This Exam P sample reference tests Normal Approximation to Binomial. For Binomial(40,0.5), the continuity-corrected tails at cutoffs 23 and 24 are approximately 0.134 and 0.077. The 0.10 crossing therefore occurs at N=23, choice A.
How to solve this Normal Approximation to Binomial question
Setup
Setup
The binomial count has mean 20 and variance 10. The requested integer is the point at which the approximated upper tail first passes from above 0.10 to below 0.10.
μ=20,σ=10
Model
Model
Apply a continuity correction to the event above 23 by using the boundary 23.5. Its normal upper-tail approximation is 0.134.
Pr(C>23)≈Pr(Z>1023.5−20)=0.134
Compute
Compute
The next boundary, 24.5, gives an upper tail of 0.077. Since 0.10 lies between these adjacent tails, the required integer is 23.
Pr(C>24)≈Pr(Z>1024.5−20)=0.077
Answer
Answer
The adjacent-tail condition selects N=23.
23(A)
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BAt cutoff 25 the upper tail is already below 0.10, so it is beyond the adjacent crossing established at 23 and 24.
CA cutoff of 32 is far above the mean and cannot satisfy the required above-0.10 first inequality.
DA cutoff of 33 is also far beyond the first crossing below 0.10.
EA cutoff of 35 is still farther into the upper tail and is not the first qualifying boundary.
Original practice · fully worked
Original variant: normal approximation for a quality score
A quality test has 100 independent items, each passed with probability 0.40. Find the integer N for which the normal approximation with continuity correction gives P(X>N)>0.05 but P(X>N+1)<0.05.
A 45
B 46
C 47
D 48
E 49
Variant answer in brief
The mean is 40 and standard deviation √(24). Corrected cutoffs 47.5 and 48.5 give upper tails about 0.063 and 0.042, so N=47, choice C.
Setup
Setup
The count has mean 40 and variance 24. The target integer is located by comparing two adjacent continuity-corrected upper tails with 0.05.
μ=40,σ=24
Model
Model
The event above 47 uses boundary 47.5 and has approximated upper tail 0.063.
Pr(X>47)≈Pr(Z>1.531)=0.063
Compute
Compute
The event above 48 uses boundary 48.5 and has approximated upper tail 0.0415. The crossing therefore occurs at 47.
Pr(X>48)≈Pr(Z>1.735)=0.0415
Answer
Answer
The probability crosses 0.05 between the two tails at N=47.
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