Question

Given a dataset of random numbers, use Python to calculate:

(a) Estimates for the sample mean and sample variance without using inbuilt functions.

(b) Suppose now that the standard deviation is known to be 0.25. Compute a 90% confidence interval for the population mean. (python libraries can be used to calculate the confidence interval)

Data：

8.91E+00

1.53E+01

5.98E+00

6.39E+00

1.54E+00

1.05E+01

1.87E+00

2.14E+00

6.50E+00

3.20E+00

3.10E+00

5.16E+00

6.04E-01

1.75E+00

1.27E+00

3.15E+00

5.09E+00

5.19E-01

5.23E+00

4.17E+00

2.34E+01

4.94E+00

8.85E+00

1.66E-01

6.38E+00

1.09E+00

2.02E+00

4.06E-02

2.44E+00

8.56E+00

Answer 1

# Putting data into a List in order to process it
data = [8.91E+00,1.53E+01,5.98E+00,6.39E+00,1.54E+00,1.05E+01,1.87E+00,2.14E+00,6.50E+00,3.20E+00,
                3.10E+00,5.16E+00,6.04E-01,1.75E+00,1.27E+00,3.15E+00,5.09E+00,5.19E-01,5.23E+00,4.17E+00,
                2.34E+01,4.94E+00,8.85E+00,1.66E-01,6.38E+00,1.09E+00,2.02E+00,4.06E-02,2.44E+00,8.56E+00]

# Get number of observations
N = len(data)

# Varaible required to calculate mean
sumOfData = 0

# Loop to calculate sum of all values
for x in data:
        sumOfData+=x

# Formula to calulate mean
mean  =  sumOfData/N

sumForVar = 0
# Loop to store diffrences of number and mean
for x in range(N):
        sumForVar+= ( (data[x]-mean)**2)

sampleVaraince =  sumForVar/(N-1)

# To calculate confidence interval
# we use mean, Stadard deviation and condidence lebel 90
# The z* value for 90% is 1.645

# Taking required variables to calcualte 
# Confidence intervals
Z_star = 1.645
stdDeviation = 0.25
N_root = N**(1/2)

# Formuala to calulate margin of error for confidence intervals
marginOfError = Z_star * (stdDeviation/N_root)

# Both Confidence intervals
CI_upperBound = mean + marginOfError
CI_lowerBound = mean -  marginOfError


print("Lower Bound of condidence interval: ",CI_lowerBound)
print("Upper Bound of condidence interval: ",CI_upperBound)

print("Mean of the data: ",mean)
print("Varaiance of data :",sampleVaraince)

Screenshots of the code:

Screenshot of the output:

Explanation:

• At first, I have created a list of all the values that you have given in the question.
• After that, the size of the data is calculated by using len function of Python.
• Then, a loop will calculate the sum of all values and after that mean is calculated by dividing the size of data by the sum of data.
• Then, a variable for the sum of all variance of the values is defined. A loop will calculate the difference of the mean and the values and it will be added to a variable.
• Then sample variance is calculated by using this variable and size of data.
• For (b) part of the question, we will use the mean, standard deviation that is given in the question, and Z* value of 90% confidence.
• After assigning all values to the required variables, the margin of error is calculated.
• This margin will be used to get upper bound and lower bound of confidence intervals.
• After subtracting and addition of the margin of error we have our required confidence intervals with 90% confidence value.
  

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