Question

Using Normal Approximation. In Exercises 5–8, do the following: If the requirements of $np⩾5$and $nq⩾5$are both satisfied, 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 should not be used.

Births of Boys With n = 20 births and p = 0.512 for a boy, find P(fewer than 8 boys).

Short answer

The probability of fewer than 8 boys is equal to 0.1093.

Step-by-step solution

Given information

A sample of 20 births is considered. The proportion of boys (p) is equal to 0.512.

Check the requirement necessary for normal approximation

It is required that$np⩾5$ and $nq⩾5$.

The values are computed below:

$$\begin{gathered} np=\left(20\right)\left(0.512\right) \\ =10.24 \\ ⩾5 \end{gathered}$$

$$\begin{gathered} nq=\left(20\right)\left(1-0.512\right) \\ =9.76 \\ ⩾5 \end{gathered}$$

As the requirement is fulfilled, the normal approximation can be applied to compute the probability value.

Mean and Standard Deviation

The mean value is equal to:

$$\begin{gathered} \mu =np \\ =20\times 0.512 \\ =10.24 \end{gathered}$$

The standard deviation is equal to

$$\begin{gathered} \sigma =\sqrt{npq} \\ =\sqrt{20\times 0.512\times 0.488} \\ =2.235 \end{gathered}$$

Continuity Correction

Let x represent the number of boys in the sample.

Here, x is equal to 8.

The value of x is transformed as follows:

$$\begin{gathered} \left(x-0.5,x+0.5\right)=\left(8-0.5,8+0.5\right) \\ =\left(7.5,8.5\right) \end{gathered}$$

It is required to compute the probability of fewer than 8 boys. Thus, the probability to the left of 7.5 needs to be computed.

z-score

The z score value is equal to:

$$\begin{gathered} z=\frac{x-\mu }{\sigma } \\ =\frac{7.5-10.24}{2.235} \\ =-1.23 \end{gathered}$$

The z-score is -1.23.

Referring to the standard normal distribution, the area to the left of -1.23 is equal to 0.1093.

Therefore, the probability of fewer than 8 boys is equal to 0.1093.