Question

Although studies continue to show smoking leads to significant health problems, 20% of adults in the United States smoke. Consider a group of 300 adults.

If required, round your answers to four decimal places. Use Table 1 in Appendix B. If your answer is zero, enter "0".

c. What is the probability that from 35 to 70 smoke? Used continuity correction factor.

A binomial probability distribution has p = .20 and n = 100.

If required, round your answers to four decimal places. Use “Continuity correction factor, if necessary”. Use Table 1 in Appendix B.

c. What is the probability of exactly 22 successes?

Answer 1

Using Normal Approximation to Binomial
Mean = n * P = ( 300 * 0.2 ) = 60
Variance = n * P * Q = ( 300 * 0.2 * 0.8 ) = 48
Standard deviation = √(variance) = √(48) = 6.9282

Part a)
P ( 35 <= X <= 70 )
Using continuity correction
P ( n - 0.5 < X < n + 0.5 ) = P ( 35 - 0.5 < X < 70 + 0.5 ) = P ( 34.5 < X < 70.5 )

X ~ N ( µ = 60 , σ = 6.9282 )
P ( 34.5 < X < 70.5 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 34.5 - 60 ) / 6.9282
Z = -3.68
Z = ( 70.5 - 60 ) / 6.9282
Z = 1.52
P ( -3.68 < Z < 1.52 )
P ( 34.5 < X < 70.5 ) = P ( Z < 1.52 ) - P ( Z < -3.68 )
P ( 34.5 < X < 70.5 ) = 0.9357 - 0.0001
P ( 34.5 < X < 70.5 ) = 0.9356

Part b)
P ( X = 22 )
Using continuity correction
P ( n - 0.5 < X < n + 0.5 ) = P ( 22 - 0.5 < X < 22 + 0.5 ) = P ( 21.5 < X < 22.5 )

X ~ N ( µ = 60 , σ = 6.9282 )
P ( 21.5 < X < 22.5 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 21.5 - 60 ) / 6.9282
Z = -5.56
Z = ( 22.5 - 60 ) / 6.9282
Z = -5.41
P ( -5.56 < Z < -5.41 )
P ( 21.5 < X < 22.5 ) = P ( Z < -5.41 ) - P ( Z < -5.56 )
P ( 21.5 < X < 22.5 ) = 0 - 0
P ( 21.5 < X < 22.5 ) = 0