Question

In a recent Gallup poll, 52% of parents with children under 18 years of age gave themselves a grade of B for the job they are doing bringing up their kids. Assume that this percentage is true for the current population of all parents with children under 18 years of age. You take a random sample of 1044 such parents and ask them to give themselves a grade for the job they are doing bringing up their kids.

a. The probability that 560 or more will give themselves a grade of B is approximately: (Round the answer to 4 decimal places.)

b. The probability that 550 or less will give themselves a grade of B is approximately: (Round the answer to 4 decimal places.)

c. The probability that 520 to 555 will give themselves a grade of B is approximately: (Round the answer to 4 decimal places.)

Answer 1

Using Normal Approximation to Binomial
Mean = n * P = ( 1044 * 0.52 ) = 542.88
Variance = n * P * Q = ( 1044 * 0.52 * 0.48 ) = 260.5824
Standard deviation = √(variance) = √(260.5824) = 16.1426

Condition check for Normal Approximation to Binomial
n * P >= 10 = 1044 * 0.52 = 542.88
n * (1 - P ) >= 10 = 1044 * ( 1 - 0.52 ) = 501.12

Part a)

P ( X >= 560 )
Using continuity correction
P ( X > n - 0.5 ) = P ( X > 560 - 0.5 ) =P ( X > 559.5 )

X ~ N ( µ = 542.88 , σ = 16.1426 )
P ( X > 559.5 ) = 1 - P ( X < 559.5 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 559.5 - 542.88 ) / 16.1426
Z = 1.03
P ( ( X - µ ) / σ ) > ( 559.5 - 542.88 ) / 16.1426 )
P ( Z > 1.03 )
P ( X > 559.5 ) = 1 - P ( Z < 1.03 )
P ( X > 559.5 ) = 1 - 0.8485
P ( X > 559.5 ) = 0.1515

Part b)

P ( X <= 550 )
Using continuity correction
P ( X < n + 0.5 ) = P ( X < 550 + 0.5 ) = P ( X < 550.5 )

X ~ N ( µ = 542.88 , σ = 16.1426 )
P ( X < 550.5 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 550.5 - 542.88 ) / 16.1426
Z = 0.47
P ( ( X - µ ) / σ ) < ( 550.5 - 542.88 ) / 16.1426 )
P ( X < 550.5 ) = P ( Z < 0.47 )
P ( X < 550.5 ) = 0.6808

Part c)

P ( 520 <= X <= 555 )
Using continuity correction
P ( n - 0.5 < X < n + 0.5 ) = P ( 520 - 0.5 < X < 555 + 0.5 ) = P ( 519.5 < X < 555.5 )

X ~ N ( µ = 542.88 , σ = 16.1426 )
P ( 519.5 < X < 555.5 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 519.5 - 542.88 ) / 16.1426
Z = -1.45
Z = ( 555.5 - 542.88 ) / 16.1426
Z = 0.78
P ( -1.45 < Z < 0.78 )
P ( 519.5 < X < 555.5 ) = P ( Z < 0.78 ) - P ( Z < -1.45 )
P ( 519.5 < X < 555.5 ) = 0.7823 - 0.0735
P ( 519.5 < X < 555.5 ) = 0.7088