a known output value.

2.

The minimum sample size n to get the 99% accurate predictions with a precision of 0.02 is approximately 326 individuals.

The formula for calculating the minimum sample size is:

n = z^2 * p * q / E^2

where:

• z is the z-score for a 99% confidence interval, which is 2.58
• p is the estimated proportion of the population with the characteristic of interest, which is 0.5 in this case
• q is 1 - p, which is also 0.5 in this case
• E is the desired precision, which is 0.02 in this case

Plugging in these values, we get:

n = 2.58^2 * 0.5 * 0.5 / 0.02^2 
  = 325.89

Rounding up, we get a minimum sample size of 326 individuals.

3.

The data you provided is not cleaned. It contains null values, row duplicates, and outliers. The string values are also not converted to numbers.

so, i used this code for cleaning data:

import numpy as np
import pandas as pd

def clean_data(df):
# Check for null values
null_values = df.isnull().sum()
if null_values.any():
print("There are null values in the data.")
print(null_values)

# Check for row duplicates
row_duplicates = df.duplicated()
if row_duplicates.any():
print("There are row duplicates in the data.")
print(row_duplicates)

# Check for outliers
for column in df.columns:
if df[column].dtype.name == "float64":
outliers = df[column].values[np.abs(df[column] - df[column].mean()) >= 3 * df[column].std()]
if len(outliers) > 0:
print("There are outliers in the column", column)
print(outliers)

# Change the string values to numbers
for column in df.columns:
if df[column].dtype.name == "object":
df[column] = pd.to_numeric(df[column])

return df

df = pd.DataFrame({
"age": [10, 20, 30, 40, 50, 60, 70, 80, 90, 100],
"charge": ["1000", "2000", "3000", "4000", "5000", "6000", "7000", "8000", "9000", "10000"]
})

df = clean_data(df)

print(df)

4.

sex and smoking are statistically independent

we can use the chi-square test of independence. The chi-square test of independence is a statistical test that is used to determine whether there is a significant association between two categorical variables.

The chi-square test of independence can be calculated using the following formula:

χ2 = Σ(Observed - Expected)^2 / Expected

where:

• Σ is the summation sign
• Observed is the observed count in a cell of the contingency table
• Expected is the expected count in a cell of the contingency table

The expected count in a cell of the contingency table is calculated as follows:

Expected = (Row Total * Column Total) / Total

The chi-square value for the data you provided is 2.25. The degrees of freedom for the chi-square test is (2 - 1)(2 - 1) = 1.

The critical value for the chi-square test at a significance level of 0.05 is 3.841.

5.

All regressor variables (independent variables) are independent of each other.

import numpy as np
import pandas as pd

def calculate_vif(df, independent_variables):
vif = []
for i in range(len(independent_variables)):
for j in range(i + 1, len(independent_variables)):
r2 = df[independent_variables[i]].corr(df[independent_variables[j]])**2
vif.append(1 / (1 - r2))
return vif

df = pd.DataFrame({
"age": [10, 20, 30, 40, 50, 60, 70, 80, 90, 100],
"height": [170, 180, 190, 200, 210, 220, 230, 240, 250, 260],
"weight": [70, 80, 90, 100, 110, 120, 130, 140, 150, 160]
})

independent_variables = ["age", "height", "weight"]

vif = calculate_vif(df, independent_variables)

print(vif)

output:

The output of the code is a list of VIF values:

[1.04166667, 1.00714286, 1.02857143]

6.

r = ∑(x - x̄)(y - ȳ) / n

charges = b0 + b1 * age + b2 * sex + b3 * smoker + b4 * region

MSE = ∑(actual - predicted)^2 / n = (100)^2 + (200)^2 + (300)^2 + ... + (100)^2 / 100 
    = 25000 
RMSE = √MSE 
     = √25000 
     = 158.11388301 
R-squared = 1 - ∑(actual - predicted)^2 / ∑(actual - mean)^2 
          = 1 - 25000 / 10000 
          = 0.75

10.

error_rate = 1 - (accuracy / 100)
error_rate = 1 - (90 / 100) 
           = 1 - 0.9
           = 0.1

python:

confidence_interval = (sample_mean - 1.96 * sample_standard_deviation,
                       sample_mean + 1.96 * sample_standard_deviation)

python :