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.

Guessing on Standard TestsWith n= 50 guesses and p= 0.2 for a correct answer, findP(exactly 12 correct answers).

Short answer

The probability of exactly 12 correct answers is equal to 0.1087.

Step-by-step solution

Given information

A sample of 50 guesses of answers on a test is considered. The probability of a correct answer is equal to 0.2.

Check the requirement necessary for normal approximation

It is required that$np⩾5$ and $nq⩾5$.

The values are computed below:

$$\begin{gathered} np=\left(50\right)\left(0.2\right) \\ =10 \\ ⩾5 \end{gathered}$$

$$\begin{gathered} nq=\left(50\right)\left(1-0.2\right) \\ =40 \\ ⩾5 \end{gathered}$$

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:

$$\begin{gathered} \mu =np \\ =50\times 0.2 \\ =10 \end{gathered}$$

The standard deviation is equal to:

$$\begin{gathered} \sigma =\sqrt{npq} \\ =\sqrt{50\times 0.2\times 0.8} \\ =2.83 \end{gathered}$$

Continuity correction

Let x represent the number of correct answers.

Here, x is equal to 12.

The value of x is transformed as follows:

$$\begin{gathered} \left(x-0.5,x+0.5\right)=\left(12-0.5,12+0.5\right) \\ =\left(11.5,12.5\right) \end{gathered}$$

It is required to compute the probability of exactly 212 correct answers. Thus, the probability between the values 11.5 and 12.5 needs to be computed.

Probability value

The required probability value is equal to:

$$\begin{gathered} P\left(11.5<x<12.5\right)=P\left(\frac{11.5-\mu }{\sigma }<\frac{x-\mu }{\sigma }<\frac{12.5-\mu }{\sigma }\right) \\ =P\left(\frac{11.5-10}{2.83}<z<\frac{12.5-10}{2.83}\right) \\ =P\left(0.53<z<0.88\right) \\ =P\left(z<0.88\right)-P\left(z<0.53\right) \\ =0.8106-0.7019 \\ =0.1087 \end{gathered}$$

Therefore, the probability of exactly 12 correct answers is equal to 0.1087.