Let X represent a binomial random variable with n = 380 and p = 0.78. Find...

Let X represent a binomial random variable with n = 380 and p = 0.78. Find the following probabilities. (Round your final answers to 4 decimal places.) Probability a. P(X ≤ 300) b. P(X > 320) c. P(305 ≤ X ≤ 325) d. P( X = 290)

Expert Solution

Using Normal Approximation to Binomial
Mean = n * P = ( 380 * 0.78 ) = 296.4
Variance = n * P * Q = ( 380 * 0.78 * 0.22 ) = 65.208
Standard deviation = √(variance) = √(65.208) = 8.0751

Part a)

P ( X <= 300 )
Using continuity correction
P ( X < n + 0.5 ) = P ( X < 300 + 0.5 ) = P ( X < 300.5 )

X ~ N ( µ = 296.4 , σ = 8.0751 )
P ( X < 300.5 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 300.5 - 296.4 ) / 8.0751
Z = 0.51
P ( ( X - µ ) / σ ) < ( 300.5 - 296.4 ) / 8.0751 )
P ( X < 300.5 ) = P ( Z < 0.51 )
P ( X < 300.5 ) = 0.6950

Part b)

P ( X > 320 )
Using continuity correction
P ( X > n + 0.5 ) = P ( X > 320 + 0.5 ) = P ( X > 320.5 )

X ~ N ( µ = 296.4 , σ = 8.0751 )
P ( X > 320.5 ) = 1 - P ( X < 320.5 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 320.5 - 296.4 ) / 8.0751
Z = 2.98
P ( ( X - µ ) / σ ) > ( 320.5 - 296.4 ) / 8.0751 )
P ( Z > 2.98 )
P ( X > 320.5 ) = 1 - P ( Z < 2.98 )
P ( X > 320.5 ) = 1 - 0.9986
P ( X > 320.5 ) = 0.0014

Part c)

P ( 305 <= X <= 325 )
Using continuity correction
P ( n - 0.5 < X < n + 0.5 ) = P ( 305 - 0.5 < X < 325 + 0.5 ) = P ( 304.5 < X < 325.5 )

X ~ N ( µ = 296.4 , σ = 8.0751 )
P ( 304.5 < X < 325.5 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 304.5 - 296.4 ) / 8.0751
Z = 1
Z = ( 325.5 - 296.4 ) / 8.0751
Z = 3.6
P ( 1 < Z < 3.6 )
P ( 304.5 < X < 325.5 ) = P ( Z < 3.6 ) - P ( Z < 1 )
P ( 304.5 < X < 325.5 ) = 0.9998 - 0.8413
P ( 304.5 < X < 325.5 ) = 0.1585

Part d)

P ( X = 290 )
Using continuity correction
P ( n - 0.5 < X < n + 0.5 ) = P ( 290 - 0.5 < X < 290 + 0.5 ) = P ( 289.5 < X < 290.5 )

X ~ N ( µ = 296.4 , σ = 8.0751 )
P ( 289.5 < X < 290.5 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 289.5 - 296.4 ) / 8.0751
Z = -0.85
Z = ( 290.5 - 296.4 ) / 8.0751
Z = -0.73
P ( -0.85 < Z < -0.73 )
P ( 289.5 < X < 290.5 ) = P ( Z < -0.73 ) - P ( Z < -0.85 )
P ( 289.5 < X < 290.5 ) = 0.2327 - 0.1977
P ( 289.5 < X < 290.5 ) = 0.0350

milcah answered 4 months ago

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