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  1. Prepare
  2. Artificial Intelligence
  3. Probability & Statistics - Foundations
  4. The Central Limit Theorem #4
  5. Discussions

The Central Limit Theorem #4

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  • vai_suliafu
     + 0 comments

    norm.interval(0.95, 500, 80/sqrt(100))

    Then round each endpoint to 2 decimal places.

  • julians_suarez
     + 1 comment
    #How to find the confidense level 
Cl = 0.95
df = 100 - 1 #degrees of freedom = sampleSize - 1
mean = 500
std_error = 80/np.sqrt(100)
confidence_interval = stats.t.interval(Cl,df,mean,std_error)
print(confidence_interval)

    I used thi approach but got the decimal places wrong, what am I doing wrong

  • deankiss
     + 0 comments
    import math

n=100
mu = 500
sigma=80
sample_mu=mu
sample_sigma=sigma/math.sqrt(n)
zed = 1.96

A = sample_mu - zed*sample_sigma
B = sample_mu + zed*sample_sigma
print(A,B, sep='\n', end='')

  • MrStubby007
     + 0 comments

    Here are the steps to find the confidence interval(CI):

    1. Select Confidence level(CL): 95%(given)

    2. Calculate Margin of Error(ME):

      Follow below steps -

      • Find Standard Error(SE)

        SE = σ/sqrt(n) = 80/10 = 8

      • Find Critical Value:

        1. Calculate alpha(α)

          α = 1-CL/100 = 1-0.95 = 0.05

        2. Calculate Critical Probability(p*)

          p* = 1-α/2 = 1-0.025 = 0.975

        3. Compute degree of freedom(df)

          df = n-1 = 100-1 = 99

          Therefore, critical value is t-statistics having 99 df and cumulative prob. equals to 0.975 Or, We can also expressed the critical value as a z-score since the
          sample size is large, so a z-score analysis produces the same
          result.

          critical value = 1.96.

      ME = critical value * SE In our case, ME = 1.96*8

    3. The range of the confidence interval is (sample statistic - margin of error, sample statistic + margin of error)

      In our case, CI = (500-1.96*8, 500+1.96*8)

      or, CI = (484.32, 515.68)

  • ComradeAndrew
     + 4 comments

    Here is how I did figure it out. If somebody want to dig into this task and learn step by step solution.

    from math import erf
n = 100
# map mean and sigma to normal distribution with
# the following parameters by The Centeral Limit Theorem
mean = n*500
sigma = n**0.5*80
# Defining Phi function for CDF of our 
# retrieved Normal Distribution 
phi = lambda x: 0.5*(1+erf((x-mean)/(2**0.5*sigma)))
target = 0.95  # initiating target of 95 %
# Probability of our confidance interval
# given z-score value
ci_prob = lambda z: phi(z*sigma+mean)-phi(-z*sigma+mean)
# True if our CI of z-score achieved desired probability
found = lambda z: target <= round(ci_prob(z), 3)
z = 0.01  # initial z-score
# interating z until our CI satisfies our target
while not found(z): z += 0.01 
# getting left and right interval values and mapping
# back from normal distribution to initial distribution 
# by multiplying both values by 1/n, i.e. values in
# initial distribution where n times less due to additive
# property of an EV
print(round((-z*sigma+mean)/n, 2))
print(round((z*sigma+mean)/n, 2))

    Additionaly you may need those sources: