5 P ROBABILITY D ENSITIES
5.3 The Normal Approximation to the Binomial Distribution
Unlike the Poisson approximation that applies when p is small, the normal distribu-tion approximates the binomial distribudistribu-tion when n is large and p, the probability of a success, is not close to 0 or 1. We state, without proof, the following theorem:
Normal approximation to binomial distribution
Theorem 5.1 If X is a random variable having the binomial distribution with the parameters n and p, the limiting form of the distribution function of the stan-dardized random variable
Z= X− np
√np(1− p)
as n→ ∞, is given by the standard normal distribution F (z)=
z
−∞
√1
2π e−t2/2dt − ∞ < z < ∞
Note that although X takes on only the values 0, 1, 2, . . . , n, in the limit as n → ∞, the distribution of the corresponding standardized random variable is continuous, and the corresponding probability density is the standard normal density.
A good rule of thumb for
the normal approximation Use the normal approximation to the binomial distribution only when np and n(1− p) are both greater than 15.
Note that in Example 11, which is an application of Theorem 5.1, we use again the continuity correction given on page 147.
EXAMPLE 11 The current consensus is that there are three types of neutrinos and each is accom-panied by an antimatter version. Further, any single neutrino can change (oscillate) from one type to another. When one type of antimatter neutron, called an electron antineutrino, travels two kilometers from a reactor to the detector it will disappear if it interacts with an electron neutrino and changes into another type. At one site, physicists have performed a path breaking experiment2that measured an important constant for this change. At a specific detector, with electron antineutrinos of average energy, this constant translates into probability .056 of disappearing.
Consider the outcomes of the next 300 electron antineutrinos leaving the reactor and heading toward the detector. Assuming that the conditions for Bernoulli trials hold,
(a) find the mean and standard deviation of the number which will disappear.
(b) Approximate the probability that 12 or more will disappear.
(c) Approximate the probability of exactly 12.
(d) Comment on a possible violation of independence.
2F. P. An et. al (2013) An improved measurement of Electron Antineutrino disappearances at Day Bay, Chin.Phys. C37
Sec 5.3 The Normal Approximation to the Binomial Distribution 149
Solution We take the probability p = 0.056 which is the value estimated from the physics experiment.
(a) Using the formulas for mean and standard deviation, we find Mean = n p = 300 × 0.056 = 16.80 Standard deviation = √
n p ( 1 − p ) = √
300 × 0.056 × 0.944 = 3.982 (b) Since n p > 15, the normal distribution provides a good approximation to the
probability
as illustrated in Figure 5.16. Over ninety percent of the time there will be 12 or more disappearances among the 300.
The exact value .9142 is obtained using 1 - pbinom(11,300,.056) in R.
(c) F (d) If two or more electron antineutrinos are so close that they interfere with each
other, or even collide, independence is violated.
The exact calculation is always preferrable when p is given but the
approxima-tion is important for inference when it is not. j
Exercises
5.19 If a random variable has the standard normal distribu-tion, find the probability that it will take on a value (a) less than 1.75;
(b) less than−1.25;
(c) greater than 2.06;
(d) greater than−1.82.
5.20 If a random variable has the standard normal distribu-tion, find the probability that it will take on a value (a) between 0 and 2.3;
(b) between 1.22 and 2.43;
(c) between−1.45 and −0.45;
(d) between−1.70 and 1.35.
5.21 The nozzle of a mixing vibrator is tested for its number of vibrations. The vibration frequency, for each nozzle sample, can be modeled by a normal
distribution with mean 128 and standard deviation 16 PdM.
(a) If engineering specifications require the sample to have a vibration frequency of 100 PdM, what is the probability that a sample will fail to meet specifications?
(b) In the long run, what proportion of samples will fail? Explain your answer.
(c) The mean vibration frequency can be increased by using different materials. What new mean is quired, when the standard deviation is 16, to re-duce the probability of not meeting specifications to 0.05?
5.22 If a random variable has a normal distribution, what are the probabilities that it will take on a value within (a) 1 standard deviation of the mean;
(b) 2 standard deviations of the mean;
(c) 3 standard deviations of the mean;
(d) 4 standard deviations of the mean?
5.23 Verify that
(a) z0.005= 2.575;
(b) z0.025= 1.96.
5.24 Given a random variable having the normal distribu-tion withμ = 16.2 and σ2= 1.5625, find the proba-bilities that it will take on a value
(a) greater than 16.8;
(b) less than 14.9;
(c) between 13.6 and 18.8;
(d) between 16.5 and 16.7.
5.25 The time for oil to percolate to all parts of an engine can be treated as a random variable having a normal distribution with mean 20 seconds. Find its standard deviation if the probability is 0.25 that it will take a value greater than 31.5 seconds.
5.26 Butterfly-style valves used in heating and ventilat-ing industries have a high flow coefficient. Flow co-efficient can be modeled by a normal distribution with mean 496 Cv and standard deviation 25 Cv. Find the probability that a valve will have a flow coefficient of
(a) at least 450 Cv;
(b) between 445.5 and 522 Cv.
5.27 Refer to Exercise 5.26 but suppose that a large po-tential contract contains the specification that at most 7.5% can have a flow coefficient less than 420 Cv. If the manufacturing process is improved to meet this specification, determine
(a) the new mean μ if the standard deviation is 25 Cv;
(b) the new standard deviation if the mean is 496 Cv. 5.28 Find the quartiles
−z0.25 z0.50 z0.25
of the standard normal distribution.
5.29 The daily high temperature in a computer server room at the university can be modeled by a normal distribu-tion with mean 68.7◦F and standard deviation 1.2◦F.
Find the probability that, on a given day, the high tem-perature will be
(a) between 68.3 and 70.3◦F (b) greater than 71.5◦F.
5.30 With reference to the preceding exercise, for which temperature is the probability 0.05 that it will be ex-ceeded during one day?
5.31 A machine produces soap bars with a weight of 80± 0.10 g. If the weight of the soap bars manufactured by
the machine may be looked upon as a random vari-able having normal distribution withμ = 80.05 g and σ = 0.05 g, what percentage of these bars will meet specifications?
5.32 The number of teeth of a 12% tooth gear produced by a machine follows a normal distribution. Verify that if σ = 1.5 and the mean number of teeth is 13, 74% of the gears contain at least 12 teeth.
5.33 The quantity of aerated water that a machine puts in a bottle of a carbonated beverage follows a normal dis-tribution with a standard deviation of 0.25 g. At what
“normal” (mean) weight should the machine be set so that no more than 8% of the bottles have more than 20 g of aerated water?
5.34 An automatic machine fills distilled water in 500-ml bottles. Actual volumes are normally distributed about a mean of 500 ml and their standard deviation is 20 ml.
(a) What proportion of the bottles are filled with water outside the tolerance limit of 475 ml to 525 ml?
(b) To what value does the standard deviation need to be increased if 99% of the bottles must be within tolerance limits?
5.35 If a random variable has the binomial distribution with n = 25 and p = 0.65, use the normal ap-proximation to determine the probabilities that it will take on
(a) the value 15;
(b) a value less than 10.
5.36 From past experience, a company knows that, on aver-age, 5% of their concrete does not meet standards. Use the normal approximation of the binomial distribution to determine the probability that among 2000 bags of concrete, 75 bags contain concrete that does not meet standards.
5.37 The probability that an electronic component will fail in less than 1,000 hours of continuous use is 0.25. Use the normal approximation to find the probability that among 200 such components fewer than 45 will fail in less than 1,000 hours of continuous use.
5.38 Workers in silicon factories are prone to a lung dis-ease called silicosis. In a recent survey in a factory, about 11% of the workers have been infected by it.
Assume the same rate of infection holds everywhere.
Use the normal distribution to approximate the prob-ability that, out of a random sample of 425 workers, the numbers that are prone to infection at present will be
(a) 30 or more;
(b) 28 or less.
Sec 5.5 The Uniform Distribution 151
5.39 Refer to Example 11 concerning the experiment that confirms electron antineutrinos change type. Suppose instead that there are 400 electron antineutrinos leav-ing the reactor. Repeat parts (a)–(c) of the example.
5.40 To illustrate the law of large numbers mentioned on Page 116, find the probabilities that the proportion of drawing a club from a fair deck of cards will be any-where from 0.24 to 0.26 when a card is drawn (a) 100 times;
(b) 10,000 times.
5.41 Verify the identity F (−z) = 1 − F(z) given on page 141.
5.42 Verify that the parameterμ in the expression for the normal density on page 140, is, in fact, its mean.
5.43 Verify that the parameterσ2in the expression for the normal density on page 140 is, in fact, its variance.
5.44 Normal probabilities can be calculated using MINITAB. Let X have a normal distribution with mean
11.3 and standard deviation 5.7. The following steps yield the cumulative probability of 9.31 or smaller, or P(X≤ 9.31).
Dialog box:
Calc> Probability Distribution > Normal Choose Cumulative Distribution. Choose Input constantand enter 9.31.