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E19) Normal Approximation To Binomial Distribution.Doc
For each of the following exercises, if np ≥ 5 and nq ≥ 5, estimate the indicated probability by using the normal distribution
as an approximation to the binomial distribution; if np < 5 or nq < 5, then state that the normal approximation is not
suitable.
1.
Check in each case whether the conditions for the normal approximation to the binomial distribution are satisfied:
n = 16 and p = ¼;
n = 150 and p = 0.97;
a)
e)
h)
n = 135 and p = 1
b)
n = 65 and p = 0.10;
f)
c)
n = 120 and p = 0.98.
g)
d)
n = 200 and p = 0.01;
1
8
n = 100 and p =
n = 40 and p =
1
5
.
45
1
25
i)
n = 150 and p =
j)
n = 50 and p = 0.95
2.
Use the normal distribution to approximate the probability that at most 40 of 225 loan applications received by a bank
will be refused, if the probability is 0.20 that any such loan applications will be refused.
3.
Use the normal distribution to approximate the probability that at most 12 of 50 patients will get a headache from using a
certain kind of medication, if the probability is 0.22 that any one patient will get a headache from using medication.
4.
Use the normal distribution to approximate the probability that at least 60 of 90 persons flying across the Atlantic Ocean
will feel the effect of the time difference for at least 24 hours, if the probability is 0.70 that any one person flying across
the Atlantic Ocean will feel the effect of the time difference for at least 24 hours.
5.
Use the normal distribution to approximate the probability that at least 150 of 400 persons who have cable television will
watch a certain movie, if the probability is 0.34 that any one person who has cable television will watch that movie.
6.
Use the normal distribution to approximate the probability that more than 90 of 100 scorpion stings will cause extensive
discomfort, if the probability is 0.85 that any one of them will cause extensive discomfort.
7.
Estimate the probability of getting at least 55 girls in 100 births.
8.
Estimate the probability of passing a true/false test of 50 questions if 60% (or 30 correct responses) represents a passing
grade and all responses are random guesses.
9.
In a survey, the Canadian Automobile Association (CAA) found that 38 % of respondents had bought used vehicles. A
car dealership surveys 150 people at random and finds that 45 of them have bought used vehicles. Estimate the
probability of 45 or more respondents being used vehicle purchasers. Based on that value, does the CAA survey result
seem too high?
10. A manufacturer of automobile headlights claims that 10% of a batch of these headlights are defective. A quality control
inspector tests a random selection of 300 headlights in the batch. The number of defective headlights in this random
sample is 35, which is more than expected. Estimate the probability of getting at least 35 defective lights in a random
sample of 300. Based on the result, does it appear that the manufacturer's claim of 10% defectiveness is accurate? Was
there a problem with the sample?
11. Sunnybrook Hospital is conducting a blood drive because its supply of group O blood is low, and it needs 177 donors of
group O blood. If 400 volunteers donate blood, estimate the probability that the number with group O blood is at least
177. Forty-six percent of us have group O blood, according to data provided by Canadian Blood Services.
12. Replacement times for TV sets are normally distributed with a mean of 8.2 years and a standard deviation of 1.1 years
(based on data from "Getting Things Fixed," Consumer Reports). Estimate the probability that for 250 randomly selected
TV sets, at least 15 of them have replacement times greater than 10.0 years.
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ANSWERS:
1.
a) Not satisfied since
np = 4 < 5
b) Satisfied.
c) Not satisfied since
n(1-p) = 2.4 < 5
d) Not satisfied since
np = 2 < 5
e) Not satisfied since
n(1-p) = 4.5 < 5
f) Satisfied.
g) Satisfied.
h) Not satisfied since
np = 3 < 5
i) Satisfied.
j) Not satisfied since
np = 2.5 < 5
2.
0.2266
3.
0.6950
4.
0.7881
5.
0.0764
6.
0.0618
7.
0.1841
8.
0.1020
9.
0.9817 ; no
10.
0.1922 ; not unusual.
The claim is accurate.
11.
0.7734
12.
0.2946