Oevelse 6

In [2]:

import math
from statistics import NormalDist

# --------------------------------------------------
# Given data
# --------------------------------------------------
lower = 45781          # lower end of "typical" interval
upper = 59892          # upper end of "typical" interval
n_class = 15
threshold_55000 = 55000
threshold_max = 67476

# --------------------------------------------------
# 1) Estimate mu and sigma assuming interval is mu ± sigma
# --------------------------------------------------
mu = (lower + upper) / 2
sigma = (upper - lower) / 2

print("1) Estimated parameters")
print(f"mu    = {mu:.4f}")
print(f"sigma = {sigma:.4f}")
print()

# Standard normal distribution
std_normal = NormalDist()

# Survival function for X ~ N(mu, sigma)
def normal_sf(x, mu, sigma):
    z = (x - mu) / sigma
    return 1 - std_normal.cdf(z)

# --------------------------------------------------
# 2) Probability that one software engineer earns more than 55,000
# --------------------------------------------------
p_single_55000 = normal_sf(threshold_55000, mu, sigma)

print("2) Probability that a randomly selected software engineer")
print(f"   earns more than {threshold_55000:,} DKK:")
print(f"   P(X > {threshold_55000:,}) = {p_single_55000:.4f}")
print()

# --------------------------------------------------
# 3) Probability that the average salary of 15 students exceeds 55,000
# --------------------------------------------------
se = sigma / math.sqrt(n_class)
p_mean_55000 = normal_sf(threshold_55000, mu, se)

print("3) Probability that the average salary of the 15 software engineers")
print(f"   in your class exceeds {threshold_55000:,} DKK:")
print(f"   Standard error = {se:.4f}")
print(f"   P(X_bar > {threshold_55000:,}) = {p_mean_55000:.4f}")
print()

# --------------------------------------------------
# 4) Probability that one software engineer earns more than 67,476
# --------------------------------------------------
p_single_max = normal_sf(threshold_max, mu, sigma)

print("4) Probability that a randomly selected software engineer")
print(f"   earns more than {threshold_max:,} DKK:")
print(f"   P(X > {threshold_max:,}) = {p_single_max:.4f}")
print()

# --------------------------------------------------
# 5) Expected proportion and expected number in a class of 15
#    who earn more than 67,476
# --------------------------------------------------
expected_proportion = p_single_max
expected_number = n_class * p_single_max

print("5) For a class of 15 software engineers:")
print(f"   Expected proportion earning more than {threshold_max:,} DKK = {expected_proportion:.4f}")
print(f"   Expected number earning more than {threshold_max:,} DKK     = {expected_number:.4f}")

import math from statistics import NormalDist # -------------------------------------------------- # Given data # -------------------------------------------------- lower = 45781 # lower end of "typical" interval upper = 59892 # upper end of "typical" interval n_class = 15 threshold_55000 = 55000 threshold_max = 67476 # -------------------------------------------------- # 1) Estimate mu and sigma assuming interval is mu ± sigma # -------------------------------------------------- mu = (lower + upper) / 2 sigma = (upper - lower) / 2 print("1) Estimated parameters") print(f"mu = {mu:.4f}") print(f"sigma = {sigma:.4f}") print() # Standard normal distribution std_normal = NormalDist() # Survival function for X ~ N(mu, sigma) def normal_sf(x, mu, sigma): z = (x - mu) / sigma return 1 - std_normal.cdf(z) # -------------------------------------------------- # 2) Probability that one software engineer earns more than 55,000 # -------------------------------------------------- p_single_55000 = normal_sf(threshold_55000, mu, sigma) print("2) Probability that a randomly selected software engineer") print(f" earns more than {threshold_55000:,} DKK:") print(f" P(X > {threshold_55000:,}) = {p_single_55000:.4f}") print() # -------------------------------------------------- # 3) Probability that the average salary of 15 students exceeds 55,000 # -------------------------------------------------- se = sigma / math.sqrt(n_class) p_mean_55000 = normal_sf(threshold_55000, mu, se) print("3) Probability that the average salary of the 15 software engineers") print(f" in your class exceeds {threshold_55000:,} DKK:") print(f" Standard error = {se:.4f}") print(f" P(X_bar > {threshold_55000:,}) = {p_mean_55000:.4f}") print() # -------------------------------------------------- # 4) Probability that one software engineer earns more than 67,476 # -------------------------------------------------- p_single_max = normal_sf(threshold_max, mu, sigma) print("4) Probability that a randomly selected software engineer") print(f" earns more than {threshold_max:,} DKK:") print(f" P(X > {threshold_max:,}) = {p_single_max:.4f}") print() # -------------------------------------------------- # 5) Expected proportion and expected number in a class of 15 # who earn more than 67,476 # -------------------------------------------------- expected_proportion = p_single_max expected_number = n_class * p_single_max print("5) For a class of 15 software engineers:") print(f" Expected proportion earning more than {threshold_max:,} DKK = {expected_proportion:.4f}") print(f" Expected number earning more than {threshold_max:,} DKK = {expected_number:.4f}")

1) Estimated parameters
mu    = 52836.5000
sigma = 7055.5000

2) Probability that a randomly selected software engineer
   earns more than 55,000 DKK:
   P(X > 55,000) = 0.3796

3) Probability that the average salary of the 15 software engineers
   in your class exceeds 55,000 DKK:
   Standard error = 1821.7223
   P(X_bar > 55,000) = 0.1175

4) Probability that a randomly selected software engineer
   earns more than 67,476 DKK:
   P(X > 67,476) = 0.0190

5) For a class of 15 software engineers:
   Expected proportion earning more than 67,476 DKK = 0.0190
   Expected number earning more than 67,476 DKK     = 0.2850