Question

According to medical data, the probability than an adult US citizen has been diagnosed with high blood pressure is 13%. Suppose you randomly select 69 adults and ask if he or she has been diagnosed high blood pressure

. a. Decide whether you can use the normal distribution to approximate the binomial distribution. If so, find the mean and standard deviation. Round to the nearest hundredth when necessary. If not, please explain why a normal distribution should not be used.

b. Find the probability that no more than 10 of the sampled adults have been diagnosed with high blood pressure using an appropriate method

Answer 1

Given: n = number of adults selected = 69

p = proportion of adults having high blood pressure = 13% = 0.13

Normal approximation to the Binomial:

When n * p and n * (1 - p) both are greater than or equal to 5 then normal approximation to the binomial is applicable.

n * p = 69 * 0.13 = 8.97 and n* (1 - p) = 69 * (1 - 0.13) = 60.03

Both are greater than 5, so the normal approximation is useful.

a) Mean and standard deviation

b) P(no more than 10)

No more than 10 mean less then or equal to 10 that is

Here the binomial distribution is discrete and the normal is continuous distribution, so to find the probability have to do the continuity correction.

That is

First convert the x into z score,

The formula of z score is,

That is P(X < 10.5) becomes P(Z < 0.55)

The probability using z score table for z = 0.55 is 0.7088

Therefore, the probability that no more than 10 of the sampled adults have been diagnosed with high blood pressure is 0.7088