Compute the required probability using the normal distribution.

The area under the normal curve between x =18.5 and x =19.5 will give us the (approximate) probability that 19 persons possess at least one credit card. We calculate this probability as follows:

For x =18.5; ;

For x =19.5; .

The required probability is given by the area under the standard normal curve between z =1.28 and z =1.64. (Fig 4.21).

The required probability is

.

Thus, based on the normal approximation, the probability that 19 persons in a sample of 30 will possess at least one credit card is approximately 0.0498. Earlier, using the binomial formula, we obtained the exact probability 0.0509. The error due to using the normal approximation is

. Thus, the exact probability is underestimated by 0.0011 if the normal approximation is used.

Definition:

Let x be the number of successes from n independent trials, each with probability of success p. Then number of successes, x, is a binomial random variable and if (where ) a good approximation is

(4.1)

or if we can use the continuity correction factor to obtain

(4.2)

where Z is a standard normal random variable.

Example:

Let X have a binomial distribution with p =0.6 and n =150. Approximate the probability that

a) x lies between 82 and 101;

b) x is greater than 97.

Solution:

Since , then we will use approximation without using the continuity correction.

Since , we obtain:

a)

.

b)

.

Example:

A large-scale survey conducted two years ago revealed that 30% of the adult population were regular users of alcoholic beverages. If this is still the current rate, what is the probability that in a random sample of 40 adults the number of users of alcoholic beverages will be

a) less than 15

b) 10 or more?

Solution:

For this example

n =40, p =0.3; q =0.7.

Since we must use continuity correction factor to obtain necessary probabilities.

a) .

The probability that 15 out of 40 adults use alcoholic beverages regularly

is 0.8869.

b)

.

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