Question

If np >= 5 and nq >= 5 , estimate P(at least 10) with n = 13 and p = 0.5 by using the normal distribution as an approximation to the binomial distribution ; if np < 5 or nq < 5 then state that the normal approximation is not suitable.

Answer 1

The question provides that:

$\begin{gathered} n=13 \\ p=0.5 \end{gathered}$

Therefore, we have that:

$q=1-p=1-0.5=0.5$

To check if we can use the normal distribution as an approximation, we will check the values of np and nq:

$\begin{gathered} np=13\times0.5=6.5 \\ nq=13\times0.5=6.5 \end{gathered}$

Since,

$\begin{gathered} np\ge5 \\ \text{and} \\ nq\ge5 \end{gathered}$

then we can use the normal distribution as an approximation.

To evaluate P (at least 10), we are evaluating:

$P(X\ge10)$

The standard deviation of the distribution is gotten to be:

$\sigma=\sqrt[]{np}=\sqrt[]{6.5}=2.550$

The mean is 6.5.

Therefore, the Z-score is gotten to be:

$Z=\frac{x-\bar{x}}{\sigma}$

Hence, it is calculated to be:

$Z=\frac{10-6.5}{2.550}=1.37$

The probability is therefore given to be:

$P(Z\ge1.37)=Pr(0\le Z)-Pr(0\le Z\le1.37)$

Using the Probability Distribution Table, we have:

$P(Z\ge1.37)=0.5-0.4147=0.0853\approx0.085$

Therefore, the answer is:

$P(at\text{ }least\text{ }10)=0.085$