Answer to Question #227081 in Statistics and Probability for Mike

Question #227081

Suppose a census was conducted to determine the impact of COVID-19 on the 

operations of local vendors and it found that the mean and variance of the daily 

income of all local vendors is N$ 600 and 156250 (N$ squared), respectively. Assume 

a sample of 100 local vendors is selected, what is the probability that their sample 

mean daily income is within N$ 100 of the population mean daily income.


1
Expert's answer
2021-08-18T14:14:42-0400

Let X=X= sample mean daily income: XN(μ,σ2/n).X\sim N(\mu, \sigma^2/n).  

Given μ=600,σ2=156250,n=100\mu=600, \sigma^2=156250, n=100


P(μ100<X<μ+100)P(\mu-100<X<\mu+100)

=P(X<μ+100)P(Xμ100)=P(X<\mu+100)-P(X\leq \mu-100)

=P(Z<100156250100)P(Z100156250100)=P(Z<\dfrac{100}{\sqrt{\dfrac{156250}{100}}})-P(Z\leq\dfrac{-100}{\sqrt{\dfrac{156250}{100}}})

P(Z<2.5298)P(Z2.5298)\approx P(Z<2.5298)-P(Z\leq -2.5298)

0.994290.005710.9886\approx0.99429-0.00571\approx0.9886




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