What is the True Normal Human Body Temperature?

In [1]:

    

# Question 1) Is the distribution of body temperatures normal? Although this is not a requirement for 
#             CLT to hold (read CLT carefully), it gives us some peace of mind that the population may 
#             also be normally distributed if we assume that this sample is representative of the population.
    
# Answer:  I'll try to answer this visually by constructing a normal probability plot (to help assess whether or not
#          the data set is normally distributed) and also a histogram to estimate the probability distribution of the
#          data set.

%matplotlib inline
import pandas as pd
import matplotlib.pyplot as plt
import numpy as np
from scipy import stats

df = pd.read_csv('data/human_body_temperature.csv')

temp_df = sorted(df.temperature)

# Generate a probability plot of sample data.
res = stats.probplot(df.temperature, plot=plt)
plt.show()


    

In [2]:

    

# Generate a probability density function plot (pdf) of the sample data

temp_df = sorted(df.temperature)
plt.plot(temp_df, stats.norm.pdf(temp_df, np.mean(temp_df), np.std(temp_df)))
plt.hist(temp_df, normed=True, bins = 20)
plt.xlabel('Body Temperature')
plt.ylabel('Frequency')
plt.show()


    

In [3]:

    

#  Based on the graphs, it appears that the data approximates a normal distribution.
#  The normal probability plot show a linear correlation between actual and theoretically 
#  derived normal data which indicate that the normal distribution is a good model for this data set.
    
#  However, there are other special functions I can use to test whether a sample could have
#  been drawn from a normal distribution.  First test if the skew and kurtosis of sample data
#  differ significantly from a normal distribution.

print("Normal skew test stat = %6.3f pvalue = %6.4f" % stats.skewtest(temp_df))

print("Normal kurtosis test stat = %6.3f pvalue = %6.4f" % stats.kurtosistest(temp_df))

print("Normal test stat = %6.3f pvalue = %6.4f" % stats.normaltest(temp_df))


    

    




Normal skew test stat = -0.021 pvalue = 0.9830
Normal kurtosis test stat =  1.644 pvalue = 0.1001
Normal test stat =  2.704 pvalue = 0.2587

In [4]:

    

# Since the p-values of the tests are greater than 0.01, we can not reject the null
# hypothesis which means that our sample has no significant skew and kurtosis of 
# the normal distribution.
        
# In conclusion, the probability density function and normal probability plot
# suggest a normal distribution, and further tests indicate that the data lacks
# significant skewedness nor kurtosis.
    
# Question 2) Is the sample size large? Are the observations independent? Remember that this is 
#             a condition for the CLT, and hence the statistical tests we are using, to apply.
        
# Answer:  The rule of thumb is that the sample N >= 30 in order for the sample size to be
#          considered large. So let's check for the number of data points:
    
total_rows = len(df)
print("Total number of data points is %3i" % total_rows)


    

    




Total number of data points is 130

In [5]:

    

# Determine the z-score of temperatures using the ztest function
from statsmodels.stats.weightstats import ztest

zscore = ztest(temp_df, value=98.6)
print("Z-score is equal to : %5.4F  p-value equal to: %6.9F" % (abs(zscore[0]),zscore[1]))


    

    




Z-score is equal to : 5.4548  p-value equal to: 0.000000049

In [6]:

    

#  Since the p-value calculated for a two-tailed hypothesis < 0.025
#  we can reject the null hypothesis and therefore the true population 
#  mean temperature ≠ 98.6 F.

#     Question 4) At what temperature should we consider someone's temperature to be "abnormal"? 
#                 Start by computing the margin of error and confidence interval. 

# Calculate the Margin of Error using a 95% confidence level which
# gives a z-score of 1.96
zscore = 1.96
n = 130

# First, calculate the mean and standard deviation of sample data
temp_mean = df.temperature.mean()
temp_std = np.std(temp_df)
prop_err = temp_std/np.sqrt(n)

MoE = abs(zscore*prop_err)
CI = [temp_mean + MoE, temp_mean - MoE]
print("The mean population human body temperature of sample data is %0.2F F +/- %0.2F F" % (temp_mean,MoE))
print("Calculated normal human body temperatures is between %0.2F F and %0.2F F" % (CI[1],CI[0]))


    

    




The mean population human body temperature of sample data is 98.25 F +/- 0.13 F
Calculated normal human body temperatures is between 98.12 F and 98.37 F

In [7]:

    

#  Question 5) Is there a significant difference between males and females in normal temperature? 
#              What test did you use and why? Write a story with your conclusion in the context 
#              of the original problem.
            
#  Answer: Again as before, I'd like to set up the NULL hypothesis first, that there is
#          no significant difference between males and females in normal temperature.
#          The alternative hypothesis would state there is a significant difference between the two.
            
#          Ho: True normal male temperature = normal female temperature 
#          Ho: μ1 = μ2 
#          H1: True normal male temperature ≠ normal female temperature
#          H1: μ1 ≠ μ2
                
#          Also, I will use a t-test in this case since we are clearly comparing
#          two different groups.

# Group by gender and then apply a t-test.
from scipy.stats import ttest_ind

t_test = ttest_ind(df.temperature[df.gender == 'M'], df.temperature[df.gender == 'F'],equal_var=False)

print("T-test p-value is %0.5F " % t_test[1])

male_mean = df.temperature[df.gender=='M'].mean()
female_mean = df.temperature[df.gender=='F'].mean()
print("Male body temperature mean value is %0.5F " % male_mean)
print("Female body temperature mean value is %0.5F " % female_mean)


    

    




T-test p-value is 0.02394 
Male body temperature mean value is 98.10462 
Female body temperature mean value is 98.39385