Tuesday, 11/5
➢ Continuity Correction-convert discrete random variable r(number of successes) to a
continuous normal variable x by...
→ If r is a left point of an interval, subtract 0.5 to get corresponding x
→ If r is a right point of an interval, add 0.5 to get corresponding x
➢ For example, P(6≤r≤10) is approximated by P(5.5≤x≤10.5)
➢ Np and nq must both exceed 5
➢ Unless n is fairly large, a continuity correction may be necessary to improve
approximation
Sampling distribution for p̂ = r/n
➢ n = number of binomial trials, r = number of successes, p = probability of success, q =
probability of failure(1-p)
𝑝𝑞
➢ µp̂ = p, δp̂ = √( 𝑛 )
Example
➢ Annual crime rate in neighborhood is 111 victims per 1000 residents. The Arms
apartment building has 50 residents. Suppose each n=50 residents is a binomial trial. Let
r equal the number of crime victims in the next year.
➢ What is the population probability(p) that a neighborhood resident will be a crime
victim? Probability that they will not be a victim?
→ P = 111/1000 = 0.111, so q = 1-0.111 = 0.889
➢ Consider p̂ = r/n = r/50. Can we approximate the p̂ distribution with a normal
distribution?
→ np = 50(0.111) = 5.55, and nq = 50(0.889) = 44.45, so yes(np and nq both greater
than 5)
➢ What are the mean and standard deviation here?
𝑝𝑞 (0.98)
→ µp̂ = p = 0.111, δ = √( 𝑛 ) = √ = √0.0019 = 0.044
50
Chapter 7
➢ Continuity Correction-convert discrete random variable r(number of successes) to a
continuous normal variable x by...
→ If r is a left point of an interval, subtract 0.5 to get corresponding x
→ If r is a right point of an interval, add 0.5 to get corresponding x
➢ For example, P(6≤r≤10) is approximated by P(5.5≤x≤10.5)
➢ Np and nq must both exceed 5
➢ Unless n is fairly large, a continuity correction may be necessary to improve
approximation
Sampling distribution for p̂ = r/n
➢ n = number of binomial trials, r = number of successes, p = probability of success, q =
probability of failure(1-p)
𝑝𝑞
➢ µp̂ = p, δp̂ = √( 𝑛 )
Example
➢ Annual crime rate in neighborhood is 111 victims per 1000 residents. The Arms
apartment building has 50 residents. Suppose each n=50 residents is a binomial trial. Let
r equal the number of crime victims in the next year.
➢ What is the population probability(p) that a neighborhood resident will be a crime
victim? Probability that they will not be a victim?
→ P = 111/1000 = 0.111, so q = 1-0.111 = 0.889
➢ Consider p̂ = r/n = r/50. Can we approximate the p̂ distribution with a normal
distribution?
→ np = 50(0.111) = 5.55, and nq = 50(0.889) = 44.45, so yes(np and nq both greater
than 5)
➢ What are the mean and standard deviation here?
𝑝𝑞 (0.98)
→ µp̂ = p = 0.111, δ = √( 𝑛 ) = √ = √0.0019 = 0.044
50
Chapter 7
