Question

In studies for a​ medication, 7 percent of patients gained weight as a side effect. Suppose 403 patients are randomly selected. Use the normal approximation to the binomial to approximate the probability that ​(a) exactly 29 patients will gain weight as a side effect. ​(b) no more than 29 patients will gain weight as a side effect. ​(c) at least 37 patients will gain weight as a side effect. What does this result​ suggest?

Answer 1

Let X be the random variable that number of patients will gain weight as a side effect.

X ~ Binomial (n = 403, p = 7% = 0.07)

Now here we can see that,

np = 403 * 0.07 = 28.21 > 5

n(1-p) = 403 * (1-0.07) = 374.79

Here n is too large and p is too small so we will use here normal approximation.

mu = np = 403 * 0.07 = 28.21

X ~ N(mu = 28.21, sd = 5.12)

Use the normal approximation to the binomial to approximate the probability that :

​(a) exactly 29 patients will gain weight as a side effect.

To find P(X = 29)

Using continuity correction we have to find P(29-0.5 < X < 29+0.5).

i.e. we have to find P(28.5 < X < 29.5)

Convert Xbar = 28.5 and 29.5 into z-score.

z-score is defined as,

Now we have to find P(0.06 < Z < 0.25)

P(0.06 < Z < 0.25) = P(Z < 0.25) - P(Z < 0.06)

We can find this probability in excel.

syntax :

=NORMSDIST(z)

where z is z-score

P(0.06 < Z < 0.25) = 0.5994 - 0.5226 = 0.0768

​(b) no more than 29 patients will gain weight as a side effect. ​(c) at least 37 patients will gain weight as a side effect. What does this result​ suggest?

To find P(X < 29)

By using continuity correction we have to find P(X < 29-0.5).

To find P(X < 28.5)

z-score for Xbar = 28.5 is 0.06.

Now we have to find P(Z < 0.06).

P(Z< 0.06) = 0.5226

​(c) at least 37 patients will gain weight as a side effect. What does this result​ suggest?

To find P(X 37)

By using continuity correction to find P(X > 37-0.5)

To find P(X > 36.5)

z-score for x = 36.5 is,

z = (36.5 - 28.21) / 5.12 = 1.62

Now we have to find P(Z > 1.62)

P(Z > 1.62) = 1 - P(Z < 1.62) = 1 - 0.9472 = 0.0528