Question

Random samples of size n = 75 were selected from a binomial population with p = 0.8. Use the normal distribution to approximate the following probabilities. (Round your answers to four decimal places.)

P(p̂ ≤ 0.83)

P(0.75 ≤ p̂ ≤ 0.83) =

Answer 1

SOLUTION:

From given data,

Random samples of size n = 75 were selected from a binomial population with p = 0.8. Use the normal distribution to approximate the following probabilities. (Round your answers to four decimal places.)

Let denotes the sample proportion for a random sample of size n = 75

Here,

Let Normal (0.8 , (0.8*(1-0.8) /75)) or Normal (0.8 , 0.046188^2)

P(p̂ ≤ 0.83)

P(p̂ ≤ 0.83) = P( (-0.83) / 0.8 < (0.05-0.83) / 0.8 )

P(p̂ ≤ 0.83) = P( Z < (0.83​​​​​​​-0.80) /0.046188 )

P(p̂ ≤ 0.83) = P( Z < 0.65)

P(p̂ ≤ 0.83) = 0.74215

P(p̂ ≤ 0.83) = 0.7422

P(0.75 ≤ p̂ ≤ 0.83)

P(0.75 ≤ p̂ ≤ 0.83) = P((0.75-0.80) /0.046188 < Z < (0.83​​​​​​​-0.80) /0.046188 ))

P(0.75 ≤ p̂ ≤ 0.83) = P(-0.05 /0.046188 < Z < 0.03/0.046188 )

P(0.75 ≤ p̂ ≤ 0.83) = P(-1.08 < Z < 0.65 )

P(0.75 ≤ p̂ ≤ 0.83) = P(Z < 0.65 ) - P(Z < -1.08)

P(0.75 ≤ p̂ ≤ 0.83) = 0.74215- 0.14007

P(0.75 ≤ p̂ ≤ 0.83) = 0.60208

P(0.75 ≤ p̂ ≤ 0.83) = 0.6021

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