At Westonci.ca, we provide clear, reliable answers to all your questions. Join our vibrant community and get the solutions you need. Experience the ease of finding quick and accurate answers to your questions from professionals on our platform. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
Sagot :
The probability that the sample mean is less than 5, is 0.92135.
Let X1,X2,...,X32 be a random sample from an exponential distribution with a mean of 4.
We have to find an approximate probability that the sample mean is less than 5.
Let X = [tex]\frac{\Sigma_{i=1}^{n}x_{i}}{n}[/tex]
X = [tex]\frac{\Sigma_{i=1}^{32} x_{i}}{32}[/tex]
Then required probability is P(X<5).
Now we have to find the distribution of X where [tex]x_{i}[/tex]……. x_{32}.
Follow exponential with mean 4.
[tex]x_{i}[/tex]→exponent(λ=1/mean)
[tex]x_{i}[/tex]→exponent(λ=1/4)
[tex]x_{i}[/tex]→exponent(λ=0.25)
Now we have result:
If [tex]x_{1}[/tex] ………… x_{n} are exponential with α.
Then [tex]\Sigma{x_{i}}[/tex] has Gamma Distribution with (α=0.25, n=32)
Now we have result
X→G(α, λ) then c X→G(α/c, λ)
Here, we have
[tex]\frac{\Sigma x_{i}}{n}[/tex]→Gamma(α=0.25, n=32)
[tex]\frac{\Sigma x_{i}}{n}[/tex]→Gamma(α/(1/n), n)
[tex]\frac{\Sigma x_{i}}{n}[/tex]→Gamma(nα, n)
X→Gamma(32*0.25, 32)
X has Gamma distribution with α=8, λ=32.
Required probability is;
P(X<5) = [tex]\int_{0}^{5}f(x) dx[/tex]
P(X<5) =[tex]\int_{0}^{5}\frac{\alpha }{\sqrt{\lambda}}e^{-\alpha x}x^{\lambda-1} dx[/tex]
Now to approximate this probability using normal as
X-mean(X)/SD(X) = N(0,1)
X→Gamma(α=8, λ=32)
E(X) = λ/α Var(X)= λ/α^2
E(X) = 32/8 Var(X)= 32/(8)^2
E(X) = 4 Var(X)= 0.5
Now P(X<5)=P[{(X-E(X))/√Var(X)}<{ (5-4)/√0.5}]
P(X<5)=P[{z< 1/√0.5}]
P(X<5) = P(z<1.4142
From the normal probability table
P(X<5) = 0.92135
Hence, the probability that the sample mean is less than 5, is 0.92135.
To learn more about probability link is here
brainly.com/question/11234923
#SPJ4
Thanks for stopping by. We are committed to providing the best answers for all your questions. See you again soon. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.